{\Large The Monoid Structure on Homotopy Obstructions
Satya Mandal\footnote{Partially supported by a General Research Grant (no 2301857) from U. of Kansas
1612.00749.tex
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\title{{\Large The Monoid Structure on Homotopy Obstructions}} 
%\\Nori Conjecture for Free: 16 Septemeber 2016} 
\author{
 Satya Mandal\footnote{Partially supported by a General Research Grant (no 2301857) from U. of Kansas}
 ~~~and ~~Bibekananda Mishra
 \\ %$^\dagger$\\ 
{\small University of Kansas, Lawrence, Kansas 66045, USA}\\
{\small {\it  mandal@ku.edu, bibekanandamishra@ku.edu} 
 %$^\dagger$
  }\\
 } 
\begin{document}
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\date{2 December 2016}
%16 November 2016}
%\date{23 October 2016 \TCP{(For Preview)}}
%\date{5 October 2016\\ Groups Structure 8: October 2016}%Starting on Nori Free: 16 Sept 2016} 
%May-June 2015, 12 Sept 2015\\ Closed 18 Sept 2015}
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\newtheorem{construction}[theorem]{Construction}
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\newtheorem{example}[theorem]{Example}
\newtheorem{definition}[theorem]{Definition} 
\newtheorem{conjecture}[theorem]{Conjecture} 
\newtheorem{remark}[theorem]{Remark} 
\newtheorem{statement}[theorem]{Statement} 





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\section{Introduction}

This article can be viewed as a Projective module version of \cite{MM}, on Homotopy Obstructions. 
The  free module case of the same was dealt with in 
\cite{MM}, which also has some overlap with \cite{AF}.

Suppose $A$ is a commutative noetherian ring, with $\dim A=d$ and $P$ is projective $A$-module of rank $n$. Let 
 ${\mathcal LO}(P)$ denote the set of all pairs $(I, \omega)$ such that $I$ is an ideal and $\omega: P \sur \frac{I}{I^2}$ is a surjective homomorphisms.
 Given such a pair $(I, \omega)\in {\mathcal LO}(P)$, 
this article concerns obstructions for $\omega$ to lift to a surjective homomorphism $P\sur I$. Given such a projective $A$-module $P$ and 
 a polynomial extension $A\hra A[T]$, by substituting $T=0, 1$,  two set theoretic maps  $\diagram {\mathcal LO}(P) 
 & {\mathcal LO}(P\otimes A[T])\ar[l]_{T=0\quad}\ar[r]^{\qquad T=1} &
{\mathcal LO}(P)\enddiagram$ are obtained. This leads to a chain equivalence relation on ${\mathcal LO}(P)$. The set  
of all equivalence classes would be denoted by $\pi_0\left({\mathcal LO}(P)\right)$ and referred to as the homotopy obstruction set. 
Around 1990, Madhav
V. Nori  laid out a set of ideas, on such homotopy obstructions and implicitly the set  $\pi_0\left({\mathcal LO}(P)\right)$, 
to deal with the  question of  lift ability of surjective maps $\omega: P \sur \frac{I}{I^2}$. This set of ideas were referred to as the Homotopy Program in 
\cite{MM} and a background was provided, some of which we recap.


This set of ideas were verbally communicated by Nori  to some 
in a very informal and open ended manner. Because of the nature of these communications, not everyone heard the same thing and
 these ideas took the form of some folklores. 
The
 versions of this set of ideas available  (or not) in the literature (e. g. \cite{M3, MV, MS,BS1,BS2, BS3, BK})
 may differ and 
 %written statements of these ideas 
 have been up to the interpretations and 
adaptations by  the  recipients of these communications, much to their credit.
%
Nori never classified these  as conjectures or otherwise.
However, some results followed too quickly \cite{M3},
to treat them as anything less than conjectures.
%Most authors referred to these problems as "questions". 
%
%
Analogy to the Obstruction Theory for vector bundles (\cite{St})  was the main backdrop behind this program
and  central to this Program was the Homotopy conjecture of Nori. The following is the statement of the Homotopy Conjecture from \cite{M3},
which would most likely be an adaptation by the respective author.

\begin{conjecture}[Homotopy Conjecture]\label{homoConj}
Suppose $X=\spec{A}$ is a smooth affine variety, with $\dim X=d$. Let $P$ be a projective $A$-module of rank $r$ and $f_0:P\sur I$ 
be  a surjective 
homomorphism, onto an ideal $I$ of $A$. Assume $Y=V(I)$ is smooth with $\dim Y=d-r$.
Also suppose $Z=V(J)\subseteq \spec{A[T]}=X\times \BA^1$ is a smooth subscheme, such that $Z$ intersects $X\times 0$ transversally in $Y\times 0$.
Now, suppose that $\varphi: P[T] \sur \frac{J}{J^2}$ is a surjective map such that $\varphi_{|T=0}=f_0\otimes \frac{A}{I}$.  Then, there is a
surjective 
map $F:P[T]\sur J$ such that (i) $F_{|T=0}=f_0$ and (ii) $F_{|Z}=\varphi$.
\end{conjecture}

%
While the Conjecture  \ref{homoConj} would fail without the regularity  hypothesis   \cite[Example 6.4]{BS1},
existing results (see \cite{M3, BS1, BK}) indicate that with suitable  hypotheses the regularity  and/or transversality hypotheses may be spared.
However, the Conjecture  \ref{homoConj}, as stated,  would  even fail in some cases when $A$ is regular (see \cite[Example 3.15]{BS1}), which are, conjecturally, the exceptions.
The best result, up to date, on this Conjecture \ref{homoConj} is due to Bhatwadekar and Keshari \cite{BK}.
%
%
%
In this article, we mainly investigate various aspects of the Homotopy Program that concerns the  structure of the obstruction set $\pi_0\left({\mathcal LO}(P)\right)$.
Under some additional  hypotheses, we prove that $\pi_0\left({\mathcal LO}(P)\right)$ has a natural structure of Monoid, which is a groups structure, when $P\cong Q\oplus A$.
We give a definition of the Euler Class groups $E(P)$, which coincides with the same in \cite{BS2} when $P=A^n$ and, likewise that in \cite{MY}. Then, we compare 
$\pi_0\left({\mathcal LO}(P)\right)$ with the Euler class group $E(P)$.

%
Now on, we assume that $A$ contains a field $k$, with $1/2\in k$.
The results in this article are extension of the results in \cite{MM, AF}, where the case of free modules $P=A^n$ was dealt with. 
When $P=A^n$ is a free $A$-module, ${\mathcal LO}(P)$ was denoted by ${\mathcal LO}(A, n)$ in \cite{MM}. 
In some literature \cite{F, AF},
driven by the desire to   view the same  in a functorial manner, one writes $Q_{2n}(A)=\{(s; f_1, \ldots, f_n; g_1, \ldots, g_n)\in A^{2n+1}:
\sum_{i=1}^nf_ig_i+s(s-1)=0 \}$ and the homotopy obstruction set $\pi_0\left(Q_{2n}\right)(A)$ was defined.
 It was established in \cite{MM} that $\pi_0\left({\mathcal LO}(A, n)\right)\cong \pi_0\left(Q_{2n}\right)(A)$. By analogy or due to the same desire,
 for a projective $A$-module $P$, we consider 
 $$
 \left\{
 \begin{array}{l}
 \widetilde{Q}(P)=\left\{(f, p, s)\in P^*\oplus P\oplus A: f(p)+s(s-1)=0 \right\}, \\
  \widetilde{Q}'(P)=\left\{(f, p, s)\in P^*\oplus P\oplus A: f(p)+s^2=1 \right\}.\\
  \end{array}
  \right.
 $$ 
 and define homotopy obstruction sets
 $\pi_0\left(\widetilde{Q}(P)\right)$, $\pi_0\left(\widetilde{Q}'(P)\right)$. Note that there is a bijection 
 $\widetilde{Q}(P)\iso   \widetilde{Q}'(P)$.
 We establish (see Theorem \ref{2pi0SamNov26}) that there is a bijection $\pi_0\left({\mathcal LO}(P)\right)\iso\pi_0\left(\widetilde{Q}(P)\right)$.
 Recall, there is a bijection $Q_{2n}(A)\cong Hom\left(\spec{A}, \spec{\SA_{2n+1}}\right)$, where 
 $\SA_{2n+1}=\frac{k[X_1, \ldots, X_n; Y_1, \ldots, Y_n, Z]}{\left(\sum_{i=1}^nX_iY_i+Z(Z-1)\right)}$. 
 By analogy, define 
 $\SA(P)=\frac{S(P\oplus P^*\oplus A)}{(B-1)}$,
where  $S(P\oplus P^*\oplus A)$ denotes the symmetric algebra and $B\in S_2(P\oplus P^*\oplus A)$ 
is the bilinear form of the quadratic space $P^*\oplus P\oplus A$.
In Section \ref{motivicSec}, an interpretation of $\widetilde{Q}'(P)$, analogous to that of  $Q_{2n}'(A)$, given by establishing a bijection  $\tilde{Q}'(P) \iso Hom(\SA(P), A)$.

% is given, by defining  $\SA(P)=\frac{S(P\oplus P^*\oplus A)}{(B-1)}$,
%where  $S(P\oplus P^*\oplus A)$ denotes the symmetric algebra and $B\in S_2(P\oplus P^*\oplus A)$ 
%is the bilinear form of the quadratic space $P^*\oplus P\oplus A$, and   establishing a bijection  $\tilde{Q}'(P) \iso Hom(\SA(P), A)$.
 
 



 



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Among the main results in this article, is the following reinterpretation (see (\ref{formalizeBSNori}))
 of \cite[Theorem 4.13]{BK} on the Homotopy Conjecture \ref{homoConj}, 
in terms of vanishing of obstruction classes.

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\bT\label{IntroformalizeBSNori}
Suppose $A$ is an essentially smooth ring over an infinite filed $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$, with $2n\geq d+3$. 
Let ${\bf v}_0=(f_0, 0, 0)\in \widetilde{Q}(P)$ be a base point, as in (\ref{basePoint}).
%
Suppose $(I, \omega_I)\in {\mathcal LO}(P)$, with $height(I)\geq n$. Then, $\omega_I$ lifts to a surjective map $P\sur I$ if and only if 
$\zeta(I, \omega_I)=\zeta_0({\bf v}_0)$.
\eT

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In Section \ref{secHomotopy}, we develop some machinery regarding Homotopy. In particular, we establish the  homotopy relation 
on $\widetilde{Q}(P)$ is, in deed, an equivalence relation ( see (\ref{directHomo}), (\ref{equivHomRelDec2})),
 when $A$ is a regular ring over a filed $k$, with $1/2\in k$.

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In Section \ref{secInvolution}, we consider the involution $\widetilde{Q}(P) \lra \widetilde{Q}(P)$ sending 
$(f, p, s) \mapsto (f, p, 1-s)$. 
For any commutative noetherian rings $A$, this involution induces an involution on $\pi_0\left(\widetilde{Q}(P) \right)$,
which serves as a key tool, for the subsequent definition of the Monoid structure on $\pi_0\left({\mathcal LO}(P) \right))
=\pi_0\left(\widetilde{Q}(P) \right))$. In Section \ref{secGroupStru}, we establish that,
when $A$ is a  regular ring over a filed $k$, with $1/2\in k$, with $2rank(P)\geq d+2$, 
$\pi_0\left(\widetilde{Q}(P) \right))$ has the structure of
a natural Monoid (see Theorem ref{abelianGroup}).
There are two distinguished elements in $\widetilde{Q}(P)$, namely ${\bf e}_0=\zeta_0(0, 0, 0), {\bf e}_1=\zeta_0(0, 0, 1)
\in \pi_0\left(\widetilde{Q}(P) \right)$. It turns out that, if ${\bf e}_0={\bf e}_1$ then the Monoid structure 
on $\pi_0\left(\widetilde{Q}(P) \right)$ becomes an abelian group, where ${\bf e}_0={\bf e}_1$ would be the identity.
It is easy to see that, if $P=Q\oplus A$ for some projective $A$-module $Q$, then ${\bf e}_0={\bf e}_1$.

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It had always remained   conjectural , as a part of  
 the Homotopy Program, that the Homotopy obstruction sets $\pi_0\left(\widetilde{Q}(P)\right)$ would have a group structure, 
  when $rank(P)$ is hight enough. However, when $P=A^n$ is free, an obstruction group $E^n(P)$ was explicitly given by 
  Nori, when $n=\dim A$. Subsequently, this definition was extended in \cite{BS2}. This was further extended in \cite{MY}, when 
  $P=L\oplus A^{n-1}$, where $L$ is rank one projective $A$-module. These groups came to be known as Euler class groups. 
  In Section \ref{secular}, we extend these and give a natural definition of an Euler class groups $E(P)$,
  for any projective $A$-module $P$ over a commutative noetherian ring $A$. It is helpful that certain superfluous aspect of the definition in 
  \cite{BS2} was pointed out in \cite{MM}. in analogy the the results in \cite{BS2, MY}, we prove the following:
  
   \bT\label{IntrooldGoldMethod}
 Suppose $A$ is a commutative noetherian ring with $\dim A=d$ and $P$ is a projective $A$-module, with $rank(P)=n$.
 Assume $2n\geq d+3$ and $P\cong Q\oplus A$. 
 \TCP{If $A=R[X]$ is a polynomial ring over a regular ring $R$, over an infinite field $k$, assume $2n\geq \dim R[T]+2$.} 
 Let $(J, \omega_J) \in {\mathcal LO}^n(P)$ and $\overline{\varepsilon}\left(J, \omega_J\right)=0\in E(P)$.
 Then, $\omega_J$ lifts to a surjective map $P\sur J$.
 \eT
  
  
  
When $\pi_0\left(\widetilde{Q}(P)\right)$ is a group, one can naturally define a group homomorphism $E(P) \lra \pi_0\left(\widetilde{Q}(P)\right)$.
Consequently, we prove the following.

  \bT\label{IntrouseBKNov14}
Suppose $k$ is a  field, with $1/2\in k$ and $\dim A=d$. % and $A$ is an essentially smooth ring over $k$, with $\dim A=d$. 
Suppose $P=Q\oplus A$ is a projective $A$-module with $rank(P)=n$
and  $2n\geq d+2$. Then, there is a surjective group homomorphism 
$\rho: E(P) \sur \pi_0\left(\widetilde{\CQ}(P)\right)$.

In particular,  if  $A$ is an essentially smooth ring over an infinite perfect field $k$ and $2n\geq d+3$ then,
%Then, $\pi_0\left(\widetilde{\CQ}(P)\right) =\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$  and 
the homomorphism 
$\rho: E(P) \lra \pi_0\left(\widetilde{\CQ}(P)\right)$ is an 
isomorphism.
{\rm (We make further comments when $\pi_0\left(\widetilde{\CQ}(P)\right)$ fails to be a group  or when $P$ fails to have a unimodular element.)}
\eT

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%\section{Notations and Preliminaries} \label{prelimSec}
\section{Homotopy Obstructions}\label{homoSecAll}

Before we proceed, we introduce some notations.
%We establish some basic notations and preliminaries.
\begin{notations}\label{nota}{\rm 
Throughout, $k$ will denote a field, with $1/2\in k$. Also  $A$ will denote a commutative ring with $\dim A=d$.
For  $A$-modules $M, N$, we denote $M[T]:=M\otimes A[T]$ and $M^*=Hom(M,A)$.
For
$f\in Hom(M,N)$, denote $f[T]:=f\otimes 1\in Hom(M[T], N[T])$. 
% 
Homomorphisms  $f: M\lra \frac{I}{I^2}$ would be identified with the induced 
maps $\frac{M}{IM} \lra \frac{I}{I^2}$.

 For surjective homomorphisms $\omega_1: M\sur \frac{I_1}{I_1^2}$, $\omega_2: M\sur \frac{I_2}{I_2^2}$, where 
 $I_1, I_2$ be two ideals, with $I_1+I_2=A$,  $\omega_1\star\omega_2: M\sur \frac{I_1I_2}{(I_1I_2)^2}$ will denote the unique surjective map 
induced by $\omega_1, \omega_2$. 
%

For a projective $A$-module $P$,   $\BQ(P)=(\BQ(P), q)$ will denote the quadratic space  $\BH(P)\perp A$, where $\BH(P)=P^*\oplus P$ is the hyperbolic space.
So, $P^*\oplus P\oplus A$ id the underlying projective module of  $\BQ(P)$ and, for $(f, p, s)\in P^*\oplus P\oplus A$, $q(f, p, s)=f(p)+s^2$.

The category of schemes over $\spec{A}$ will be donated by $\Sch_A$.
The category of sets will be denoted by $\Sets$.
%
%%%%%
Given a pre-sheaf $\CF:\Sch_A \to \Sets$, and a scheme $X\in \Sch_A$, define $\pi_0(\CF)(X)$ by the pushout 
\begin{equation}\label{Defpi0CF}
\diagram 
\CF(X\times \BA^1) \ar[r]^{T=0}\ar[d]_{T=1} & \CF(X)\ar[d]\\
\CF(X) \ar[r] & \pi_0(\CF)(X)\\ 
\enddiagram 
\qquad {\rm in}~~\Sets
\end{equation}
 For  an affine scheme $X=\spec{B}\in \Sch_A$ and a pre-sheaf $\CF$, as above, we write $\CF(A):=\CF(\spec{B})$ and $\pi_0(\CF)(B):=\pi_0(\CF)(\spec{B})$.
 ({\it For our purpose, we could restrict ourselves to affine schemes $\spec{A}$ and its polynomial extensions $\spec{A[T]}$}.)
 }
\end{notations}
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%\subsection{Homotopy Obstructions}
Given a projective module $P$ over a noetherian commutative ring $A$, 
 we define a homotopy obstruction set $\pi_0({\mathcal LO}(P))$ 
and establish various descriptions of the same. These are analogous to similar obstruction sets available in \cite{F, M2, MM},
when $P=A^n$ is free.


%
\bD\label{defpizero2Oct}{\rm 
 Let $A$ be a noetherian commutative ring, $X=\spec{A}$ and $P$ be a projective $A$-module. 
 By a \TCP{local $P$-orientation}, we mean 
a pair $(I, \omega)$ where $I$ is an ideal of $A$ and $\omega:P \sur \frac{I}{I^2}$ is a surjective homomorphism, which is 
identified with 
 surjective homomorphism $\frac{P}{IP} \sur \frac{I}{I^2}$, induced by $\omega$. 
%The set of all such local orientations will be denoted by $\CL\CO(P)$.
