A specialization property of index
Ananyo Dan\
\
Inder Kaur
In \cite{koll Koll\'{ar defined $i$-th index of a proper scheme over a field. In this note we study how index behaves under specialization, in any characteristic.
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\title{A specialization property of index}
\author{Ananyo Dan\, \, Inder Kaur}
\date{\today}

\begin{document}

 \maketitle


\begin{abstract}
 In \cite{koll} Koll\'{a}r defined $i$-th index of a proper scheme over a field. 
In this note we study how index behaves under specialization, in any characteristic.
\end{abstract}
\let\thefootnote\relax\footnotetext{Mathematics Subject Classification($2010$):Primary $14$G$05$, $14$C$25$, Secondary $13$H$15$, $14$D$06$

Keywords: Index of varieties, Semistable sheaves, Specialization}

 \section{Introduction}

 In \cite{koll}, Koll\'{a}r generalized the classical definition of index of a variety. For a proper scheme $X$ defined over
 a field $k$ (of any characteristic), he defines $\mr{elw}_i(X)$ to be  the ideal in $\mb{Z}$ generated by $\chi(\mc{F})$ as $\mc{F}$ ranges over
 all coherent sheaves on $X$ supported in a subscheme of dimension at most $i$. One can easily
 notice, \[\mr{elw}_0(X) \subset \mr{elw}_1(X) \subset ... \subset \mr{elw}_{\dim X}(X).\]
 Among other important properties, it is known that $\mr{elw}_i(-)$ is birational invariant.
 In this note we ask whether this property is locally constructible i.e., given a flat family $\pi:X \to B$ of proper $k$-schemes and any closed irreducible 
 subset $Y \subset B$, does there exist a non-empty open subset $U$ of $Y$ on which the function $u \mapsto \mr{elw}_i(\pi^{-1}(u))$ is constant? We observe that this is not 
 the case in general (see Example \ref{sp1}). However, we prove that the function $\mr{elw}_i(-)$ is invariant under specialization.
 In particular, we prove:
\begin{thm}\label{elw1}
Let $R$ be a discrete valuation ring, $\pi:X_R \to \mr{Spec}(R)$ be a flat, projective morphism and $X_R$ is an integral scheme. Suppose that the dimension of the 
fibers is equal to $n$. Denote by $K$ (resp. $k$) the fraction field
(resp. residue field) of $R$. Then, for any $0 \le i  \le n$, $\mr{elw}_i(X_K)= \mr{elw}_i(X_k)$.
%Furthermore, if $R$ is an excellent, Henselian DVR and $k$ is algebraically closed  of characteristic $0$ then
%$\mr{elw}_i(X_K)=\mr{elw}_i(X_k)$. 
\end{thm}
We note that there is no assumption on the characteristic of $K$ or $k$ in this article.

We now discuss the strategy of the proof. The inequality $\mr{elw}_i(X_K) \ge \mr{elw}_i(X_k)$ is a direct consequence of \cite[Proposition $4$]{koll}.
To prove the reverse inequality we introduce a concept of semistable elw-indices. The key observation is that 
this coincides with Koll\'{a}r's definition of the elw-index (see Proposition \ref{elw2}).
We then check that any semistable sheaf on the special fiber can be lifted to the generic fiber (see Proposition \ref{elw3}).
This gives the reverse inequality, hence proves the theorem.

 
\vspace{0.2 cm}
\emph{Acknowledgements}
This work was done when the first author was a post-doctoral fellow 
and the second author was a visiting researcher at the Tata Institute of Fundamental Research, Mumbai. The second author thanks TIFR for their hospitality. 
The first author is currently supported by ERCEA Consolidator Grant $615655$-NMST and also
by the Basque Government through the BERC $2014-2017$ program and by Spanish
Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa
excellence accreditation SEV-$2013-0323$ 

 
 \section{Semi-stable Esnault-Levine-Wittenberg index}

 Recall, the definition of the $i$-th index as given in \cite{koll}:
 \begin{defi}
  Let $Y$ be a projective scheme over a field $k$. For $0 \le i \le \dim Y$, the $i$-th \emph{Esnault-Levine-Wittenberg index},
  denoted by $\mr{elw}_i(Y)$ is defined as the ideal of $\mb{Z}$ generated by $\chi(\mc{F})$ as 
  where $\mc{F}$ runs through all coherent sheaves on $Y$ of dimension $\dim \mc{F}:=\dim \mr{Supp} \mc{F} \le i$.
 \end{defi}
 
 We now define a new semi-stable version of the above index:
 \begin{defi}
  Let $Y$ be as before. For $0 \le i \le \dim Y$, the \emph{semi-stable Esnault-Levine-Wittenberg index},
  denoted by $\mr{elw}^{ss}_{i} (Y)$ is defined as the ideal of $\mb{Z}$ generated by $\chi(\mc{F})$ as 
 $\mc{F}$ runs through all coherent semi-stable sheaves  over $Y$ of dimension $\dim \mc{F} \le i$.
 \end{defi}

 \begin{abs}
 We will sometimes abuse notation $\mr{elw}_i(Y)$ to mean the generator of the ideal. This usage will be clear from the context.
 \end{abs}


\begin{prop}\label{elw2}
Let $Y$ be a projective scheme of pure dimension $n$. 
Then, for all  $0 \le i \le \dim Y$, $\mr{elw}^{ss}_i(Y)=\mr{elw}_i(Y)$.
\end{prop}

\begin{proof}
Clearly, $\mr{elw}_m^{ss}(Y) \subset \mr{elw}_m(Y)$ for all $0 \le m \le n$. We now prove the converse.
Let $\mc{F}$ be a coherent sheaf of dimension $m$. There exists an unique torsion filtration:
\[0 \subset T_0(\mc{F}) \subset ... \subset T_m(\mc{F})=\mc{F}\]
where $T_i(\mc{F})$ is the maximal subsheaf of $\mc{F}$ of dimension at most equal to $i$. Note by definition, $T_i(\mc{F})/T_{i-1}(\mc{F})$ is zero or 
pure of dimension $i$. If $T_i(\mc{F})/T_{i-1}(\mc{F})$ is non-zero then, by \cite[Theorem $1.3.4$]{huy}, there exists an unique
Harder-Narasimahan filtration \[0=E_{-1} \subsetneq E_0 \subsetneq E_1 \subsetneq ... \subsetneq E_l=T_i(\mc{F})/T_{i-1}(\mc{F})\] 
of the pure sheaf $T_i(\mc{F})/T_{i-1}(\mc{F})$ such that $E_j/E_{j-1}$ is semistable of dimension $i$, for every $j=1,...,l$.
As Euler characteristic is additive, \[\chi(T_i(\mc{F})/T_{i-1}(\mc{F}))=\sum_j \chi(E_j/E_{j-