%
Denote 
$$
\left\{
\begin{array}{l}
{\mathcal LO}(P)=\left\{(I, \omega): (I, \omega)~{\rm is~a~local}~P~{\rm orientation} \right\}\\
\CQ(P)=\left\{(f, s)\in P^*\oplus A: s(1-s)\in f(P)  \right\}\\
\widetilde{\CQ}(P)=\{(f, p, s)\in P^*\oplus P\oplus A: f(p)+s(s-1)=0 \}\\
\widetilde{\CQ}'(P)=\{(f, p, z)\in P^*\oplus P\oplus A: f(p)+z^2=1 \}\\
\end{array}
\right.
$$
There is a commutative diagram of set theoretic maps, as denoted:
\begin{equation}\label{diaEtasNU}
\diagram
\widetilde{\CQ}(P) \ar@{->>}[r]^{\nu} \ar@{->>}[d]_{\eta} & \CQ(P)\ar@{->>}[dl]^{\eta'}\\
{\mathcal LO}(P)&\\
\enddiagram~~
{\rm where,~for}~(f, p, s)\in \widetilde{\CQ}(P), \quad \nu(f, p, s)=(f, s)\quad  
\end{equation}
%
and $\eta'(f, s)=\eta(f, p, s)= (I, \omega)$, where $I=f(P)+As$ and $\omega: P \sur \frac{I}{I^2}$
is the homomorphism
is induced by $f$. These maps $\eta, \eta', \nu$ are surjective. There is also a bijection 
\begin{equation}\label{QPQprimeNov19}
\kappa: \widetilde{\CQ}(P)\iso \widetilde{\CQ}'(P)\quad{\rm sending}\quad 
(f, p, s) \mapsto \left(2f, 2p, 2s-1\right) %\frac{f}{2}, \frac{p}{2}, \frac{s-1}{2} \right)
\end{equation}
Now, when $P$ is fixed projective $A$-module, and schemes $Y\in \underline{Sch}_A$ ,with $\pi: Y\to \spec{A}$, the associations 
$Y \mapsto \CQ(\pi^*P)$,  $Y \mapsto \widetilde{\CQ}(\pi^*P)$, $Y \mapsto \widetilde{\CQ}'(\pi^*P)$ is a pre-sheaves.
By the pushout diagram (\ref{Defpi0CF}), applied to these pre-sheaves defines, the Homotopy Obstructions sheaves 
$$
\pi_0\left(\widetilde{\CQ}(P)\right),\quad  \pi_0\left(\widetilde{\CQ}'(P)\right), \quad \pi_0\left(\CQ(P)\right).
$$
For an ideal $I\subseteq A$,  and a ring homomorphism $A\lra B$, $I\otimes B$ is  not necessarily identifiable with an ideal of $B$.
In particular, for ideals $I\subseteq A[T]$,  $I\otimes \frac{A[T]}{(T)}=\frac{I}{TI}$ does not necessarily inject in to $A$.
Therefore,  the pre-sheaf behavior of ${\mathcal LO}(P)$ on 
$\underline{Sch}_A$, is to be understood with some care. 
 However, we  explicitly define the  
Homotopy Obstruction sheaf
$\pi_0\left({\mathcal LO}(P)\right)$, by the 
pushout diagrams, in $\underline{Sets}$, as follows:
$$
%
%
\diagram
{\mathcal LO}(P[T])\ar[r]^{T=0}\ar[d]_{T=1} &{\mathcal LO}(P)\ar[d]\\
{\mathcal LO}(P)\ar[r] & \pi_0\left({\mathcal LO}(P)\right)\\ 
\enddiagram 
\qquad \qquad {\rm in}\qquad \underline{Sets}.
$$
In deed, $\pi_0\left({\mathcal LO}(P)\right)$ was the Homotopy obstruction explicitly envisioned by Nori (around 1990).
}
\eD

We record, the following basic lemma 

\bL\label{basicMaps}
Use the notations, as above (\ref{defpizero2Oct}). 
The bijections $\kappa$, induces a bijection
$$
\overline{\kappa}: \pi_0\left(\widetilde{\CQ}(P)\right)\iso \pi_0\left(\widetilde{\CQ}'(P)\right).
$$
Further, the maps $\eta, \nu, \eta'$ (in diagram \ref{diaEtasNU})
 %are homotopy invariant and will 
 induce set theoretic maps, as denoted in the commutative diagram od surjective maps:
$$
\diagram
\pi_0\left(\widetilde{\CQ}(P)\right) \ar@{->>}[r]^{\overline{\nu}} \ar@{->>}[d]_{\overline{\eta}} 
& \pi_0\left(\CQ(P)\right)\ar@{->>}[dl]^{\overline{\eta}'}\\
\pi_0\left({\mathcal LO}(P)\right)&\\
\enddiagram
$$
%Note that all these maps are surjective. 
\eL
\pf It follows from the fact that $\kappa, \eta, \nu, \eta'$ are homotopy invariant. 
$\eop$

\vspace{2mm}
We proceed to prove that these maps are bijective.
We fix notations, for $(f, p, s)\in \widetilde{\CQ}(P)$, its equivalence class in $\pi_0\left(\widetilde{\CQ}(P)\right)$ will be denoted by 
$[(f, p, s)]$ and similar notations will be used for $(f, s)\in \CQ(P)$ and $(I, \omega)\in {\mathcal LO}(P)$.
%
Note, given $(I, \omega)\in \CL\CO(P)$, $\omega$ lifts to a homomorphism $f$, as follows:
\begin{equation}\label{aLiftOmega}
\diagram 
P \ar[r]^f\ar@{->>}[dr]_{\omega} & I \ar@{->>}[d]\\
& \frac{I}{I^2}\\
\enddiagram
\end{equation}
By Nakayama's lemma there is an element $s\in I$ such that $(1-s)I\subseteq f(P)$.
Consequently, $(f,s)\in \CQ(P)$ and $I=(f(P), s)$. This association would not be unique.
Such a pair $(f,s)\in \CQ(P)$ will be referred to as a \TCP{lift} of $(I,\omega)$ in $\CQ(P)$.
%
Now define an the \TCP{Obstruction class} map:
\begin{equation}
\label{defOfZetaOct2}
\chi: \CL\CO(P) \lra \pi_0\left(\CQ(P) \right) \quad{\rm by~~~} \chi\left(I, \omega\right)=[(f, s)]\in \pi_0\left(\CQ(P) \right)
\end{equation}
where $(f, s) \in \CQ(P)$ be any lift of $(I,\omega)$ in $\CQ(P)$,  ({\it as in diagram \ref{aLiftOmega}}) and 
$[(f, s)]$ is its equivalence class.
%
In several lemmas, we establish that $\zeta$ is well defined. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%Now, we proceed to give an alternate description of $\pi_0({\mathcal Q}(P)$, in analogy to $Q_{2n}(A)$.
\bL%[LindOfs]
\label{LindOfs2Oct} {\rm
Use the  notations as in (\ref{defpizero2Oct}).
Let $(I, \omega_I)\in \CL\CO(P)$  and  $(f,s)\in \CQ(P)$ be a lift, as in diagram (\ref{aLiftOmega}). 
Further, assume that $t(1-t) \in f(P)$, with $I=(f(P), s)=(f(P), t)$.
%(\TCB{we need to have this conditions. for example, we could switch $s, 1-s$}).
%
Then 
$$
[(f, s)]=[(f, t)]\in  \pi_0\left(\CQ(P)\right).
$$
}
\eL
\pf 
First note, \TCP{$(1-s)I \subseteq f(P)$ and $(1-t)I\subseteq f(P)$}.
Write $I[T]=IA[T]$. So,
$$
I[T]=f(P)A[T]+sA[T]=f(P)A[T]+tA[T].
$$
%
Let   $S(T)=t+T(s-t)$. Clearly, $S(T)\in I[T]$. Further,
$$
{\bf Claim}: \qquad (1-S(T))I[T] \subseteq f(P)A[T] 
$$
We have $(1-S(T))I[T]=(1-S(T))[f(P)A[T]+sA[T]$. 
So, we only need to prove that $(1-S(T))s \in f(P)A[T]$. But
$$
(1-S(T))s= (1-t)s-T(s-t)s= (1-t)s+T[(s-1)s+(1-t)s] \in f(P)A[T] 
$$
So, the claim is established. Therefore, $(1-S(T))S(T)\in \in f(P)A[T]$.
Denote $f[T]:=f\otimes 1_{A[T]}$. Then, $f[T]: P[T]\sur f(P)A[T]$ is a surjection.
\TCP{Clearly, $(f[T], S(T))\in \CQ(P[T])$}. Now,
$(f[T], S(T))_{T=0}=(f, t)$ and $(f[T], S(T))_{T=1}=(f, s)$. 
 \pic $\eop$  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bL%[2Lifts]
\label{2Lifts2Oct}
Use the  notations as in (\ref{defpizero2Oct}).
Suppose $(I, \omega)\in \CL\CO(P)$ and and $f, g$ be two lifts of $\omega$ as follows:
$$
\diagram 
P\ar@{->>}[r]^f\ar@{->>}[dr]_{\omega} & f(P)\ar@{->>}[d]\\ 
  & \frac{I}{I^2}\\
\enddiagram \quad {\rm and} \quad 
\diagram 
P\ar@{->>}[r]^g\ar@{->>}[dr]_{\omega} & g(P)\ar@{->>}[d]\\ 
  & \frac{I}{I^2}\\
\enddiagram
$$
%
$$
\ni ~~I=(f(P), s)=(g(P), t) ~{\rm and} ~s(1-s) \in f(P), ~t(1-t) \in g(P).
$$
Then 
$$
[(f, s)]=[(g, t)]\in  \pi_0\left(\CQ(P) \right)
$$
\eL
\pf Note,  $(g-f)(P)\subseteq I^2$. Let $F=f[T]+T(g[T]-f[T]) \in P[T]^*$. 
%
It is obvious that
$$
I[T]=F(P[T])+I[T]^2
$$
For completeness, we give a proof.
$$
\forall ~x\in I, x=(1-s)x+sx =f(p)+ sx \quad {\rm where} \quad p\in P~sx\in I^x
$$
So,
$$
{\rm (modulo ~}I[T]^2{\rm )}
\quad x \equiv f(p) \equiv F[T](p). 
$$
So,
$$
\exists \quad S(T)\in I[X]~\ni (1-S(T))I[T]\subseteq  F[T](P[T]) 
$$
So, $(F[T], S(T))\in \CQ(P[T])$. Therefore,
$$
[(f, S(0))]=[(F(0), S(0))]=[(F(1), S(1))]=[(g, S(1))] 
$$
Now, the proof is complete by (\ref{LindOfs2Oct}). $\eop$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bT
Use the  notations as in (\ref{defpizero2Oct}).
Let $(I,\omega)\in \CL\CO(P)$. The $\chi(I, \omega)$ as defined in equation (\ref{defOfZetaOct2}) is well defined.
\eT
\pf Follows from Lemmas \ref{2Lifts2Oct}. $\eop$
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\vspace{2mm}
Now, we prove that $\overline{\nu}$ is a bijection, as follows.
\bT\label{2pi0SamNov26}
Use the  notations as in (\ref{defpizero2Oct}).
The map 
$$
\overline{\nu}: \pi_0\left(\widetilde{\CQ}(P)\right) \sur \pi_0\left(\CQ(P)\right)
\qquad {\rm is~a~bijection}. 
$$
\eT 
\pf %
First, we define a map 
 $\Psi_0: \CQ(P) \to \pi_0\left(\widetilde{\CQ}(P)\right)$ as follows:
 Given $(f,s)\in \CQ(P)$,  $\exists~p\in P~\ni ~f(p)=s(1-s)$. Define 
 $$
 \Psi_0(f, s):=[(f, p, s)]\in  \pi_0\left(\widetilde{\CQ}(P)\right).
 $$
We show that this association is a well defined map. To show this, suppose there is another 
$q\in P$ such that $f(q)=s(1-s)$. Note $f(p-q)=0$. So, $f[T](p+T(q-p))= f(p)+Tf(q-p)= f(p)+0=s(1-s)$.
Therefore,
$$
H(T):=\left(f[T], p+T(q-p), s) \right)\in \widetilde{\CQ}(P[T])
$$
and, hence
$$
H(0)=(f, p, s)\sim H(1)=(f, q, s).
$$
This establishes that $\Psi_0$ is well defined. Now, we   show that $\Psi_0$ is homotopy invariant. 
To see this, suppose $H(T)=(F, S(T))\in {\CQ}(P[T])$. Then, $S(T)(1-S(T))=F(p(T))$, for some 
$p(T)\in P[T]$. 
Write $\widetilde{H}=(F, p(T), S(T))\in \widetilde{\CQ}(P[T])$. 
So, 
$$
\Psi_0(F(0), s(0))=[\widetilde{H}(0)]= [\widetilde{H}(1)]= \Psi_0(F(1), S(1)) 
$$
This establishes that $\Psi_0$ factors through a map
$$
\Psi:  \pi_0\left(\CQ(P)\right) \to \pi_0\left(\widetilde{\CQ}(P)\right).
$$
It is easy to check that $\overline{\nu}$ and $\Psi$ are inverse of each other. 
\pic $\eop$ 

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%%
\vspace{2mm}
\bL\label{3reePINot}
{\rm 
Use the  notations as in (\ref{defpizero2Oct}). Then, the 
obstruction class map $\chi: {\mathcal LO}(P) \lra \pi_0\left(\CQ(P)\right)$ (see Lemma \ref{defpizero2Oct}) induces a well defined 
map $ \overline{\chi}: \pi_0\left({\mathcal LO}(P)\right) \lra \pi_0\left(\CQ(P)\right)$,
 which  is a bijection.
 
In fact, $\overline{\chi}$ is the inverse of $\overline{\eta}'$. Consequently, $\overline{\eta}$, $\overline{\eta}'$, $\overline{\nu}$,
$\overline{\kappa}$ are bijective maps.
}\eL
%
\pf Given a homotopy $H(T)\in {\mathcal LO}(P[T])$, it lifts to a homotopy $\widetilde{H}(T)=(F(T), S(T))\in {\CQ}(P[T])$.
So, $\chi(H(0))=[(F(0), S(0))]= [(F(1), S(1))]= \chi(H(1))$. So, $\chi$ is homotopy invariant, hence $\overline{\chi}$ 
is well defined. 

 Now, we have a commutative diagram of set theoretic maps:
$$
\diagram 
\pi_0\left(\widetilde{Q}(P) \right)\ar[d]_{\overline{\eta}}\ar[dr]^{\overline{\nu}} & \\
\pi_0\left({\mathcal LO}(P) \right)\ar[r]_{\overline{\chi}} & \pi_0\left(\CQ(P) \right)\\
\enddiagram
$$
Since $\eta$ is an on to, so is $\overline{\eta}$. Since $\overline{\nu}$ is bijective, $\overline{\eta}$ is one to one and hence bijective.
This also establishes that $\overline{\chi}$ is bijective. It is easy to check, $\overline{\chi}$ is the inverse of $\overline{\eta}'$. 
It is established that $\overline{\eta}$, $\overline{\nu}$ are bijective and so is $\overline{\eta}'$. We already established that 
$\overline{\kappa}$ is bijective.
\pic 
$\eop$

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\bC
Use the  notations as in (\ref{defpizero2Oct}).
With $P=A^r=\oplus Ae_i$ is free,  then $Q_{2n}(A))\cong \widetilde{Q}(P)$ is a bijection.
This bijection induces a bijection $\pi_0\left(Q_{2n}\right)(A))\cong \pi_0\left(\widetilde{Q}(P)\right)$.
\eC
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Before we proceed, we introduce the following notions.
\begin{notations}\label{notZetas}{\rm
Suppose $A$ is a commutative noetherian ring, with $\dim A=d$ and $P$ is a projective $A$-module, with $rank(P)=n$.
Denote $\zeta= \overline{\nu}^{-1}\chi:{\mathcal LO}(P) \lra \pi_0\left(\widetilde{Q}(P)\right)$ and $\zeta_0:\widetilde{Q}(P) \lra 
 \pi_0\left(\widetilde{Q}(P)\right)$. So, we have a commutative diagram:
 $$
 \diagram 
 \widetilde{Q}(P) \ar[rd]^{\zeta_0}\ar[d]_{\eta} & \\
 {\mathcal LO}(P)  \ar[r]_{\zeta\qquad} &  \pi_0\left(\widetilde{Q}(P)\right)\\
 \enddiagram
 $$
}\end{notations}
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\subsection{Homotopy Triviality} 
In this subsection, we establish that, for $(I, \omega_I)\in {\mathcal LO}(P)$, under some additional conditions,
that the triviality of  $\zeta(I, \omega_I)$ implies that $\omega_I$ lifts to a surjective map $P\sur I$.
First, we fix a base point, as follows.
%
\bD\label{basePoint}{\rm
Suppose $A$ is a commutative noetherian ring, with $\dim A=d$ and $P$ is a projective $A$-module, with $rank(P)=n$.
If $P=P_0\oplus A$, let  $f_0\in P^*$ be the projection map $P_0\oplus A \sur A$. If $P$ does not split, let $f_0\in P^*$ be such that,
$height(f_0(P))=n$. Recall, existence of such a  map $f_0$ is assured by basic element theory.
 We fix   ${\bf v}_0:=(f_0, 0, 0) \in \widetilde{Q}(P)$ and treat is it as the {\bf base point} of $\widetilde{Q}(P)$.
 We will see later that $\zeta_0({\bf v}_0)=\zeta_0(0, 0, 0)\in \pi_0\left(\widetilde{Q}(P)\right)$ (see \ref{listHomoNov21}).
%So, $\eta({\bf v}_0)$ will be treated as the base point of ${\mathcal LO}(P)$.

}
\eD
%
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The following is the final result in this section. 
% \TCB{This will move elsewhere}.
%
\bT\label{formalizeBSNori}
Suppose $A$ is an essentially smooth ring over an infinite perfect field $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$, with $2n\geq d+3$. 
Let ${\bf v}_0=(f_0, 0, 0)\in \widetilde{Q}(P)$ be a base point, as in (\ref{basePoint}).
%
Suppose $(I, \omega_I)\in {\mathcal LO}(P)$, with $height(I)\geq n$. Then, $\omega_I$ lifts to a surjective map $P\sur I$ if and only if 
$\zeta(I, \omega_I)=\zeta_0({\bf v}_0)$.
\eT
\pf Denote $\eta({\bf v}_0)=(I_0, \omega_{I_0})$. 
Suppose $\omega_I$ lifts to a surjective map $f_1:P \sur I$.   Let $f(T)=(1-T)f_0+Tf_1\in P[T]^*$. 
Write $H(T)= (f(T), 0, 0)\in \widetilde{Q}(P[T])$. Then, $\zeta(I, \omega_I)=\zeta_0(H(1))=  \zeta_0({\bf v}_0)$.


Conversely, suppose $\zeta(I, \omega_I)=\zeta_0({\bf v}_0)$. 
There is an element ${\bf u}=(f_1, p_1, s_1) \in \widetilde{Q}(P)$ such that $\eta({\bf u})=(I, \omega_I)$.
By Moving Lemma argument \ref{movingLemm} (below), we can assume $height( f_1(P))\geq n$.
%KEEP proof
%We would have $F=f+s^2g$ is basic on $ht(\wp)\leq n-1$ and $s\notin \wp$. Now, suppose $F(P)\subseteq \wp$.
%If $s\notin \wp$, then $ht(\wp)\geq n$. If $s\in \wp$ then $I=(f(P), s) \subseteq (F(P), s)\subseteq \wp$. So, $ht(\wp)\geq n$.
Therefore $\zeta_0({\bf u})=\zeta_0({\bf v}_0)$. By (\ref{directHomo}), there is a homotopy $H(T)=(f(T), p(T), S(T))\in \widetilde{Q}(P[T])$ such that 
$H(0) = {\bf v}_0$ and $H(1)={\bf u}$. Write $\eta(H(T))=(J, \Omega)$.  We would apply \cite[Theorem 4.13]{BK}, for which we would need $height(J)
\geq n$. So, we modify $H(T)$, as follows. Denote $Z(T)=1-S(T)$. Write 
$\SP=\{\wp\in \spec{A[T]}: height(\wp)\leq n-1, T(1-T)Z(T)\notin \wp \}$.
%
 Define a generalized dimension $\delta:\SP \lra \BN$ by $\delta(\wp)=height(\wp)$ for $\wp\in \SP$.
 Then,
$\forall~\wp\in \SP$, we have $\delta(\wp)\leq n-1$.
Now, $(f(T), T(1-T)Z(T)^2)\in P[T]^*\oplus A[T]$ is basic on $\SP$. 
 So, there is an element $g(T)\in P[T]^*$ such that
 \TCP{ $F(T)=f(T)+T(1-T)Z(T)^2g(T)$}
  is basic on $\SP$.
 It follows, $F(0)=f(0)$ and $F(1)=f(1)$.
 
We have $Z(T)(1-Z(T))=(1-s(T))s(T)=$
%
$$
f(T)(p(T))=F(T)(p(T))-T(1-T)Z(T)^2g(T)(p(T))
$$
Write $\CJ=(f(T)(P[T]), Z(T))$. Then $\CJ=(F(T)(P[T]), Z(T))$. Write $M=\frac{\CJ}{F(T)(P[T])}$. Let $p_1, \ldots, p_m$ be a set of 
generators of $P$. So, $\CJ$ is generated by the images of $f(T)(p_1), \ldots, f(T)(p_m), Z(T)$. 
Use "overline" to denoted images in $M$ and repeat the proof of Nakayama's Lemma, as follows:
$$
\left(
\begin{array}{c}
\overline{f(T)(p_1)}\\
\overline{f(T)(p_2)}\\
\cdots\\
\overline{f(T)(p_m)}\\
\overline{Z(T)}\\
\end{array}
 \right)
 =
 \left(
\begin{array}{ccccc}
0 & 0 & \cdots & 0 & -T(1-T)Z(T)g(T)(p_1)\\
0 & 0 & \cdots & 0 & -T(1-T)Z(T)g(T)(p_2)\\
0 & 0 & \cdots & 0 & \cdots\\
0 & 0 & \cdots & 0 & -T(1-T)Z(T)g(T)(p_m)\\
%0 & 0 & \cdots & 0 & Z(T)[1-T(1-T)g(T)(p(T))]\\
0 & 0 & \cdots & 0 & Z(T)-T(1-T)Z(T)g(T)(p(T))\\
\end{array}
 \right)
 \left(
\begin{array}{c}
\overline{f(T)(p_1)}\\
\overline{f(T)(p_2)}\\
\cdots\\
\overline{f(T)(p_m)}\\
\overline{Z(T)}\\
\end{array}
 \right)
$$
So,
$$
 \left(
\begin{array}{ccccc}
1 & 0 & \cdots & 0 & T(1-T)Z(T)g(T)(p_1)\\
0 & 1 & \cdots & 0 & T(1-T)Z(T)g(T)(p_2)\\
0 & 0 & \cdots & 0 & \cdots\\
0 & 0 & \cdots & 1 & T(1-T)Z(T)g(T)(p_m)\\
0 & 0 & \cdots & 0 &1-Z(T)+ T(1-T)Z(T)g(T)(p(T))\\
\end{array}
 \right)
 %
 \left(
\begin{array}{c}
\overline{f(T)(p_1)}\\
\overline{f(T)(p_2)}\\
\cdots\\
\overline{f(T)(p_m)}\\
\overline{Z(T)}\\
\end{array}
 \right)
 =
 \left(
\begin{array}{c}
0\\
0\\
\cdots\\
0\\
0\\
\end{array}
 \right)
 $$
With $Z'(T)=Z(T)-T(1-T)Z(T)g(T)(p(T))$, the determinant of this matrix is $1-Z'(T)$. It follows, $(1-Z'(T))\CJ\subseteq F(T)(P[T])$.
So,\\
 \TCP{$(1-Z'(T))Z'(T)=F(T)(q(T))$} for some $q(T)\in P[T]$. Note, \TCP{$Z'(0)=Z(0)$ and $Z'(1)=Z(1)$}.
Therefore, $(F(T), q(T), Z'(T))\in \widetilde{Q}(P[T])$. Also, with
\TCP{ $S'(T)=1-Z'(T)$}, $(F(T), q(T), S'(T))\in \widetilde{Q}(P[T])$. We have  \\
$S'(T)(1-S'(T))=(1-Z'(T))Z'(T)=F(T)(q(T))$\\
$S'(0)=1-Z'(0)=1-Z(0)=S(0)=0$ and\\
$S'(1))=1-Z'(1)=1-Z(1)=S(1)$.\\
Write $\CH(T)= (F(T), q(T), S'(T))$ and $\eta(\CH(T))=(J', \Omega')$. It is clear $\CH(0)=(f_0, q(0), 0)$, $\CH(1)=(f_1, q(1), S(1))$.
So, $\eta(\CH(0))=\eta({\bf v}_0)$ and $\eta(\CH(1))=\eta({\bf u})=(I, \omega_I)$.

 We have $J'=(F(T)(P[T]), S'(T))$. We claim that $height(J')\geq n$. To see this,
 let $J'\subseteq \wp\in \spec{A[T]}$. If $T\in \wp$, then $I_0 \subseteq \wp$ and hence $height(\wp)\geq n$. Likewise, 
  if $1-T\in \wp$, then $I \subseteq \wp$ and hence
 $height(\wp)\geq n$. So, we assume $T(1-T)\notin \wp$. 
 If $Z(T)\in \wp$, then $\CJ=(F(T)(P[T]), Z'(T))= (F(T)(P[T]), Z(T))\subseteq \wp$, which is impossible because $S'(T)\in \wp$.
 So, $T(1-T)Z(T)\notin \wp$. Since $F$ is basic on $\SP$, $height(\wp)\geq n$. This establishes the claim.
 
 So, $\CH(T)=(F(T), q(T), S'(T))\in \widetilde{Q}(P[T])$ is such that $\eta(\CH(0))=(I_0, \omega_{I_0})$, 
 $\eta(\CH(1))=(I, \omega_{I})$ and with $\eta(\CH(T))=(J', \Omega')$, we have $height(J') \geq n$. 
 If $T\in \wp \in Ass\left(\frac{A[T]}{J'}\right)$ then $(J'(0), T)=(I_0, T)\subseteq \wp$. Then, $height(\wp)\geq n+1$.
 This is impossible because $A[T]$ is regular (Cohen-Macaulay) and $J'$ is local complete intersection ideal.
 Hence, 
 $$
 \diagram
 \frac{A[T]}{TJ'}\ar[r]^{T=0}\ar[d] & A\ar[d]\\
 \frac{A[T]}{J'}\ar[r] & \frac{A}{J'(0)}\\
 \enddiagram
 $$
 is a patching diagram (see (\ref{patchBK}) below). So, the map $\Omega': P[T] \sur \frac{J'}{(J')^2}$ and $f_0:P\sur I_0$ combines to give a surjective 
 maps $\phi:P[T] \sur \frac{J'}{T(J')^2}$. Now, by \cite[Theorem 4.13]{BK}, there is a surjective homomorphism 
 $\varphi:P[T] \sur J'$ such that $\varphi(0)=f_0$ and $\varphi\otimes \frac{A[T]}{J'}=\Omega'$. Now, it follows that 
 $\varphi(1)$ is a lift of $\omega_I$.
 \tcp $\eop$
 
 
 
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%\subsection{On Patching (\TCP{to be removed})}
%Recall from \cite[\S 2]{Mil}
%\bD\label{MilnotDef}{\rm
%Consider a diagram of commutative ring
%$$
%\diagram 
%\Lambda \ar[r]^{i_1}\ar[d]_{i_2} & \Lambda_1\ar[d]^{j_1}\\
%\Lambda_2 \ar[r]_{j_2} & \Lambda'\\
%\enddiagram
%$$
%such that
%\bE
%\item (Hyp 1) given $(x_1, x_2)\in \Lambda_1\times \Lambda_2$ with $j_1(x_1)=j_2(x_2)$ there is one and only one element 
%$x\in \Lambda$ such that $i_1(x)=x_1$ and $i_2(x)=x_2$.
%\item (Hyp 2). At least one of $j_1$ or $j_2$ is surjective.
%\eE
%}
%\eD
%

%
%\bD\label{generalDef}{\rm
%Given any ring $R$ we can consider such patching diagram of $R$-modules
%$$
%\diagram 
%M \ar[r]^{i_1}\ar[d]_{i_2} & M_1\ar[d]^{j_1}\\
%M_2 \ar[r]_{j_2} & M'\\
%\enddiagram
%$$
%such that
%\bE
%\item (Hyp 1) given $(x_1, x_2)\in M_1\times M_2$ with $j_1(x_1)=j_2(x_2)$ there is one and only one element 
%$x\in \Lambda$ such that $i_1(x)=x_1$ and $i_2(x)=x_2$.
%\eE
%}
%\eD
%
%\bL\label{patchProj}{\rm
%Given a patching diagram, as in (\ref{MilnotDef}), patching of projectives work great. They are even patching diagrams as in (\ref{generalDef}).
%}
%\eL
%
%\begin{example}\label{IJDiagram}{\rm
%Suppose $I, J$ be two ideal in $R$.
%Then, 
%$$
%\diagram 
%\frac{A}{I\cap J} \ar[r]\ar[d] & \frac{A}{J}\ar[d]\\
%\frac{A}{I} \ar[r] & \frac{A}{I+J}\\
%\enddiagram
%\quad {\rm is ~a~patching~diagram~as~in}\quad (\ref{MilnotDef}). 
%$$
%In this case,
%$$
%\diagram 
%\frac{I}{I^2\cap (JI)} \ar[r]\ar[d] & \frac{I}{JI}\ar[d]\\
%\frac{I}{I^2} \ar[r] & \frac{I}{I^2+JI}\\
%\enddiagram
%\quad {\rm ~\TCP{is~a~Patching~Diagram~(\ref{generalDef})}}
%$$
%}
%\end{example}
We used the following lemma above, while it needs a proof. The standard references for Patching diagrams are \cite{Mil, R, O2}. 
We will be specific in the following statement, because
the 
literature does not seem complete regarding definitions Patching diagrams of modules that are not projective. 
%
\bL\label{patchBK}
Let $R$ be a noetherian commutative ring and
 $A=R[T]$. Take $J=AT$ and  $I$ is locally complete intersection ideal of height $r$ and $T:\frac{A}{I} \hra \frac{A}{I}$ is injective 
(i. e.
$T\notin \wp \in Ass\left( \frac{A}{I}\right)$). Then,
$$
\diagram 
\frac{I}{TI^2} \ar[r]\ar[d] & \frac{I}{TI}\ar[d]\\
\frac{I}{I^2} \ar[r] & \frac{I}{I^2+TI}\\
\enddiagram
%\quad {\rm ~\TCP{is~a~Patching~Diagram~(\ref{generalDef})}}
$$
is a Patching diagram, in the sense that it is a Cartesian square.
Further,
\bE
\item $\frac{I}{TI} \iso I(0)$.
\item $\frac{I}{I^2+TI}\iso \frac{I(0)}{I(0)^2}$.
\eE
\eL
\pf The patching diagram follows from (\ref{IJDiagram}), because 
 $I^2\cap (TI)=TI^2$.
 
 To see this, first
we have $TI^2 \subseteq I^2\cap (TI)$. Suppose $f\in I^2\cap (TI)$. Then, $f=Tg$ with $g\in I$. Now, consider the map
$$
T: \frac{I}{I^2} \lra \frac{I}{I^2}
$$
Since $\frac{I}{I^2}$ projective and $T:\frac{A}{I} \hra \frac{A}{I}$ is injective, $T$ is also injective on $\frac{I}{I^2}$. So, $g\in I^2$.
So, $f=Tg\in TI^2$. 

Now, we prove $\frac{I}{TI} \iso I(0)$. Obviously, the map is on to. Suppose $f(T)\in I$ and $f(0)=0$.Then, $f=Tg$. Since $T$ is non zero divisor on 
$\frac{A}{I}$, $g\in I$. SO, $f\in TI$.

Finally, we prove $\frac{I}{I^2+TI}\iso \frac{I(0)}{I(0)^2}$. Again, the map is on to. Suppose $f(T)\in I$ and $f(0)\in I(0)^2$.
Then, $f(0)=\sum f_i(0)g_i(0)$. Then, $f-\sum f_ig_i\in (T)\cap I=TI$ (by the above, if we like). So, $f\in I^2+TI$.

$\eop$


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We close this section with the following  "moving lemma argument", which  is fairly standard. A number of  variations of the same (\ref{movingLemm}) 
would be among the frequently used tools for the rest of our discussions.
\bL[Moving Lemma]\label{movingLemm}
Suppose $A$ is a commutative noetherian ring with $\dim A=d$ and $P$ is a projective $A$-module with $rank(P)=n$.
Assume $2n\geq d+1$.
Let $K\subseteq A$ be an ideal with $height(K)\geq n$ and $(I, \omega_I)\in {\mathcal LO}(P)$. 
Then, there is an element ${\bf v}=(f, p, s)\in \widetilde{Q}(P))$ such that %$I=(f_1, \ldots, f_n, s)$, 
$\eta({\bf v}) =(I, \omega_I)$. Further, with $J=f(P)+A(1-s)$, we have $height(J)\geq n$ and $J+K=A$.
\eL
\pf 
Let $f_0:P \sur I$ be any lift of $\omega_I$. Then, $I=f_0(P)+I^2$. By Nakayama's Lemma, 
there is an element $t\in I$,
such that $(1-t)I \subseteq f_0(P)$. Therefore, $t(1-t)=f_0(p_0)$ for some $p_0\in P$.
%
({\it Readers are referred to \cite{M1} regarding generalities on Basic Element Theory and generalized
dimension functions}.)
%
Write 
$$
\SP=\left\{\wp\in \spec{A}: t\notin \wp, ~{\rm and~either}~K\subseteq \wp~{\rm or}~height(\wp)\leq n-1 \right\}
$$
There is a generalized dimension function (see \cite{M1})  $\delta: \SP \lra \BN$, such that $\delta(\wp)\leq n-1~\forall~\wp\in \SP$.
Now $(f_0, t^2)\in P^* \oplus A$ is basic on $\SP$. So, there is an element $g\in P^*$ such that $f:=f_0+t^2g$ is basic on $\SP$.
It follows, $f(P)+At=f_0(P)+At=I$ and $I=f(P)+I^2$. 
%
By Nakayama's Lemma, 
there is an element $s\in I$,
such that $(1-s)I \subseteq f(P)$ and hence $f(p)=s(1-s)$, for some $s\in I$. Hence, $I=(f(P), s)$.
%
Now, write $J=f(P)+A(1-s)$. For $J \subseteq \wp\in \spec{A}$, $s\notin \wp$ and hence $t\notin \wp$.
Since, $f$ is basic on $\SP$, $height(\wp)\geq n$. This establishes, $height(J)\geq n$. 

Now suppose  $J+K \subseteq \wp\in \spec{A}$. By the same argument above, $t\notin \wp$. Hence,
$\wp\in \SP$. This is Impossible, because $f$ is basic on $\SP$. So,  $J+K=A$. Now, ${\bf v}=(f, p, s)
\in \widetilde{Q}(P)$, satisfies the requirement.
$\eop$


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%\newpage 


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\section{Homotopy}\label{secHomotopy}

In this section, we prove the following key homotopy theorem.
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\bT\label{PopThm3p4M2} %\label{1p0p5F}
Let $A$ be a regular ring over a  field $k$, with $1/2\in k$. Let $P$ be a projective $A$-module, with $rank(P)=n\geq 2$, 
and $(\BQ(P), q)=\BH(P) \perp A$
{\rm (see \ref{nota})}. Recall $\widetilde{Q}'(P)\subseteq \BQ(P)=P^*\oplus P \oplus A$.
Suppose $H(T)\in \widetilde{Q}'(P[T])$. Then, there is an orthogonal matrix $\sigma(T)\in O(\BQ(P), q)$, such that 
$$
H(T)=\sigma(T)(H(0))\qquad {\rm and}\qquad \sigma(0)=1.
$$
%
%$$ 
\eT
\pf 
Let $H(T)=(f(T), p(T), s(T)  \in \widetilde{Q}'(P[T])$ be a homotopy, as above. So, $H(0) \in \widetilde{Q}'(P)$. Then,
$$
A[T]H(T) \cong A[T]H(0) \cong (A[T], q_0) \qquad {\rm are ~isometric},
$$
where $q_0$ is the trivial quadratic space of rank one.
The bilinear inner product  in $\BQ(P)$ will be denoted $\langle -, -\rangle$.
We have the following split exact sequences of quadratic spaces:
$$
\diagram
0\ar[r] & K \ar[r] & \BQ(P[T]) \ar[rr]^{\langle H(T), -\rangle} & & A[T] \ar[r] & 0\\ 
0\ar[r] & K_0 \ar[r] & \BQ(P) \ar[rr]_{\langle H(0), -\rangle} & & A \ar[r] & 0\\ 
\enddiagram
$$
%
Therefore, 
$K=\left(A[T]H(T)\right)^{\perp}$,  $K_0=\left(AH(0)\right)^{\perp}$
are orthogonal complements. 
%
Write $\overline{K}:=K\otimes \frac{A[T]}{(T)}$. 
Note, for $\wp\in \spec{A}$, $\BQ(P)_{\wp} \cong (A, q_{2n+1})$, where $q_{2n+1}=\sum_{i=1}^nX_iY_i+Z^2$.
So, $\overline{K}_{\wp}\cong (K_0)_{\wp}$ are isometric. It is standard (see \cite[Lemma 4.1]{MM}),
 that $(K_0)_{\wp}=(A_{\wp}H(0))^{\perp}\cong (A, q_{2n})_{\wp}$
where 
$q_{2n}=\sum_{i=1}^nX_iY_i$. In other words, $\overline{K}$ is locally trivial. By the the Quadratic version \cite[Theorem 3.4]{MM} of  Lindel's 
theorem \cite{L},
%
there is an isometry $\tau:K \iso \overline{K}\otimes A[T]$. 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Further, it follows 
%
$\overline{K}=\left(RH(0)\right)^{\perp}\cong K_0$. 
Therefore, there is an isometry $\sigma_0:  \overline{K} \iso K_0$, which extends to an isometry
$\sigma_0\otimes 1: \overline{K}\otimes A[T] \iso K_0\otimes A[T]$.
Then,  $\sigma_1:=(\sigma_0\otimes 1)\tau: K \iso  K_0\otimes A[T]$ is an isometry.
%
Finally, note 
$$
(A[T]H(T), q_{|A[T]H(T)}) \cong (A[T], q_0) \cong (A[T]H(0), q_{|A[T]H(0)}).
$$
Now, consider the  diagram
\begin{equation}\label{linearMaps}
\diagram
0\ar[r] & K\ar[r]\ar[d]_{\sigma_1} & \BQ(P[T])  \ar@{-->}[d]^{\sigma(T)} \ar[rr]^{\langle H(T), -\rangle} & & A[T] \ar@{=}[d]\ar[r] & 0\\ 
0\ar[r] & K_0\otimes A[T] \ar[r] & \BQ(P[T])  \ar[rr]_{\langle H(0), -\rangle} & & A[T] \ar[r] & 0\\ 
\enddiagram
\end{equation}
of quadratic spaces. In this diagram, the horizontal lines are split exact sequences of quadratic spaces. Hence, there is an 
isometry $\sigma(T)\in O\left(A[T], q\right)$, such that the diagram commutes. That means, for all ${\bf v}\in  \BQ(P[T])$, we have
$\langle H(T), {\bf v}\rangle= \langle H(0), \sigma(T){\bf v}\rangle$. Replacing $\sigma(T)$ by $\sigma(T)^{-1}$, we have
 $\sigma(T)H(0)=H(T)$. So, we have $\sigma(0)H(0)=H(0)$. Again,
 by replacing $\sigma(T)$ by $\sigma(T)\sigma(0)^{-1}$, we have $\sigma(0)=1$.
\pic 
%
 $\eop$ 
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The following Corollary would be of some use for our future discussions.
\bC\label{directHomo}
Let $A$ be a regular ring over a  field $k$, with $1/2\in k$. Let $P$ be a projective $A$-module, with $rank(P)=n\geq 2$, 
and $(\BQ(P), q)=\BH(P) \perp A$.
Let ${\bf u}, {\bf v}\in \widetilde{Q}'(P)$ such that $[{\bf u}]= [{\bf v}]\in \pi_0\left(\widetilde{Q}'(P)\right)$.
Then, there is a homotopy $H(T)\in \widetilde{Q}'(P[T])$ such that $H(0)={\bf u}$ and $H(1)={\bf v}$.
%
Equivalently, for 
${\bf u}, {\bf v}\in \widetilde{Q}'(P)$ if $\zeta_0({\bf u})= \zeta_0({\bf v})\in  \pi_0\left(\widetilde{Q}(P)\right)$, then
there is a homotopy $H(T)\in \widetilde{Q}(A[T])$ such that $H(0)={\bf u}$ and $H(1)={\bf v}$.
%
\eC
\pf Suppose ${\bf u}, {\bf v}\in Q_{2n}'(A)$ such that $[{\bf u}]= [{\bf v}]\in \pi_0\left(\widetilde{Q}'(P)\right)$.
Then, there is a sequence of homotopies $H_1(T), \ldots , H_m(T) \in \widetilde{Q}'(P[T])$
such that ${\bf u}=:{\bf u}_0:=H_1(0)$, ${\bf u}_m:=H_m(1)={\bf v}$ and $\forall~i=1, \ldots, m-1$, we have ${\bf u}_i:=H_i(1)=H_{i+1}(0)$.
By Theorem \ref{PopThm3p4M2}, for $i=1, \ldots, m$ there are orthogonal matrices $\sigma_i(T)\in O(\BH(P[T]), q)$ such that
$\sigma_i(0)=1$ and 
$H_i(T)=\sigma_i(T)H_i(0) =\sigma_i(T){\bf u}_{i-1}$. Therefore, ${\bf u}_i=H_i(1) = \sigma_i(1){\bf u}_{i-1}$.

Write $H(T)=\sigma_m(T)\cdots \sigma_1(T){\bf u}_0$. Then, $H(T)\in Q_{2n}'(A[T])$ and $H(0)= {\bf u}_0$ and $H(1)= {\bf u}_m$.
%
%
This establishes first part of the statement on $\pi_0\left(\widetilde{Q}'(P)\right)$. The latter assertion on $\pi_0\left(\widetilde{Q}(P)\right)$ follows from the former, 
by the bijective correspondences 
$\widetilde{Q}'(P)  \llra \widetilde{Q}(P)$ and $\widetilde{Q}'(P[T]) \llra \widetilde{Q}(P[T])$. 
\tcp $\eop$
%%%%
\begin{remark}\label{equivHomRelDec2}{\rm
Another way to state (\ref{directHomo}) would be to say that the homotopy relation on $\widetilde{Q}(P)$ is actually an equivalence relation.
}
\end{remark}

In a slightly more formal language, the above is summarized as follows.
\bT\label{summHomT}
Let $A$ be a regular ring over a  field $k$, with $1/2\in k$. Let $P$ be a projective $A$-module, with $rank(P)=n\geq 2$, 
and $(\BQ(P), q)=\BH(P) \perp A$.
For, $\sigma(T)\in O\left(\BQ(P), q\right)$ and ${\bf u}\in \widetilde{Q}'(P)$, define the  {\rm (left)} action 
$\sigma(T){\bf u}:=\sigma(1){\bf u}\in \widetilde{Q}'(P)$. 
Denote $O\left(\BQ(P), q, T\right)=\{\sigma(T)\in O\left(\BQ(P), q\right): \sigma(0)=1 \}$.
%
Then, the map
$$
\frac{\widetilde{Q}'(P)}{O\left(\BQ(P), q, T\right)} 
\lra \pi_0\left(\widetilde{Q}'(P)\right) \quad {\rm is~a~bijection}.
$$
\eT
\pf Similar to the proof of (\ref{directHomo}). $\eop$
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\section{The Involution}\label{secInvolution}
%{Algebra of Homotopy Obstructions}
 
In analogy to the Involution on on $Q_{2n}(A)$, defined in \cite{MM}, we introduce an involution on $\widetilde{Q}(P)$.

\bD\label{invoDefi}
Suppose $A$ is a commutative ring and $P$ is a projective $A$-module, with $rank(P)=n$.
For $(f, p, s) \in \widetilde{Q}(P)$, define $\Gamma(f, p, s)= (f, p, 1-s)$.
This 
 association, ${\bf v}\mapsto \Gamma({\bf v})$, establishes a bijective correspondence
 $$
 \Gamma:  \widetilde{Q}(P)\iso  \widetilde{Q}(P), \quad{\rm such ~that}\quad \Gamma^2=1.
 $$
 We say $\Gamma$ is an involution on 
 on $\widetilde{Q}(P)$, which will be a key instrument in the subsequent discussions. {\rm (This notation $\Gamma$ will be among the standard notations throughout this article.)}
\eD
We record the following  obvious lemma.
\bL\label{obvious}
Suppose $A$ is a commutative ring and $P$ is a projective $A$-module, with $rank(P)=n$ and 
%
 $\Gamma: \widetilde{Q}(P)\iso  \widetilde{Q}(P)$ is the involution.
Let 
 ${\bf v}=(f, p, s)\in \widetilde{Q}(P)$ and denote $\eta({\bf v})=(I, \omega_I)$ and $\eta(\Gamma({\bf v}))=(J, \omega_J)$.
Then,
\bE
\item $I\cap J=f(P)$.
 
\item For $H(T)\in \widetilde{Q}(P[T])$, we have
 $\Gamma(H(T))_{T=t}= \Gamma(H(t))$.
\item\label{obNov1216} Therefore,
$
\forall~{\bf v}, {\bf w}\in Q_{2n}(S)\qquad  \zeta_0({\bf v}) =\zeta_0({\bf w}) \Llra  \zeta_0(\Gamma({\bf v})) =\zeta_0(\Gamma({\bf w})).
$
\eE
\eL
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We also record the following obvious observation.
\bL\label{Bibekananda}
Suppose $A$ is a commutative noetherian ring with $\dim A=d$ and $P=P_0\oplus A$ is a projecttive $A$-module. 
Denote $p_0=(0, 1)\in P_0\oplus A$ and $f_0:P \sur A$ be the projection map $P_0\oplus A\lra A$. Let 
${\bf u}_0=(f_0, 0, 0)\in \widetilde{Q}(P)$ be the base point and ${\bf u_1}=(0, 0, 1) $. Then, $\zeta_0({\bf u}_0)
= \zeta_0({\bf u}_1) \in \pi_0\left(\widetilde{Q}(P)\right)$
\eL
\pf 
Write $H(T)=\left((1-T)f_0, Tp_0, T \right)$. Then, $(1-T)f_0(Tp_0)=T(1-T)$. So, $H(T)\in \widetilde{Q}(P[T])$.
We have $H(0)= {\bf u}_0$ and $H(1)= {\bf u}_1$. \pic $\eop$

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\vspace{2mm}
In deed, $\Gamma$ factors through an involution on $\pi_0(Q_{2n})(A)$, as follows.
 
\bC\label{theInvolution} 
Suppose $A$ is a commutative ring and $P$ is a projective $A$-module, with $rank(P)=n$.
Then, the involution $\Gamma:  \widetilde{Q}(P)\iso  \widetilde{Q}(P)$ induces a bijective map
$\widetilde{\Gamma}:  \pi_0\left(\widetilde{Q}(P)\right) \iso  \pi_0\left(\widetilde{Q}(P)\right)$, such that 
$\widetilde{\Gamma}^2=1$ and $\zeta_0\Gamma= \widetilde{\Gamma}\zeta_0$. We say $\widetilde{\Gamma}$  is  an involution.
{\rm (The notation $\widetilde{\Gamma}$ will also be among our standard notations throughout this article.)}
\eC 
\pf  First, consider the map $\zeta_0\Gamma: \widetilde{Q}(P) \lra \pi_0\left(\widetilde{Q}(P)\right)$.
For, $H(T)\in \widetilde{Q}(P[T])$, we have $\zeta_0\Gamma(H(0))= \zeta_0\Gamma(H(1))$. Therefore, $\zeta_0\Gamma$
is homotopy invariant. Hence, it induces the a well defined map 
$\widetilde{\Gamma}:  \pi_0\left(\widetilde{Q}(P)\right) \iso  \pi_0\left(\widetilde{Q}(P)\right)$. 
Clearly, $\widetilde{\Gamma}^2=1$ and $\widetilde{\Gamma}$ is a bijection.
\pic $\eop$
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\vspace{2mm}
The following is a way to compute the involution.
%
\bC\label{altF_i4involution}
Suppose $A$ is a commutative ring and $P$ is a projective $A$-module, with $rank(P)=n$.
Suppose $(I, \omega)\in {\mathcal LO}(P)$. For any ${\bf v}=(f, p, s) \in \widetilde{Q}(P)$ with 
$\eta({\bf v})= (I, \omega)$, write $\eta(\Gamma({\bf v})) =(J, \omega_J)$.
%
Then, 
$$
\widetilde{\Gamma}(\zeta(I, \omega))= \zeta(J, \omega_J)\in \pi_0\left(\widetilde{Q}(P)\right).
$$ 
\eC
 \pf Obvious.
$\eop$
 
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The following is another version of the Moving Lemma \ref{movingLemm}. 
\bL[Moving Representation]\label{movingRep}
Suppose $A$ is a commutative ring, with $\dim A=d$.
Let  $P$ be a projective $A$-module, with $rank(P)=n$, with $2n\geq d+1$.
$x\in \pi_0\left(\widetilde{Q}(P)\right)$ and let $K\subseteq A$ be an ideal with $height(K)\geq n$ and . Then, there 
is a local $P$-orientation $(J, \omega_J)\in {\mathcal LO}(P)$ such that $x=\zeta(J, \omega_J)$, $height(J)\geq n$ and $J+K=A$.
\eL
\pf 
First, let $x=\zeta(I, \omega_I)$. First, $\eta({\bf u})=(I, \omega_I)$ for some ${\bf u}\in \widetilde{Q}(P)$.
 Denote   $(I_0, \omega_{I_0}):=\eta(\Gamma({\bf u}))$. Then, 
$\tilde{\Gamma}(x)=\zeta(I_0, \omega_{I_0})$.

Now, we apply Moving Lemma \ref{movingLemm}, to $(I_0, \omega_{I_0})$ and $K$. 
There is ${\bf v}\in \widetilde{Q}(P)$, such that $\eta({\bf v})=(I_0, \omega_{I_0})$, and 
with $\eta(\Gamma({\bf v}))=(J, \omega_J)$, we have $height(J)\geq n$ and $J+K=A$. Now, $x=\tilde{\Gamma} (\tilde{\Gamma} (x))
= \tilde{\Gamma} (\zeta(I_0, \omega_{I_0}))=\zeta(J, \omega_J)$.
\pic $\eop$


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\section{The Monoid Structure on $\pi_0\left(\widetilde{Q}(P)\right)$} \label{secGroupStru}
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In this section, we  establish a group structure on the homotopy obstruction set $\pi_0\left(\widetilde{Q}(P)\right)$ , when $2rank(P)\geq \dim A+2$ and $A$ is a regular ring 
over a  field $k$, with $1/2\in k$. We start with the following basic 
ingredient of the group structure. 
%
\bD\label{psudoPlus}
Let $A$ be a commutative noetherian ring and $P$ be a projective $A$-module, with $rank(P)=n\geq 2$.
 Let
$(I, \omega_I), (J, \omega_J)\in {\mathcal LO}(P)$ be such that $I+J=A$. 
Let $\omega:=\omega_I\star \omega_J:  P \sur \frac{IJ}{(IJ)^2}$ be the unique surjective map induced by $\omega_I, \omega_J$.
% 
We define a pseudo-sum 
$$
(I, \omega_I) \hat{+} (J, \omega_J) :=  \zeta(IJ, \omega)\in \pi_0\left(\widetilde{Q}(P)\right).
$$
Also, for ${\bf u}, {\bf v}\in \widetilde{Q}(P)$ with $\eta({\bf u})=(I, \omega_I)$ and $\eta({\bf v})= (J, \omega_J)$, if $I+J=A$,
define pseudo-sum
$$
{\bf u}\hat{+} {\bf v} := \eta({\bf u}) \hat{+} \eta({\bf v}) \in\pi_0\left(\widetilde{Q}(P)\right).
$$
\eD 
The rest of this section is devoted to establish that this pseudo sum respects homotopy and extends to $\pi_0(Q_{2n})(A)$,
when 
$A$ is a regular ring 
over a  field $k$, with $1/2\in k$. 




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace{2mm} 
Now we define a pseudo-difference in the spirit of (\ref{psudoPlus}).
\bD\label{psudoDiff}
Let $A$ be a commutative noetherian ring and $P$ be a projective $A$-module, with $rank(P)=n\geq 2$.
Suppose $(K, \omega_K), (I, \omega_I) \in {\mathcal LO}(P)$. % 
Assume that there is ${\bf u}=(f, p, s)\in \widetilde{Q}(P)$ such that 
$\eta({\bf u})=(I, \omega_I)$ and with $\eta(\Gamma({\bf u}))=(J, \omega_J)$, $J+K=A$. Therefore,
%
%
$$
f(P)=I \cap J \quad \ni\quad I+J=K+J=A. 
$$
%
Define the pseudo-difference
$$
(K, \omega_K) \hat{-} (I, \omega_I):=(K, \omega_K) \hat{+} (J, \omega_J) \in \pi_0\left(\widetilde{Q}(P)\right).
$$
{\rm 
We remark:
(1)
A priori, the pseudo-difference depends on the choice of $J$.  There is no conditions on $height(I)$, $height(J)$, $height(K)$; nor did we assume $I+K=A$.
%
(2) By Moving Lemma \ref{movingLemm}, such choices  ${\bf u}=(f, p, s)\in \widetilde{Q}(P)$
would be available if $2n\geq \dim A+1$ and $height(K)\geq n$.

}
 \eD
 Subsequently, under additional hypotheses, we would first prove that the definition (\ref{psudoDiff})
 of pseudo difference does not depend of the choice of $(J, \omega_J)$. Then, we  
 prove that the pseudo difference is homotopy invariant with respect to either coordinate. A key to such proofs would be the following 
 lemma that  combines ({\it i.e. "adds"}) homotopies.
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%\subsection{Homotopy Lemmas}In this subsection, we prove some homotopy lemmas.
\bL\label{HLK1hoK2}
Let $A$ be a commutative noetherian ring and $P$ be a projective $A$-module, with $rank(P)=n$, with $2n\geq d+2$.
%
%Suppose $A$ is a commutative noetherian ring with $\dim A=d$ and $n\geq 2$ is an integer with $2n\geq d+2$. 
Consider a homotopy
 $$
 H(T)=(f(T), p(T), Z(T))\in \widetilde{Q}(P[T]).
 $$
  Write $\eta(H(0))=
(K_0, \omega_{K_0})$ and $\eta(H(1))=(K_1, \omega_{K_1})$.
Further suppose $(J, \omega_J)\in {\mathcal LO}(P)$ such that $K_0+J=K_1+J=A$ and $height(J)\geq n$. Then, there is a homotopy 
$\CH(T)\in \widetilde{Q}(P[T])$ such that $\eta(\CH(0))=(K_0J, \omega_{K_0J})$ and $\eta(\CH(1))=(K_1J, \omega_{K_1J})$,
where, for $i=0, 1$ $\omega_{K_iJ}:=\omega_{K_i}\star \omega_J:A^n \sur \frac{K_iJ}{(K_iJ)^2}$.
\eL
\pf  We will write $f=f(T)$, $p=p(T)$ and $Z=Z(T)$. Dnote $Y=1-Z$ and $\eta(\Gamma(H(T))= (\BJ, \omega_{\BJ})$. Then, $\BJ= (f(P[T]), Y)$.
Write 
$$
\SP=\{\wp\in \spec{A[T]}: YT(1-T)\notin \wp,  J\subseteq \wp \}.
$$
 There is a generalized dimension function $\delta: \SP \lra \BN$ such that 
 $\forall~\wp\in \SP$,
 $\delta(\wp)\leq \dim\left(\frac{A[T]}{JA[T]}\right) \leq d+1-height(J) \leq d+1-n\leq n-1$.
%
 Further, $(f, Y^2T(1-T))$ is a basic element in $P[T]^*\oplus A[T]$, on $\SP$. Therefore, there is an element
$\lambda:=\lambda(T) \in P[T]^*$ such that
 %$\varphi_i'=\varphi_i+\lambda_iY^2T(1-T)$ for $i=1, \ldots, n$, we have 
$$
f'=f+Y^2T(1-T)\lambda
%(\varphi_1', \varphi_2', \ldots, \varphi_n')
~~ {\rm is~basic~on}~\SP.\quad 
{\rm So,} \quad \TCP{f'(0)=f(0), ~f'(1)=f(1)}.
$$
%
%
 We have $\BJ=(f(P[T]), Y)= (f'(P[T]), Y)$. Further,
$$
Z(1-Z)=Y(1-Y) =f(p)= f'(p) - Y^2T(1-T)\lambda(p).
%\sum_{i=1}^n \varphi_i'\gamma_i - Y^2T(1-T)\sum_{i=1}^n \lambda_i\gamma_i
$$
%
So,
$$
Y= f'(p) - Y^2T(1-T)\lambda(p) +Y^2
%\sum_{i=1}^n \varphi_i'\gamma_i  - Y^2T(1-T)\mu +Y^2\qquad {\rm where}\quad \mu = \sum_{i=1}^n \lambda_i\gamma_i.
$$
%
 Write $M=\frac{\BJ}{f'(P[T])}$.  Let $p_1, \ldots, p_m$ be a set of generators of $P$.
 Use "overline" to indicate images in 
$M$. We intend to repeat the proof of Nakayama's Lemma and we have 
%
$$
\left(\begin{array}{c}
\overline{f(p_1)}\\ \overline{f(p_2)}\\ \cdots \\ \overline{f(p_m)}\\   \overline{Y}\\
\end{array} \right)
=
\left(\begin{array}{ccccc}
0 & 0 & \cdots & 0 & -\lambda(p_1)YT(1-T)\\
0 & 0 & \cdots & 0 & -\lambda(p_2)YT(1-T)\\
\cdots &\cdots &\cdots &\cdots &\cdots\\
0 & 0 & \cdots & 0 & -\lambda(p_m)YT(1-T)\\
0 & 0 & 0 & 0 & Y-  \lambda(p)YT(1-T) \\
\end{array} \right)
\left(\begin{array}{c}
\overline{f(p_1)}\\ \overline{f(p_2)}\\ \cdots \\ \overline{f(p_m)}\\   \overline{Y}\\
\end{array} \right)
\Lra 
$$
%
$$
\left(\begin{array}{ccccc}
1 & 0 & \cdots & 0 & \lambda(p_1)YT(1-T)\\
0 & 1 & \cdots & 0 & \lambda(p_2)YT(1-T)\\
\cdots &\cdots &\cdots &\cdots &\cdots\\
0 & 0 & \cdots & 1 & \lambda(p_m)YT(1-T)\\
0 & 0 & 0 & 0 & 1-Y+ \lambda(p)YT(1-T) \\
\end{array} \right)
\left(\begin{array}{c}
\overline{f(p_1)}\\ \overline{f(p_2)}\\ \cdots \\ \overline{f(p_m)}\\   \overline{Y}\\
\end{array} \right)
=
\left(\begin{array}{c}
0 \\ 0 \\ \cdots \\ 0\\ 0 \\
\end{array} \right)
$$
Multiplying by the adjoint matrix and computing the determinant, with $Y'= Y- \lambda(p)YT(1-T)$, we have
$$
(1-Y')\BJ\subseteq f'(P[T]).
$$
We have \TCP{$Y'(0)=Y(0)=1-Z(0)$, $Y'(1)=Y(1)=1-Z(1)$.}
Further, 
$$
Y'(1-Y')=f'(p') %\sum_{i=1}^n\varphi'\gamma_i'
 \qquad{\rm for~some~polynomials}\quad p'\in P[T].
$$
%
%
$$
{\rm Therefore}\qquad H'(T)= (f'. p', Y) \in \widetilde{Q}(P[T]).
%(Y', \varphi_1', \ldots, \varphi_n'; \gamma'_1, \ldots, \gamma'_n) \in Q_{2n}(A[T]).
$$
We have
$$
\BJ= (f(P[T], Y)= (f'(P[T]), Y)=(f'(P[T]), Y').
$$
%
In fact, $\eta(H'(T))= (\BJ, \omega_{\BJ})$ and write $\eta(\Gamma(H'(T)))= (\BI, \omega_{\BI})$.   
Claim
%
$$
\BI+JA[T] =A[T].\qquad i.e. \qquad (f'(P[T]), 1-Y')+JA[T] =A[T].
$$
To see this, let 
$$
\BI+JA[T]\subseteq \wp \in \spec{A[T]}
$$
\bE
\item If $Y\in \wp$ then $\BJ=(f'(P[T]), Y) =(f'(P[T]), Y') \subseteq \wp$.
%$(\varphi_1', \ldots, \varphi_n', Y)=(\varphi_1', \ldots, \varphi_n', Y')\subseteq \wp$. 
So, $Y'\in \wp$, which is 
impossible, since $1-Y'\in \wp$. \TCP{So, $\wp \in D(Y)$.}
\item 
Since $f'$
%$(\varphi'_1, \ldots, \varphi'_n)$ 
is unimodular of $\SP$ and $\wp \in D(Y)$ , we must have $T(1-T)\in \wp$.
\item Now, $T\in \wp$ implies, 
$$
\BI(0)+J=(f'(0)(P), 1-Y'(0))+J= (f(0)(P), 1-Y(0))+J= K_0+J=A\subseteq \wp
$$
%
%$$
%\BJ(H'(0))+J= (\varphi_1'(0), \ldots, \varphi_n'(0), 1-Y'(0))+J= (\varphi_1(0), \ldots, \varphi_n(0), 1-Y(0))+J
%$$
%
%$$
%=(\varphi_1(0), \ldots, \varphi_n(0), Z(0))+J=K_0+J=A\subseteq \wp,
%$$
which is impossible. 
\item Likewise, $1-T\in \wp$ implies, 
$$
\BI(1)+J= 
(f'(1)(P), 1-Y'(1))+J= (f(0)(P), 1-Y(1))+J= K_1+J=A\subseteq \wp.
%(\varphi_1'(1), \ldots, \varphi_n'(1), 1-Y'(1))+J
%= (\varphi_1(1), \ldots, \varphi_n(1), 1-Y(1))+J
$$
%
%$$
%=(\varphi_1(1), \ldots, \varphi_n(1), Z(1))+J=K_1+J=A\subseteq \wp
%$$
This is also impossible.
\eE
This establishes the claim.
%
%So, 
%$$
%(\varphi_1', \ldots, \varphi_n')=\BI(H'(T))\cap \BJ(H'(T)) \quad {\rm where}\quad \BJ(H'(T))= (\varphi_1', \ldots, \varphi_n', 1-Y').
%$$
Recall, $\omega_{\BI}: P[T] \sur \frac{\BI}{\BI^2}$ is induced by $f'$. 
%Let 
%$$
%\Omega_{\BJ(H'(T))}:A[T]^n \sur \frac{\BJ(H'(T))}{\BJ(H'(T))^2} \quad {\rm be~induced~by}\quad \varphi_1', \ldots, \varphi'_n.
%$$
Extend  $\omega_J:A^n \sur \frac{J}{J^2}$ to a surjective map  $\omega_{JA[T]}:A[T]^n \sur \frac{JA[T]}{J^2A[T]}$.
Let 
$$
\Omega:=\omega_{\BI}\star\omega_{JA[T]}  : P[T]\sur \frac{J\BI}{J^2\BI^2} \quad {\rm be~induced~by}~\Omega_{\BI}, ~{\rm and}~\omega_{JA[T]}.
$$
Now, there is a homotopy$\CH(T)\in \widetilde{Q}(P[T])$, such that  $\eta(\CH(T))= (\BI JA[T], \Omega)$. Specializing at $T=0$ and $T=1$, we have
$$
\eta(\CH(0))= (K_0J, \omega_{K_0J}), \qquad \eta(\CH(1))= (K_1J, \omega_{K_1J}).
$$
The proof is complete. $\eop$

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Now we proceed to prove, in several propositions, that the pseudo-difference (\ref{psudoDiff}) is well defined and homotopy invariant. 
\bP\label{wellpsDiff}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$, with $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$. 
As in (\ref{psudoDiff}), let  $(K, \omega_K), (I, \omega_I)\in {\mathcal LO}(P)$ be given.
%
Let 
 ${\bf u}= (f, p, s) \in \widetilde{\CQ}(P)$ be such that 
 %
 %(s, f_1, \ldots, f_n; g_1, \ldots, g_n)\in Q_{2n}(A)\in Q_{2n}(A)$ be such that, 
 with 
$\eta({\bf u})=(I, \omega_I)$ and $\eta(\Gamma({\bf u}))=(J, \omega_J)$, $K+J=A$.
Likewise, let 
${\bf v}=(F, q, S) \in \widetilde{\CQ}(P)$  be such that
$\eta({\bf v})=(I, \omega_I)$, and with $\eta(\Gamma({\bf v}))=(L, \omega_L)$, $K+L=A$.
%
Assume  \TCP{$height(K)\geq n$}. 
Then, 
$$
(K, \omega_K) \hat{+} (J, \omega_J)= (K, \omega_K) \hat{+} (L, \omega_L) \in \pi_0\left( \widetilde{\CQ}(P)\right).
$$
{\rm (Recall form (\ref{psudoDiff}), there is no restriction on $height(I), height(J)$ and, nor did we assume $I+K=A$.)}
\eP 
%

\pf 
%
By hypotheses, $K+J=K+L=A$.
%.
%$$
By Corollary \ref{altF_i4involution},   
$$
\widetilde{\Gamma}(\zeta(I, \omega_I))=[\zeta(J, \omega_J)]= [\zeta(L, \omega_L)]. ~~ {\rm Hence}~
\zeta_0(\Gamma({\bf u}))=\zeta_0(\Gamma({\bf v})). 
$$
%
By Corollary \ref{directHomo}, there is a homotopy
$$
H(T)=(\varphi(T), \gamma(T), Z(T)) \in \widetilde{\CQ}(P[T])
%(Z(T); \varphi_1(T), \ldots, \varphi_n(T); \gamma_1(T), \ldots, \gamma_n(T)) \in Q_{2n}(A[T])
$$
such that $H(0)=\Gamma({\bf u})$, $H(1)=\Gamma({\bf v})$.
%
By the Homotopy Lemma \ref{HLK1hoK2} %
there is a homotopy $\CH(T)\in Q_{2n}(A[T])$ such that $\eta(\CH(0))=(KJ, \omega_K\star\omega_J)$ and $\eta(\CH(1))=(KL, \omega_K\star\omega_L)$.
This establishes,
$$
(K, \omega_K) \hat{+} (J, \omega_J)= (K, \omega_K) \hat{+} (L, \omega_L) \in  \pi_0\left( \widetilde{\CQ}(P)\right).
$$
\tcp $\eop$
%
%

\vspace{3mm}
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\bC\label{wellPseDiff}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%Let $n\geq 2$ be an integer, such that $2n\geq d+2$.  
 Let $(K, \omega_K), (I, \omega_I) \in \CL\CO(A, n)$.
Assume  \TCP{$height(K)\geq n$}.
Then, the pseudo-difference {\rm (\ref{psudoDiff})}
$$
(K, \omega_K)\hat{-} (I, \omega_I) \in \pi_0\left( \widetilde{\CQ}(P)\right)\quad {\rm is~ well ~defined}.
$$
\eC
\pf This is immediate from Proposition \ref{wellpsDiff}. $\eop$

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\vspace{3mm}

Now we prove that the pseudo-difference is homotopy invariant, with respect to the the $(I, \omega_I)$-coordiante. 
\bP\label{pseDinvHomo}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%
%
%Let $n\geq 2$ is an integer, with $2n\geq d+2$. 
Let
 $(K, \omega_K), (I_0, \omega_{I_0}), (I_1, \omega_{I_1}) \in {\mathcal LO}(P)$ and  \TCP{$height(K)\geq n$}.
Suppose
 $\zeta(I_0, \omega_{I_0})= \zeta(I_1, \omega_{I_1})\in \pi_0\left( \widetilde{\CQ}(P)\right)$. Then, 
%tje
$$
(K, \omega_K) \hat{-} (I_0, \omega_{I_0})= (K, \omega_K) \hat{-} (I_1, \omega_{I_1}) \in \pi_0\left( \widetilde{\CQ}(P)\right).
$$
In other words, pseudo differences {\rm (defined in \ref{psudoDiff})} are homotopy invariant, with respect to the latter coordinate.
\eP
\pf By Moving Lemma \ref{movingLemm}, we can find ${\bf u}_0, {\bf u}_1\in \widetilde{\CQ}(P)$ such that $\eta({\bf u}_0)= (I_0, \omega_{I_0})$,
 $\eta({\bf u}_1)= (I_1, \omega_{I_1})$, and with, 
$\eta(\Gamma({\bf u}_0))=(J_0, \omega_{J_0})$ and $\eta(\Gamma({\bf u}_1))=(J_1, \omega_{J_1})$, we have  
 %$J_0=\BJ({\bf u}_0)$, $J_1=\BJ({\bf u}_1)$,  
 $K+J_0=K+J_1=A$. 
%For $i=0, 1$, let $\omega_{J_i}=\omega_{\Gamma({\bf u}_i)}: A^n \sur \frac{J_i}{J_i^2}$. 
 By hypothesis,
 $\zeta(I_0, \omega_{I_0})= \zeta(I_1, \omega_{I_1})$ and hence
 $$
\zeta_0(\Gamma({\bf u}_0))= \zeta(J_0, \omega_{J_0})= \widetilde{\Gamma}(\zeta(I_0, \omega_{I_0})) =  \widetilde{\Gamma}(\zeta(I_1, \omega_{I_1})) =\zeta(J_1, \omega_{J_1})=\zeta_0(\Gamma({\bf u}_1)).
 $$
 By  Corollary \ref{directHomo}, there is a homotopy $H(T)\in \widetilde{\CQ}(P)$ such that $H(0)= \Gamma({\bf u}_0)$ and $H(1)=\Gamma({\bf u}_1)$. 
 Since $K+J_0=K+J_1=A$, by Lemma \ref{HLK1hoK2}, there is a homotopy $\CH(T)\in Q_{2n}(A[T])$ such that 
 $\eta(\CH(0))=(KJ_0, \omega_K\star\omega_{J_0})$ and $\eta(\CH(1))=(KJ_0, \omega_K\star\omega_{J_1})$.
Therefore,
$$
(K, \omega_K)\hat{+} (J_0, \omega_{J_0}) = (K, \omega_K)\hat{+} (J_1, \omega_{J_1})
$$
and hence
$$
(K, \omega_K)\hat{-} (I_0, \omega_{I_0}) = (K, \omega_K)\hat{-} (I_1, \omega_{I_1}).
$$
\pic$\eop$




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\vspace{3mm}

We extend the definition of pseudo difference to $\pi_0\left(\widetilde{\CQ}(P)\right)$ as follows.
\bC\label{psDiffKxpi0Q2nA}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$, with $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%
%Let $n\geq 2$ is an integer, with $2n\geq d+2$. 
Let $(K, \omega_K) \in {\mathcal LO}(P)$ and  \TCP{$height(K)\geq n$}. Then, there is a well defined   set theoretic  map 
$$
\pi_0\left(\widetilde{\CQ}(P)\right) \lra \pi_0\left(\widetilde{\CQ}(P)\right)\qquad {\rm sending}\qquad \zeta(I, \omega_I)
\mapsto (K, \omega_K) \hat{-} (I, \omega_I).
$$
\eC
\pf Immediate from Proposition \ref{pseDinvHomo}. $\eop$
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Now, we extend the pseudo-difference to $\pi_0\left(\widetilde{\CQ}(P)\right) \times  \pi_0\left(\widetilde{\CQ}(P)\right)$.

\bT\label{FINpsDiffThm}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$ and  $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%
%Let $n\geq 2$ is an integer, with $2n\geq d+2$. 
Then, there is a set theoretic  map
$$
\Theta: \pi_0\left(\widetilde{\CQ}(P)\right) \times  \pi_0\left(\widetilde{\CQ}(P)\right) \lra  \pi_0\left(\widetilde{\CQ}(P)\right)
$$
such that, for $(K, \omega_K) \in {\mathcal LO}(P)$, with  \TCP{$height(K)\geq n$}, and $(I, \omega_{I}) \in {\mathcal LO}(P)$,
\begin{equation}\label{DiaFINpsD}
\Theta\left(\zeta(K, \omega_K), \zeta(I, \omega_{I})\right)= (K, \omega_K)\hat{-}  (I, \omega_{I}).
\end{equation}
\eT
\pf Suppose $x\in \pi_0(Q_{2n})(A)$. 
By the Moving Lemma \ref{movingRep},
 we can write $x=\zeta(K, \omega_{K})$, with $height(K)\geq n$. Therefore, 
 if well defined,
 the Equation 
\ref{DiaFINpsD} applies to all $(x, y)\in \pi_0\left(\widetilde{\CQ}(P)\right) \times  \pi_0\left(\widetilde{\CQ}(P)\right)$.

Let $x, y\in \pi_0(Q_{2n})(A)$. We can write $x=\zeta(K, \omega_{K})$, with $height(K)\geq n$ and $y=\zeta(I, \omega_I)$.
Define,
$$
\Theta(x, y)= (K, \omega_K)\hat{-}  (I, \omega_{I}).
$$
We need to prove that, 
for $(K, \omega_{K}), (K', \omega_{K'})\in {\mathcal LO}(P)$,
if $x=\zeta(K, \omega_{K})= \zeta(K', \omega_{K'})$, with $height(K)\geq n$ and $height(K')\geq n$,
then 
$$
(K, \omega_{K})\hat{-}  (I, \omega_{I})= (K', \omega_{K'})\hat{-}  (I, \omega_{I}).
$$
Again,  by Moving Lemma \ref{movingLemm}, there is ${\bf u} \in\widetilde{\CQ}(P)$
%(s; f_1, f_2, \ldots, f_n; g_1, \ldots, g_n) \in Q_{2n}(A)$ 
such that 
$\eta({\bf u})=(I, \omega_I)$, and with  $\eta(\Gamma({\bf u}))=(J, \omega_J)$,
we have $J+KK'=A$.
Since $x=\zeta(K, \omega_{K})=\zeta(K', \omega_{K.})$, % 
by Corollary \ref{directHomo},  there is a homotopy $H(T)\in \widetilde{\CQ}(P)$ such that 
$\eta(H(0))=(K, \omega_K)$ and $\eta(H(1))=(K', \omega_{K'})$. By Lemma \ref{HLK1hoK2}, there is a homotopy 
$\CH(T)\in \widetilde{\CQ}(P)$ such that $\eta(\CH(0))=(KJ, \omega_K\star\omega_J)$ and $\eta(\CH(1))=(K'J, \omega_{K'}\star\omega_J)$. 
Therefore,
$$
(K, \omega_K)\hat{+}(J, \omega_J)=\zeta(KJ, \omega_K\star\omega_J)
= \zeta(K'J, \omega_{K'}\star\omega_J)= (K', \omega_{K'})\hat{+}(J, \omega_J).
$$
Therefore, by definition,
$$
(K, \omega_K)\hat{-}(I, \omega_I)= (K', \omega_{K'})\hat{-}(I, \omega_I).
$$
\tcp $\eop$

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\vspace{3mm}

Finally, we are ready to define the binary structure on $\pi_0\left(\widetilde{\CQ}(P)\right)$.
\bD\label{defnBinary}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%
%Let $n\geq 2$ is an integer, with $2n\geq d+2$. 
Then, for $x, y\in \pi_0\left(\widetilde{\CQ}(P)\right)$, the association
$$
(x, y) \mapsto \Theta\left(x, \widetilde{\Gamma}(y)\right)
$$ 
is a well defined binary operation on $\pi_0\left(\widetilde{\CQ}(P)\right)$, where $\Theta$ is as in (\ref{FINpsDiffThm}). 
This operation is well defined because so are $\Theta$ and $\widetilde{\Gamma}$
{\rm  (see Ccorollary \ref{theInvolution})}.
We denote
$$
x+y:=\Theta\left(x, \widetilde{\Gamma}(y)\right)
$$
This binary operation will be referred to an addition.
\eD
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With the help of the Moving Lemma \ref{movingLemm}, the addition operation on $\pi_0\left(\widetilde{\CQ}(P)\right)$ can be described in a more direct manner,
as follows.
\bL\label{describeSum}
Suppose $A$ is a regular ring over afield $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%
%Let $n\geq 2$ is an integer, with $2n\geq d+2$. 
Let $x, y\in \pi_0\left(\widetilde{\CQ}(P)\right)$. By the Moving Lemma \ref{movingLemm}, $x=\zeta(K, \omega_K)$ and
$y=\zeta(I, \omega_I)$, for some  $(K, \omega_K), (I, \omega_I) \in {\mathcal LO}(P)$, such that $I+K=A$ and
 $height(K)\geq n$.
 %
Then, 
$$
x+y= (K, \omega_K)\hat{+} (I, \omega_I) \qquad {\rm as~in~Definition~(\ref{psudoPlus})}.
$$
\eL
\pf 
Let ${\bf u}\in \widetilde{\CQ}(P)$ be such that $\eta({\bf u})=(I, \omega_I)$ and  write $\eta(\Gamma({\bf u}))=(J, \omega_J)$.
Then, $\widetilde{\Gamma}(\zeta(I, \omega_I))= \zeta(J, \omega_J)$.
%
By Definition \ref{defnBinary},
%
% 
$$
x+y=\Theta\left(x, \widetilde{\Gamma}(y)\right)= \Theta\left(x, \zeta(J, \omega_J)\right)= (K, \omega_K)\hat{-} (J, \omega_J)=  (K, \omega_K)\hat{+} (I, \omega_I).
$$
\tcp $\eop$

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The following is a final statement on the binary structure on $\pi_0\left(\widetilde{\CQ}(P)\right)$, which is not necessarily a group.
\bT\label{abelianGroup}
Suppose $A$ is a regular ring over a field $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+2$.
%
%Let $n\geq 2$ is an integer, with $2n\geq d+2$. 
Then, the addition operation on $\pi_0\left(\widetilde{\CQ}(P)\right)$, defined in (\ref{defnBinary})
has the following properties. Let ${\bf e}_0=\zeta_0(0, 0, 0)$ and ${\bf e}_1=\zeta_0(0, 0, 1)$.
%endows astructure of an abelian group on $\pi_0\left(\widetilde{\CQ}(P)\right)$.
\bE
\item The addition in $\pi_0\left(\widetilde{\CQ}(P)\right)$ is commutative and associative.
Further, ${\bf e}_1$ acts as the additive identity in $\pi_0\left(\widetilde{\CQ}(P)\right)$. 
In other words, $\pi_0\left(\widetilde{\CQ}(P)\right)$ has a structure of an abelian monoid.
\item For any $x\in \pi_0\left(\widetilde{\CQ}(P)\right)$, $x+\widetilde{\Gamma}(x)={\bf e}_0$.
\item If ${\bf e}_0={\bf e}_1$, then $\pi_0\left(\widetilde{\CQ}(P)\right)$ is an abelian groups under this addition.
{\rm (In particular, if $P=Q\oplus A$, then $\pi_0\left(\widetilde{\CQ}(P)\right)$ is an abelian groups.)}
\eE
\eT
\pf  
Given $x, y, z\in \pi_0\left(\widetilde{\CQ}(P)\right)$, by applications of the Moving Lemma \ref{movingLemm}, we can write 
$$
x=\zeta(K, \omega_K),~y=\zeta(I, \omega_I)~z=\zeta(J, \omega_J)\quad \ni~K+I=K+J=I+J=A
$$
and $height(K)\geq n$, $height(I)\geq n$, $height(J)\geq n$. By Lemma \ref{describeSum}, we have the following.
$$
(x+y)+z=((K, \omega_K)\hat{+}(I, \omega_I))\hat{+} (J, \omega_J)=x+(y+z).
$$
%
$$
{\rm and} \qquad \qquad 
x+y= (K, \omega_K)\hat{+}(I, \omega_I)= (I, \omega_I)\hat{+}(K, \omega_K)=y+x.
$$
So, the associativity and commutativity hold. 
It is  obvious that, for all $x\in\pi_0\left(\widetilde{\CQ}(P)\right)$, we have  $x+{\bf e}_1=x$. So, ${\bf e}_1$ acts as the additive identity.

Also, with $x=\zeta(K, \omega_K) \in \pi_0\left(\widetilde{\CQ}(P)\right)$,
as above, we have
$x=\zeta_0({\bf u})$ for some ${\bf u}=(f, p, s) \in \widetilde{Q}(P)$. So, $\eta{\bf u}=(K, \omega_{K})$ and we write 
$\eta(\Gamma({\bf u}))=(I_1, \omega_{I_1})$.
We can assume $height(I_1) \geq n$. It follows. 
$$
x+\widetilde{\Gamma}(x)=\zeta_0(f, 0, 0)={\bf e}_0.
$$
Therefore, if ${\bf e}_0={\bf e}_1$, $x$ has an additive inverse.
 \tcp $\eop$

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\vspace{3mm}
The following is a list of trivial comments on homotopy equivalences.
\begin{remark}\label{listHomoNov21}{\rm 
Let $A$ be a commutative ring with $\dim A=d$ and $P$ be a projective $A$-module with
 $rank(P)=n$. 
There are two distinguished elements in $\pi_0\left(\widetilde{\CQ}(P)\right)$, which we denote
${\bf e}_0=\zeta_0(0, 0, 0)$ and ${\bf e}_1=\zeta_0(0, 0, 1)$.
\bE

\item\label{inParNov21mid} Let $(f, p) \in P^*\oplus P$. Then, $(f, p, 1), (f, p, 0)\in \widetilde{Q}(P)$. Further,
the homotopy $H(T)= ((1-T)f, (1-T)p, 1)\in  \widetilde{Q}(P[T])$ implies 
$\zeta_0(f, p, 1) ={\bf e}_1$.
 Likewise, $H(T)= ((1-T)f, (1-T)p, 0)$ 
implies $\zeta_0(f, p, 0)= {\bf e}_0$.
%\item  In particular,  $\zeta_0(f, 0, 1) = \zeta_0(0, 0, 1)$ and $\zeta_0(f, 0, 0) =\zeta_0(0, 0, 0)$

\item Assume $P=Q\oplus A$. Write $p_0=(0, 1) \in P$, $f_0: P \sur A$ be the projection.
%
The homotopy $H(T)=(f_0, T(1-T)p_0, T)$, and   (\ref{inParNov21mid}) implies 
$$
{\bf e}_0 = \zeta_0(f_0, 0, 0) = \zeta_0(f_0, 0, 1)={\bf e}_1.
$$
A converse to this statement follows.
 %for  $(f, p) \in P^*\oplus P$, we have $\zeta_0(f, p, 0)={\bf e}_0 ={\bf e}_1= \zeta_0(f, p, 1).$  

\eE
}
\end{remark}
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\bP\label{e0e1Split}
Suppose $A$ is a essentially smooth ring over an infinite perfect  field $k$, with $1/2\in k$ and $\dim A=d$. 
Let $P$ be a projective $A$-module with $rank(P)=n$. Assume $2n\geq d+3$. Then, ${\bf e}_0 ={\bf e}_1$ if and only if 
$P\cong Q\oplus A$ for some projective $A$-module $Q$.
\eP
\pf If $P\cong Q\oplus A$, by (\ref{listHomoNov21}), ${\bf e}_0 ={\bf e}_1$. Conversely, suppose ${\bf e}_0 ={\bf e}_1$. Fix 
$f_0\in P^*$ such that $height(P)=n$. Then, $\zeta_0(f_0, 0, 0)={\bf e}_0 ={\bf e}_1$. Then, it follow from Theorem 
\ref{formalizeBSNori} that $\eta(0, 0, 1)$ lifts to a surjective map $P\sur A$. \tcp $\eop$

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%\TCB{Double check the following:}

\begin{remark} \label{wittpiZeroDef}{\rm 
%
Use the notation as in (\ref{abelianGroup}). When ${\bf e}_0\neq {\bf e_1}$, the results in (\ref{abelianGroup}) describe a situation 
similar to  what leads to the definition of Grothendieck Witt groups and then Witt groups, starting from the Monoid of symmetric spaces.
 For our  discussions, the group 
analogous to the Witt groups would be of some interest to us, where ${\bf e}_0$ is forcibly identified with the additive identity.


For  $x, y \in \pi_0\left(\widetilde{\CQ}(P)\right)$ define $x\sim y$ if $x+n{\bf e}_0= y+m{\bf e}_0$, for integers $m, n\geq 0$.
This is easily checked to be an equivalence relation. 
Let $\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ be the set of all equivalence classes and $\ell: \pi_0\left(\widetilde{\CQ}(P)\right)
\sur \BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ be the natural surjective map. It also follows that 
$\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ has a structure of an abelian group, induced by that additive structure 
on $\pi_0\left(\widetilde{\CQ}(P)\right)$.
The identity element of $\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ is  $\ell({\bf e}_0)$
and for $x\in \pi_0\left(\widetilde{\CQ}(P)\right)$, the additive inverse of $\ell(x)$ is $\ell(\widetilde{\Gamma}(x))$.

Clearly, if ${\bf e}_0= {\bf e_1}$, then $\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)=\pi_0\left(\widetilde{\CQ}(P)\right)$.

} 

\end{remark}

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%\vspace{10mm}



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 \section{The Euler Class Groups}\label{secular}

Suppose $A$ is a noetherian commutative ring with $\dim A=d$ and $P$ is a projective $A$-module, with $rank(P)=n$.
In this section, in analogy to the definition of the Euler class groups $E^n(A)$ in \cite{BS2, MY}, we define a group 
$E(P)$, which would also be called the Euler class group of $P$. Subsequently, we compare $E(P)$ with $\pi_0\left(\widetilde{\CQ}(P)\right)$.
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% \subsection{Preliminaries on Euler Class Groups} \label{eulerDefSec}
%
 Also, refer to some superfluous aspect of the definitions 
in \cite{BS2, MY}, pointed out in \cite{MM}. 
%
({\it In the sequel, for a set $S$, the free abelian group generated by $S$ will be denoted by} $\BZ(S)$).
\bD\label{newEuler}
{\rm
Suppose $A$ is a noetherian commutative ring, with $\dim A=d$ and $P$ is a projective $A$-module, with $rank(P)=n\geq 0$.
Denote,
$$
\left\{
\begin{array}{ll}
{\mathcal LO}^n(P)=\{(I, \omega_I)\in {\mathcal LO}(P):  height(I)\geq n \},\\
{\mathcal LO}_c^n(P)=\{(I, \omega_I)\in {\mathcal LO}(P):  V(I)~{\rm is~connected~and}~height(I)\geq n \}.\\
\end{array}
\right.
$$
%
Let $(I, \omega_I)\in {\mathcal LO}^n(P)$ and $I=\cap_{i=1}^mI_i$ be a decomposition, where $V(I_i)\subseteq 
\spec{A}$ are connected. 
The local orientation $(I, \omega_I) \in {\mathcal LO}^n(P)$ induce  $(I_i, \omega_{I_i})\in {\mathcal LO}^n_c(P)$, for $i=1, \ldots, m$.
Denote 
 $\varepsilon(I, \omega_I)=\sum_{i=1}^m(I_i, \omega_{I_i})\in \BZ\left({\mathcal LO}^n_c(P)\right)$.
%
%
A local orientation $(I, \omega_I)\in {\mathcal LO}(P)$ would be called global, if $\omega_I$ lifts to a surjective map $P\sur I$.
 Let $\SR(P)$ denote the subgroup of $\BZ\left({\mathcal LO}^c(P)\right)$, generated by the set 
$\left\{\varepsilon(I, \omega_I):   (I, \omega_I)\in {\mathcal LO}^n(P)~{\rm and,~is~global} \right\}$.
({\it Note $\eta(0, 0, 1)$ is global, only if $P=Q\oplus A$}.)
%
Define
$$
E(P)= \frac{\BZ\left({\mathcal LO}^n_c(P)\right)}{\SR(P)}
\quad {\rm to~be~called~the~Euler~class~group~of}~~P.
$$
Images of $\varepsilon(I, \omega_I)$ in $E(P)$ with be denoted by $\overline{\varepsilon}(I, \omega_I)$.

}
\eD

 
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We define 
 a  homomorphism  $\rho: E(P) \lra \BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$, as follows.

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\bD\label{EntopizeroNov14}{\rm 
Suppose $A$ is a regular  ring over a field $k$, with $1/2\in k$ and  $\dim A=d$, and $P$ is a projective $A$-module with $rank(P)=n$
and  $2n\geq d+2$.   ({\it Use the notations in }(\ref{wittpiZeroDef})).
%Let $\ell: \pi_0\left(\widetilde{\CQ}(P)\right) \lra \BG\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ denote the natural map.
The  restrictions of the map  $\zeta$ give the following commutative diagram:
$$
\diagram
{\mathcal LO}^n_c(P)\ar[dr]^{\zeta} \ar@{^(->}[d]& &\\
{\mathcal LO}^n(P)\ar@{->>}[r]_{\zeta} & \pi_0\left(\widetilde{\CQ}(P)\right)\ar@{->>}[r]_{\ell\qquad} &\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right) \\
\enddiagram
$$
  Since $\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ 
 has the structure of an abelian group, the composition map $\ell\zeta: {\mathcal LO}^n_c(P) \lra \BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$
  extends to a group homomorphism $\rho_0: \BZ\left({\mathcal LO}^c(A, n)\right)\lra\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$. 
  
  Now suppose $(I, \omega_I) \in {\mathcal LO}^n(P)$ be global. Then, 
  \TCP{$\varepsilon(I, \omega)={\bf e}_0$}.
  Since, $\ell({\bf e}_0)$ is the additive identity,  $\rho_0$ factors through a 
  group homomorphism $\rho: E(P)\sur \BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$. In fact, 
  $\rho$ is surjective. 
  ({\it We remark that a for a global orientation $(I, \omega)$,  $\varepsilon(I, \omega)={\bf e}_0$ would not necessarily map to zero in the Grothendieck group
  of the monoid $\pi_0\left(\widetilde{\CQ}(P)\right)$}.)
}
\eD 
\pf It is self explanatory  that  $\rho: E(P)\sur \BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$ is a well defined group homomorphism.
For $x\in \pi_0\left(\widetilde{\CQ}(P)\right)$, by Moving Lemma \ref{movingLemm}, $x=\zeta(I, \omega_I)$ for some 
$(I, \omega_I)\in {\mathcal LO}^n(P)$. Let $I =\cap_{i=1}^mI_i$ be a decomposition, with $V(I_i)$ connected and $\omega_i:
P\sur \frac{I_i}{I_i^2}$ be the surjective map induced by $\omega_I$. Then,
$$
\rho_0(\varepsilon(I, \omega_I))=\sum_{i=1}^m \ell(\zeta(I_i, \omega_i))=\ell\left(\sum_{i=1}^m\zeta(I_i, \omega_i)\right)= \ell(\zeta(I, \omega_I))=
\ell(x).
$$
So, $\rho_0$ is surjective and hence so in $\rho$.
\tcp $\eop$


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 We comment on the question of infectivity of $\rho$, in two cases, when $P\cong Q\oplus A$ and  when $P$ does not split.
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\bT\label{useBKNov14}
Suppose $k$ is an infinite perfect field, with $1/2\in k$ and $A$ is an essentially smooth ring over $k$, with $\dim A=d$. 
Suppose $P$ is a projective $A$-module with $rank(P)=n$
and  $2n\geq d+3$. Assume $P\cong Q\oplus A$.
%Let $n\geq 2$ be an integer, with $2n\geq d+3$. 
Then, $\pi_0\left(\widetilde{\CQ}(P)\right) =\BE\left(\pi_0\left(\widetilde{\CQ}(P)\right)\right)$  and the homomorphism 
$\rho: E(P) \lra \pi_0\left(\widetilde{\CQ}(P)\right)$ is an 
isomorphism.
%Then, the homomorphism $\rho$  in (\ref{EntopizeroNov14}) is an isomorphism. 
\eT
\pf We only need to prove that $\rho$ is injective. Let $\rho(x)=0$ for some $x\in E^n(A)$. We can write $x=\overline{\varepsilon}(I, \omega_I)$,
for some $(I, \omega_I) \in {\mathcal LO}^n(P)$, with $height(I)\geq n$. 
We can assume $height(I)=n$. We have $\zeta(I, \omega_I)= \zeta_0(0, 0, 1)$. 
It follows from Theorem \ref{formalizeBSNori} that $(I, \omega_I)$ and hence $x=\overline{\varepsilon}(I, \omega_I)=0$.
So, $\rho$ is an isomorphism.
 \tcp $\eop$
  
\bC\label{noCostEulerNov25}
Suppose $k$ is an infinite perfect field, with $1/2\in k$ and $A$ is an essentially smooth ring over $k$, with $\dim A=d$. 
$P$ is a projective $A$-module with $rank(P)=n$
and  $2n\geq d+3$. Assume $P\cong Q\oplus A$. Suppose $(I, \omega_I)\in {\mathcal LO}^n(P)$ and 
$\overline{\varepsilon}(I, \omega_I)=0\in E(P)$. Then, $\omega_I$ lifts to a surjective homomorphism $P\sur I$.
\eC
 \pf It is immediate from Theorem \ref{useBKNov14}. $\eop$

\vspace{3mm}

In fact, a  version of (\ref{noCostEulerNov25}), without the regularity condition,   would follow by   the same argruments in 
\cite{BS2}.
%
Before give this version, we comment on the question of injectivity of $\rho$, when $P$ does not split, as above.
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\bP\label{nonSplitE(P)Nov25}
Suppose $A$ is a noetherian commutative ring with $\dim A=n$ and $P$ is a projective $A$-module, with $rank(P)=n$.
Assume that $P$ does not have a unimodular element. Then, there is an exact sequence of abelian group
$$
\diagram
0 \ar[r] & \widetilde{E}(P) \ar[r] & E(P) \ar[r]^{\iota} & \BZ \ar[r] & 0
\enddiagram
\quad {\rm Hence}\quad E(P) \cong \widetilde{E}(P) \oplus \BZ.
$$
where $\widetilde{E}(P)$ is the subgroup of $E(P)$ generated by $\overline{\varepsilon}(I, \omega_I)$, with $(I, \omega_I)\neq \eta(0, 0, 1)=(A, *)$,
{\rm (i. e. $height(I)=n$)}.

Consequently, even when $A$ is regular over a field $k$, with $1/2\in k$ and $2n\geq d+2$, the map 
$\rho: E(P) \sur \BE\left(\pi_0\left(\widetilde{Q}(P)\right)\right)$ is not injective.
\eP
\pf Let  $\iota_0:\BZ\left({\mathcal LO}^n_n(P)\right)\lra \BZ$ be the homomorphism defined by the coefficient of $\eta(0, 0, 1)$.
It is clear that, if $(I, \omega_I)\in {\mathcal LO}(P)$ is global then $\iota(\varepsilon(I, \omega_I)=0$. So, $\iota_0$ induce a 
surjective map $\iota: E(P) \sur \BZ$. It is clear that $\ker(\iota)= \widetilde{E}(P)$.
%
The latter statement about injectivity of $\rho$, follows because $\rho\left(\overline{\varepsilon}(A, *)\right)=0$.
\tcp $\eop$

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 \subsection{The Vanishing of Euler cycles}
 We use the notations as in (\ref{newEuler}). An element $x\in E(P)$ of may be referred to as an Euler cycle. In this subsection,
 we prove a less restrictive version of Corollary \ref{noCostEulerNov25}, using the same method in \cite{BS2}.
 We will follow the arguments in the proof of \cite[Theorem 4.2]{BS2}, 
 which mainly depends on the availability of Subtraction and Addition Principles.
   Accordingly, the following is a version of \cite[Proposition 3.3]{BS2}.

 

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 \bP\label{BS2Prop3p3Nov25}
 Suppose $A$ is a noetherian commutative ring, with $\dim A=d$ 
 and $P$ is a projective $A$-module, with $rank(P)=n$.
 Assume $2n\geq d+3$ and $P\cong Q\oplus A$. 
%
\TCP{If $A=R[X]$ is a polynomial ring over a regular ring $R$, over an infinite field $k$, assume $2n\geq \dim R[T]+2$.} 
%
Let $J_0, J_1, J_2, J_3\subseteq A$ be  ideals, with $height(J_i)\geq n$ for $i=0, 1, 2, 3$, 
$J_0+J_1J_2=A$ and 
$J_0J_1J_2+J_3=A$.

Also, let
$$
\alpha: P \sur J_0\cap J_1, \quad \beta: P\sur J_0\cap J_2\quad {\rm be ~surjective~maps}~\ni \alpha\otimes \frac{A}{J_0} = \beta\otimes \frac{A}{J_0}.
$$
Further, assume that 
 there is a surjective map 
$$
\gamma: P \sur J_1\cap J_3 \quad \ni \quad \gamma\otimes \frac{A}{J_1}=\alpha\otimes \frac{A}{J_1}.
$$
Then, there is a surjective map
$$
\delta: P \sur J_2\cap J_3 \quad \ni \quad \delta \otimes \frac{A}{J_3}= \gamma \otimes \frac{A}{J_3}, \quad \delta \otimes \frac{A}{J_2}= \beta \otimes \frac{A}{J_2}.
$$
 \eP
 \pf Denote $\omega_0=\alpha\otimes \frac{A}{J_0}=\beta\otimes \frac{A}{J_0}$, $\omega_1=\alpha\otimes \frac{A}{J_1}=\gamma\otimes \frac{A}{J_1}$
 $\omega_2=\beta\otimes \frac{A}{J_2}$, $\omega_3=\gamma\otimes \frac{A}{J_3}$.
  %
% With standard basis $e_1, \ldots, e_n$ of $A^n$, write $\alpha(e_i)=a_i$, for $i=1, \ldots, n$.
 % Let $s\in J_0$ such that $1-s\in J_1$. 
 By Moving Lemma \ref{movingLemm} there is ${\bf u}=(f, p, s)\in \widetilde{Q}(P)$, such that 
 $\eta({\bf u})= (J_0, \omega_0)$ and 
 with $\eta(\Gamma({\bf u}))=(J_4, \omega_4)$, we have 
 %$J_4= \BJ({\bf u})$, $\omega_4:=\omega_{\Gamma({\bf u})}: A^n \sur \frac{J_4}{J_4^2}$, we have 
 $J_1J_2J_3+J_4=A$ and $height(J_4)\geq n$. Clearly, 
 $f(P)=J_0\cap J_4$, with $J_0+J_4=A$. 
 
 Let \TCP{$g:=f:P\sur J_0J_4$} be the 
 a surjective map defined by $f$. ({\it It makes a difference, where we look at $f\in P^*$ or $f\in Hom(P, J_0J_4)$.}) Then, $g\otimes \frac{A}{J_0}=\omega_0$.
 %
 By Addition Principle \cite[Theorems 5.6, 5.7]{BK}, to $\gamma$ and $g$, there is a surjective map
 $$
 \mu: P \sur (J_1\cap J_3)\cap (J_0\cap J_4) ~~ \ni \quad \mu \otimes \frac{A}{J_1\cap J_3}=\gamma\otimes \frac{A}{J_1\cap J_3}= \omega_1\star\omega_3,
 $$
 %
 $$
 {\rm and} \qquad \mu \otimes \frac{A}{J_0\cap J_4}=g \otimes \frac{A}{J_0\cap J_4}=  \omega_0\star\omega_4.
 $$
% It follows,
 It follows, $\mu \otimes \frac{A}{J_0\cap J_1}=\omega_0\star \omega_1= \alpha\otimes \frac{A}{J_0\cap J_1}$. 
 By Subtraction Principle  \cite[Theorems 3.7, 4.11]{BK}, applied to $\mu$ and $\alpha$, there is a surjective map 
 \TCP{$\nu: P\sur J_3\cap J_4$}
 such that $\nu\otimes \frac{A}{J_3\cap J_4} = \mu \otimes \frac{A}{J_3\cap J_4}=\omega_3\star \omega_4$.
 %
 By Addition Principle \cite[Theorems 5.6, 5.7]{BK}, applied to $\nu$ and $\beta$, there is a surjective map 
 $$
 \lambda:P \sur (J_0\cap J_2) \cap (J_3\cap J_4)
 \quad \ni \quad \lambda\otimes \frac{A}{J_0\cap J_2}= \beta \otimes \frac{A}{J_0\cap J_2}=\omega_0\star\omega_2
 $$
 %
 $$
 {\rm and,}\quad ~~\lambda\otimes \frac{A}{J_3\cap J_4}= \nu \otimes \frac{A}{J_3\cap J_4}= \omega_3\star \omega_4.
$$
Now apply Subtraction Principle \cite[Theorems 3.7, 4.11]{BK}, to $\lambda$ and $g$. There is a surjective map
$$
\delta: P\sur J_2\cap J_3 \quad \ni \quad \delta\otimes \frac{A}{J_2\cap J_3} = \lambda \otimes \frac{A}{J_2\cap J_3}=\omega_2\star \omega_3.
$$
So, $\delta\otimes \frac{A}{J_2} =\omega_2$ and  $\delta\otimes \frac{A}{J_3} =\omega_3$.
\pic $\eop$


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 The following is the version of Corollary \ref{newEuler}.
 \bT\label{oldGoldMethod}
 Suppose $A$ is a commutative noetherian ring with $\dim A=d$ and $P$ is a projective $A$-module, with $rank(P)=n$.
 Assume $2n\geq d+3$ and $P\cong Q\oplus A$. 
 \TCP{If $A=R[X]$ is a polynomial ring over a regular ring $R$, over an infinite field $k$, assume $2n\geq \dim R[T]+2$.} 
 Let $(J, \omega_J) \in {\mathcal LO}^n(P)$ and $\overline{\varepsilon}\left(J, \omega_J\right)=0\in E(P)$.
 Then, $\omega_J$ lifts to a surjective map $P\sur J$.
 \eT
 
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\pf Suppose 
$(J, \omega_J) \in {\mathcal LO}^n(P)$ and $\overline{\varepsilon}\left(J, \omega_J\right)=0\in E(P)$.
We have a set 
$$
\left\{(J_t, \omega_t) : 1 \leq t \leq r+s \right\} 
$$
such that 
\begin{enumerate} 
\item $height (J_t)=n.$ 
\item there are surjective maps 
$\alpha_t:  P \sur J_t$ such that $\alpha_t$ lifts $\omega_t.$ 

\item And 
\begin{equation}\label{eulerEqnNov25}
\varepsilon(J, \omega)+\sum_{l=r+1}^{r+s} \varepsilon(J_t, \omega_t) = \sum_{t=1}^r  \varepsilon(J_t, \omega_t) 
\quad {\rm in}\quad \BZ\left({\mathcal LO}^n_c(P)\right).
\end{equation}
holds in the free group $ \BZ\left({\mathcal LO}^n_c(P)\right)$.
\end{enumerate}
%
First assume that $J_1, J_2, \ldots, J_r$ are pairwise comaximal.
In this case, $J, J_{r+1}, \ldots, J_{r+s}$ are pairwise comaximal. 
Write 
$$
J'=\cap_{l=r+1}^{r+s} J_t, \qquad J"=\cap_{t=1}^r J_t.  
\qquad {\rm Then}\quad J\cap J' = J".
$$  
Further, by Addition Principle \cite[Theorems 5.6, 5.7]{BK}, there are surjective homomorphisms 
$\alpha': P\sur J'$  and $\alpha": P \sur J"$ such that
%\begin{enumerate}
$$
\varepsilon(J', \omega') = \sum_{t=r+1}^{r+s}\varepsilon(J_t, \omega_t),  \quad 
\varepsilon(J", \omega") = \sum_{t=r+1}^{r+s}\varepsilon(J_t, \omega_t) 
\quad {\rm in}\quad \BZ\left({\mathcal LO}^n_c(P)\right)
%where \quad \omega' =\alpha'\otimes A/J'. 
$$
where $\omega' =\alpha'\otimes A/J'$ and $\omega" =\alpha"\otimes A/J"$.
%
 So, Subtraction Principle \cite[Theorems 3.7, 4.11]{BK},
there is a surjective homomorphism $\alpha: P\sur J$ such that 
$\alpha \otimes A/J =\alpha"\otimes A/J= \omega.$  


Now, we consider that $J_1, J_2, \ldots, J_r$ are not, necessarily, pairwise 
comaximal. Given an Equation, as in (\ref{eulerEqnNov25}), we would associate an integer $n({\rm Eqn-}\ref{eulerEqnNov25})\geq 0$, as follows.
Let $S_i$ be the set of all connected components of $J_i$ and 
$S=\cup_{i=1}^rS_i.$ 
For $K \in S,$ let $n(K)+1$ be the cardinality
of the set $\{t: K+J_t \neq A \}.$ Let $n({\rm Eqn-}\ref{eulerEqnNov25})= \sum_{K\in S} n(K).$
We have $n({\rm Eqn-}\ref{eulerEqnNov25})=0$ if and only if $J_1, J_2, \ldots, J_r$ are comaximal. 
 
Now,  assume $n({\rm Eqn-}\ref{eulerEqnNov25})\geq 1$. Therefor, $n(K)\geq 1$ for some $K \in S.$ 
We can assume $K \in S_1$  and $K+J_2 \neq A.$ 
So, $\exists ~ \tilde{K}$ a connected component of $J_2$ such that 
$K+ \tilde{K} \neq A.$    

First, assume $K \neq \tilde{K}.$ Both $K,\tilde{K}$ cannot be connected component of $J.$ 
\TCP{({\it components add up to $A.$})}
Without loss of generality, assume $K$ is not a
connected component of $J.$ 
Using Eqn-\ref{eulerEqnNov25}, it follows that there is an integer $l$, with $r+1 \leq l \leq r+s$, 
%and a surjective homomorphism $\omega_K: P \sur K/K^2$
 such that (1) $K$ is a connected component of $J_l$, (2) $\alpha_l \otimes A/K = \alpha_1 \otimes A/K.$  
Assume $l=r+1$ and denote $\omega_K:= \alpha_l \otimes A/K = \alpha_1 \otimes A/K: P \sur K/K^2$.
%
%
%We can assume $l=r+1$. 
We  write $J_1=K \cap K_1$ and $J_{r+1}= K \cap K_2$
where $K+K_1=A=K+K_2.$ By Moving Lemma \ref{movingLemm}, applied to $\omega_{K_1}:=\alpha_1\otimes \frac{A}{K_1}$,
%Eisenbud Evans (\cite[Cor 2.4]{BRS3}), 
%we can find 
there is an ideal $K_3$ such that (3) $height(K_3)\geq n$, (4) $K_3$ is comaximal to $J, J_j, \forall 1 \leq j \leq r+s$,
(5) there is a surjective map  $\beta: P \sur K_3 \cap K_1$  such that $\alpha_1 \otimes A/K_1= \beta \otimes A/K_1.$
\\
The have three surjective maps:
$$
\alpha_1: P \sur K \cap K_1, \quad \alpha_{r+1}: P \sur K \cap K_2 \quad \beta: P \sur K_1\cap K_3
$$
% 
%
By proposition \ref{BS2Prop3p3Nov25}, there is a surjective map 
$$
\beta_{r+1}: P \sur K_3 \cap K_2\quad \ni ~~
\alpha_{r+1}\otimes \frac{A}{K_2}=\beta_{r+1}\otimes \frac{A}{K_2}, %A/K_2
 ~~\beta\otimes \frac{A}{K_3} %A/K_3
=\beta_{r+1}\otimes\frac{A}{K_3}. % A/K_3.
$$ 
%such that 
%$$
%$$ 
So, we have 
\begin{equation}\label{EupfEquanTwo}
\varepsilon(J, \omega)+\varepsilon\left(\widetilde{J_{r+1}}, \widetilde{\beta_{r+1}} \right)
+\sum_{l=r+2}^{r+s}\varepsilon(J_l, \omega_l) 
 =
\varepsilon\left(\widetilde{J_{1}}, \widetilde{\beta_{1}} \right) + 
\sum_{l=2}^{r}\varepsilon(J_l, \omega_l).  
\end{equation}
where $\widetilde{J_{r+1}}= K_3 \cap K_2$ and $\widetilde{J_{1}}= K_3 \cap K_1.$   
({\it $K$ is removed from both sides and $K_3$ is inserted.})
It is clear $n({\rm Eqn-}\ref{EupfEquanTwo})<n({\rm Eqn-}\ref{eulerEqnNov25}).$
Therefore, by induction, the  Equation-\ref{EupfEquanTwo}) would reduce to an Equation (*), so that 
$n(*)=0$.

Now assume $K= \tilde{K}.$ Let $\omega_K= \alpha_1 \otimes A/K.$ 
Therefore, $K=\tilde{K}$ is a component of $J_2$. We denote $\tilde{\omega}_K
=\alpha_2 \otimes A/K.$   Using Equation-\ref{eulerEqnNov25}, it follows that either  $(K, \omega_K)$ or $(K, \tilde{\omega}_K)$ 
is a summand of $\sum_{t={r+1}}^{r+s}\varepsilon(J_t, \omega_t)$. Without loss of generality, we assume that 
$(K, \omega_K)$ is a summand of $\varepsilon(J_{r+1}, \omega_{r+1})$ and complete the induction exactly in the same manner, as above.
\tcp
$\eop$




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\appendix
\section{The Motivic Approach}\label{motivicSec}
We assume $A$ is a commutative ring containing a field $k$, with $1/2\in k$.
In some literature (e. g. \cite{F, AF}), perhaps known as Motivic Approach, 
driven by the desire to   view the data in ${\mathcal LO}(A^n)={\mathcal LO}(A, n)$  in a functorial manner, one writes 
$$
\left\{
\begin{array}{l}
Q_{2n}(A)=\{(s; f_1, \ldots, f_n; g_1, \ldots, g_n)\in A^{2n+1}:
\sum_{i=1}^nf_ig_i+s(s-1)=0 \}\\
Q_{2n}'(A)=\{(s; f_1, \ldots, f_n; g_1, \ldots, g_n)\in A^{2n+1}:
\sum_{i=1}^nf_ig_i+s^2=1 \}\\
\end{array}
\right.
$$ 
and the homotopy  sets $\pi_0\left(Q_{2n}\right)(A)$, $\pi_0\left(Q_{2n}'\right)(A)$ were defined. 
It was pointed out thatt there is a bijection $Q_{2n}(A) \iso Q_{2n}'(A)$, which induces a bijection 
$\pi_0\left(Q_{2n}\right)(A)\iso \pi_0\left(Q_{2n}'\right)(A)$. So, we would comment only on $Q_{2n}'(A)$. 
 Recall, the pref-sheaf (functoriality) structure of  $A \mapsto Q_{2n}'(A)$ was obtained from 
  bijection $Q_{2n}'(A)\cong Hom\left(\spec{A}, \spec{\SB_{2n+1}}\right)$, where 
 $\SB_{2n+1}=\frac{k[X_1, \ldots, X_n; Y_1, \ldots, Y_n, Z]}{\left(\sum_{i=1}^nX_iY_i+Z^2-1\right)}$. 
 
% It was established in \cite{MM} that $\pi_0\left({\mathcal LO}(A, n)\right)\cong \pi_0\left(Q_{2n}\right)(A)$. 
By analogy, for our purpose, 
 for a projective $A$-module $P$, we considered 
 $$
 \left\{
 \begin{array}{l}
 \widetilde{Q}(P)=\left\{(f, p, s)\in P^*\oplus P\oplus A: f(p)+s(s-1)=0 \right\}, \\
  \widetilde{Q}'(P)=\left\{(f, p, s)\in P^*\oplus P\oplus A: f(p)+s^2=1 \right\}.\\
  \end{array}
  \right.
 $$ 
 and define 
 $\pi_0\left(\widetilde{Q}(P)\right)$, $\pi_0\left(\widetilde{Q}'(P)\right)$. As usual,  a bijection 
 $\widetilde{Q}(P)\iso   \widetilde{Q}'(P)$, is obtained by completing the square $s(s-1)=\left(s-\frac{1}{2}\right)^2-\frac{1}{4}$.
 %which induces a bijection  $\pi_0\left(\widetilde{Q}(P)\right)\iso\pi_0\left(\widetilde{Q}'(P)\right)$
It was established (see Theorem \ref{2pi0SamNov26}) that there is a bijection $\pi_0\left({\mathcal LO}(P)\right)\iso\pi_0\left(\widetilde{Q}(P)\right)
\iso \pi_0\left(\widetilde{Q}'(P)\right)$.
%
We clarify the pre-sheaf structure (functoriality) on $\widetilde{Q}'(P)$ as follows. 

Suppose $Q$ is a projective $A$-module and
$S(Q^*)=\bigoplus_{i\geq 0} S_i(Q^*)$ denote the symmetric algebra 
of $Q^*$. 
Let $quad(Q)=\{\varphi\in Hom(Q, Q^*): \varphi^*=\varphi \}$ denote the $A$-module of all the quadratic forms on $Q$.
Given $\varphi\in quad(Q)$, let $B(\varphi): \in Hom(Q \otimes Q, A) \cong Q^*\times Q^*$ be the corresponding bilinear map.
In fact, this association $\varphi \mapsto B(\varphi)$ induces a 
bijection $quad(Q)\iso S_2(Q^*)$ (see  \cite[\S ~2]{Sw}).
Consider the commutative diagram of bijections
 $$
 \diagram
 Q \ar[r]^{ev\qquad}_{\sim}\ar[dr]_{\lambda}^{\sim} & Hom(Q^*, A)\ar[d]^{\wr} \\
 &  Hom(S(Q^*), A) \\
 \enddiagram
 $$
  Fix $x\in Q$. For $f, g\in Q^*$, $\lambda(x)(f)=f(x)$ and $\lambda(x)(fg)=f(x)g(x)$. 
  Let "overline" denote the images of elements in $Q^*\otimes Q^*\cong Hom(Q\otimes Q, A)$ in $S_2(Q^*)$ 
%
  Given bilinear map $\beta: Q\otimes Q \lra A$,  $\beta=\sum f_i\otimes g_i$ for some $f_i, g_i\in Q^*$.
  %Denote the  image of $\beta$ in $S_2(Q^*)$ by   $\overline{\beta}$.
   So, $\lambda(x)(\overline{\beta})=\sum f_i(x)g_i(x)=\beta(x, x)$.
  
  
  Fix quadratic form $\varphi: Q\lra Q^*$ and $B(\varphi): Q\otimes Q\lra A$ 
  be the corresponding bilinear map. More precisely, $B(\varphi)(x, y)=\varphi(x)(y)$. As usual, define $q: Q \lra A$ by $q(x)=B(x, x)$.
   Then,
   $$
   {\rm for}~x\in Q\quad \lambda(x)(\overline{B(\varphi)})=B(\varphi)(x, x)=q(x).
   $$
   For our purpose, we summarize the above, as follows.
   %%%%%%%%%%%%%%%
   \bP\label{swanSphere}
   Suppose $A$ is a commutative noetherian ring, containing a field $k$, with $1/2\in k$.
   Now, let $(Q, \varphi)$ be  a quadratic space, over $A$.  Define 
   $$
   \BS(Q, \varphi)=\{x\in Q: q(x)=1 \}, \qquad \qquad
  \SB(Q, \varphi)=\frac{S(Q^*)}{(\overline{B(\varphi)}-1)}
  %\quad {\rm and} \quad \CX(Q, \varphi)=\spec{\SB(Q, \varphi)}.
   $$
   Then, there are bijections, as follows
   $$
    \BS(Q, \varphi) \iso Hom\left(\SB(Q, \varphi), A\right)=Hom\left(\spec{A}, \spec{\SB(Q, \varphi)}\right).
   $$
   \eP
   %%%%%%%%%%%
   \pf Follows from above discussions. $\eop$
   
   \vspace{3mm}
   \begin{remark}{\rm 
   Suppose $A_0$ is a commutative noetherian ring, containing a field $k$, with $1/2\in k$
   and $(Q_0, \varphi_0)$ is a quadratic space over $A_0$. There is pre-sheaf 
   $$
   \underline{Sch}_{A_0}\lra \underline{Sets} \quad {\rm sending}\quad 
   \spec{A} \mapsto \BS\left((Q_0, \varphi_0)\otimes A\right)
   $$
   In fact, there are bijections, as follows
   $$
  \BS\left((Q_0, \varphi_0)\otimes A\right)\iso Hom\left(\SB\left((Q_0, \varphi_0)\otimes A\right), A\right)
  \iso 
  Hom\left(\SB(Q_0, \varphi_0), A\right).
   $$
   }
   \end{remark}
   %
   \vspace{3mm}
   
   \bC\label{ourSphere}
     Suppose $A$ is a commutative noetherian ring, containing a field $k$, with $1/2\in k$.
   Let $P$ be a projective $A$-module, $\BH(P)=P^*\oplus P$ be  the hyperbolic space and 
   $(Q, \varphi)=\BH(P)\perp A$, and $\BH(P)=P^*\oplus P$. Let $B: Q \otimes Q\lra A$ be the bilinear form of 
   $(Q, \varphi)$.
   %
   Let $\SA(P)=\frac{S(P\oplus P^* \oplus A)}{(B-1)}$.
   %, where 
  % $B$ corresponds to the bilinear form of $(Q, \varphi)=\BH(P)\perp A$, and $\BH(P)=P^*\oplus P$ is the hyperbolic space.
  % $q(p, f, a)= f(p)+a^2$. 
   Then, there are bijections, as follows 
   $$
    \widetilde{Q}(P) \iso Hom\left(\SA(P), A\right)=Hom\left(\spec{A}, \spec{\SA(P)}\right).
   $$
   \eC
 \pf Follows from (\ref{swanSphere}).
 %, with $(Q, \varphi)=\BH(P)\perp A$, where $\BH(P)=P^*\oplus P$ is the hyperbolic space.
 \tcp $\eop$.

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%\newpage 

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\bibitem[MMu]{MMu}
 Mandal, Satya; Pavaman Murthy, M. Ideals as sections of projective modules. {\it J. Ramanujan Math. Soc.} 13 (1998), no. 1, 51-62.

\bibitem[MM]{MM} 
Satya Mandal and Bibekananda Mishra, 
The Homotopy Obstructions in Complete Intersections,
https://arxiv.org/pdf/1610.07495.pdf

\bibitem[MV]{MV}  Mandal, S.; Varma, P. L. N. On a question of Nori: the local case. {\it Comm. Algebra} 25 (1997), no. 2, 451-457.

\bibitem[MS]{MS} Mandal, Satya; Sridharan, Raja Euler classes and complete intersections. {\it J. Math. Kyoto Univ}. 36 (1996), no. 3, 453-470.

\bibitem[Mil]{Mil}
%
Milnor, John Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. 
{\it Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo}, 1971. xiii+184 pp


\bibitem[Mu]{Mu}
Murthy, M. Pavaman Zero cycles and projective modules. {\it Ann. of Math.} (2) 140 (1994), no. 2, 405-434. 
%\bibitem[PS]{PS} Petrov, V. A.; Stavrova, A. K. Elementary subgroups in isotropic reductive groups. (Russian)  {\it Algebra i Analiz} 20 (2008), no. 4, 160--188; translation in {\it St. Petersburg Math. J}. 20 (2009), no. 4, 625-644
\bibitem[MoM]{MoM}
Kumar, N. Mohan; Murthy, M. Pavaman Algebraic cycles and vector bundles over affine three-folds. {\it Ann. of Math.} (2) 116 (1982), no. 3, 579?591.

\bibitem[Mk1]{Mk1}
 Mohan Kumar, N. Stably free modules. {\it Amer. J. Math.} 107 (1985), no. 6, 1439-1444 (1986).
 
 \bibitem[Mk2]{Mk2}
 Mohan Kumar, N. Some theorems on generation of ideals in affine algebras. {\it Comment. Math. Helv.} 59 (1984), no. 2, 243-252.
 
% \bibitem[P]{P}
% Popescu, Dorin Letter to the editor: "General N\'{e}ron desingularization and approximation'' {\it Nagoya Math. J}. 118 (1990), 45-53.
 
 % \bibitem[O1]{O1} 
% Ojanguren, Manuel Formes quadratiques sur les alg�bres de polynomes. (French) {\it C. R. Acad. Sci. Paris S\'{e}r. }A-B 287 (1978), no. 9, A695-A698.
 
 \bibitem[O]{O2}
  Ojanguren, Manuel A splitting theorem for quadratic forms. {\it Comment. Math. Helv.} 57 (1982), no. 1, 145?157.
 
  \bibitem[P]{P}
 Popescu, Dorin Letter to the editor: "General N\'{e}ron desingularization and approximation'' {\it Nagoya Math. J}. 118 (1990), 45-53.
 
  \bibitem[R]{R}
  Roy, Amit Application of patching diagrams to some questions about projective modules. {\it J. Pure Appl. Algebra} 24 (1982), no. 3, 313-319.
  
%\bibitem[S]{S} Stavrova, A. Homotopy invariance of non-stable $K_1$-functors. {\it J. $K$-Theory} 13 (2014), no. 2, 199-248.
 
 \bibitem[St]{St}
 Steenrod, Norman The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. {\it Princeton University Press, Princeton, N. J.}, 1951. viii+224 pp.
 
%\bibitem[Sw]{Sw}  Swan, Richard G. K-theory of quadric hypersurfaces. {\it Ann. of Math}. (2) 122 (1985), no. 1, 113-153.

%\bibitem[Sw1]{Sw1}
 %Swan, Richard G. N\'{e}ron-Popescu desingularization. {\it Algebra and geometry (Taipei, 1995)},
  %135-192, Lect. Algebra Geom., 2, {\it Int. Press, Cambridge, MA}, 1998.
  
  \bibitem[Sw]{Sw}
   Swan, Richard G. $K$-theory of quadric hypersurfaces. 
   {\it Ann. of Math.} (2) 122 (1985), no. 1, 113?153. 

%   \bibitem[Sw2]{SwK}
%  Swan, Richard G. Higher algebraic K-theory.
 % {\it  K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992)}, 247?293, Proc. Sympos. Pure Math., 58, Part 1,
  %{\it  Amer. Math. Soc., Providence, RI,} 1995.
\end{thebibliography}

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1612.00749.extracted_bib
[AF] {AF Asok, Aravind; Fasel, Jean Euler class groups and motivic stable cohomotopy, https://arxiv.org/pdf/1601.05723.pdf %
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[BS2]{BS2 Bhatwadekar, S. M.; Sridharan, Raja On Euler classes and stably free projective modules. {Algebra, arithmetic and geometry, Part I, II (Mumbai, 2000), 139-158, Tata Inst. Fund. Res. Stud. Math., 16, {Tata Inst. Fund. Res., Bombay, 2002.
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[BS3]{BS3 Bhatwadekar, S. M.; Sridharan, Raja The Euler class group of a Noetherian ring. {Compositio Math. 122 (2000), no. 2, 183-222. %
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[CF]{CF Calm\`{es, Baptiste; Fasel, Jean Trivial Witt groups of flag varieties. {J. Pure Appl. Algebra 216 (2012), no. 2, 404-406. %
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[CF]{CF %B. Calm\`{es and F. Fasel, Groupes classiques, in {Autour des Sch\`{emas en Groupes (B. Calm\`{es, P.-H. Chaudouard, B. Conrad, C. Demarche, and J. Fasel, eds.), %{Panoramas et Synth\`{eses {46, Soc. Math. France, Paris, 2016, \'{Ecole d'\'{et\'{e "Sch�mas en Groupes," Group Schemes, A celebtraion of SGA3, Volume II, pp. 1?333.
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[F]{F %Asok, Aravind; Fasel, Jean On the number of generators of ideals in polynomial rings. {Ann. of Math. (2) 184 (2016), no. 1, 315-331. % On the number of generators of ideals in polynomial rings, arXiv:1507.05734 %Complete intersections from a cohomological viewpoint,preliminary draft.
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[K]{K Knus, Max-Albert Quadratic and Hermitian forms over rings. With a foreword by I. Bertuccioni. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 294. {Springer-Verlag, Berlin, 1991. xii+524 pp.
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[L]{L Lindel, Hartmut On the Bass-Quillen conjecture concerning projective modules over polynomial rings. {Invent. Math. 65 (1981/82), no. 2, 319-323.
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[M1]{M1 Mandal, Satya Projective modules and complete intersections. Lecture Notes in Mathematics, 1672. {Springer-Verlag, Berlin, 1997. viii+114 pp. %
1612.00749.extracted_bib
[M2]{M2 Mandal, Satya On the complete intersection conjecture of Murthy. {J. Algebra 458 (2016), 156-170.
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[M3]{M3 Mandal, Satya Homotopy of sections of projective modules. With an appendix by Madhav V. Nori. {J. Algebraic Geom. 1 (1992), no. 4, 639-646.
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[MY]{MY Mandal, Satya; Yang, Yong Intersection theory of algebraic obstructions. {J. Pure Appl. Algebra 214 (2010), no. 12, 2279-2293.
1612.00749.extracted_bib
[MMu]{MMu Mandal, Satya; Pavaman Murthy, M. Ideals as sections of projective modules. {J. Ramanujan Math. Soc. 13 (1998), no. 1, 51-62.
1612.00749.extracted_bib
[MM]{MM Satya Mandal and Bibekananda Mishra, The Homotopy Obstructions in Complete Intersections, https://arxiv.org/pdf/1610.07495.pdf
1612.00749.extracted_bib
[MV]{MV Mandal, S.; Varma, P. L. N. On a question of Nori: the local case. {Comm. Algebra 25 (1997), no. 2, 451-457.
1612.00749.extracted_bib
[MS]{MS Mandal, Satya; Sridharan, Raja Euler classes and complete intersections. {J. Math. Kyoto Univ. 36 (1996), no. 3, 453-470.
1612.00749.extracted_bib
[Mil]{Mil % Milnor, John Introduction to algebraic K-theory. Annals of Mathematics Studies, No. 72. {Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. xiii+184 pp
1612.00749.extracted_bib
[Mu]{Mu Murthy, M. Pavaman Zero cycles and projective modules. {Ann. of Math. (2) 140 (1994), no. 2, 405-434. %
1612.00749.extracted_bib
[PS]{PS Petrov, V. A.; Stavrova, A. K. Elementary subgroups in isotropic reductive groups. (Russian) {Algebra i Analiz 20 (2008), no. 4, 160--188; translation in {St. Petersburg Math. J. 20 (2009), no. 4, 625-644
1612.00749.extracted_bib
[MoM]{MoM Kumar, N. Mohan; Murthy, M. Pavaman Algebraic cycles and vector bundles over affine three-folds. {Ann. of Math. (2) 116 (1982), no. 3, 579?591.
1612.00749.extracted_bib
[Mk1]{Mk1 Mohan Kumar, N. Stably free modules. {Amer. J. Math. 107 (1985), no. 6, 1439-1444 (1986).
1612.00749.extracted_bib
[Mk2]{Mk2 Mohan Kumar, N. Some theorems on generation of ideals in affine algebras. {Comment. Math. Helv. 59 (1984), no. 2, 243-252. %
1612.00749.extracted_bib
[P]{P % Popescu, Dorin Letter to the editor: "General N\'{eron desingularization and approximation'' {Nagoya Math. J. 118 (1990), 45-53. %
1612.00749.extracted_bib
[O1]{O1 % Ojanguren, Manuel Formes quadratiques sur les alg�bres de polynomes. (French) {C. R. Acad. Sci. Paris S\'{er. A-B 287 (1978), no. 9, A695-A698.
1612.00749.extracted_bib
[O]{O2 Ojanguren, Manuel A splitting theorem for quadratic forms. {Comment. Math. Helv. 57 (1982), no. 1, 145?157.
1612.00749.extracted_bib
[P]{P Popescu, Dorin Letter to the editor: "General N\'{eron desingularization and approximation'' {Nagoya Math. J. 118 (1990), 45-53.
1612.00749.extracted_bib
[R]{R Roy, Amit Application of patching diagrams to some questions about projective modules. {J. Pure Appl. Algebra 24 (1982), no. 3, 313-319. %
1612.00749.extracted_bib
[S]{S Stavrova, A. Homotopy invariance of non-stable $K_1$-functors. {J. $K$-Theory 13 (2014), no. 2, 199-248.
1612.00749.extracted_bib
[St]{St Steenrod, Norman The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. {Princeton University Press, Princeton, N. J., 1951. viii+224 pp. %
1612.00749.extracted_bib
[Sw]{Sw Swan, Richard G. K-theory of quadric hypersurfaces. {Ann. of Math. (2) 122 (1985), no. 1, 113-153. %
1612.00749.extracted_bib
[Sw1]{Sw1 %Swan, Richard G. N\'{eron-Popescu desingularization. {Algebra and geometry (Taipei, 1995), %135-192, Lect. Algebra Geom., 2, {Int. Press, Cambridge, MA, 1998.
1612.00749.extracted_bib
[Sw]{Sw Swan, Richard G. $K$-theory of quadric hypersurfaces. {Ann. of Math. (2) 122 (1985), no. 1, 113?153. %
1612.00749.extracted_bib
[Sw2]{SwK % Swan, Richard G. Higher algebraic K-theory. % {K-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992), 247?293, Proc. Sympos. Pure Math., 58, Part 1, %{Amer. Math. Soc., Providence, RI, 1995.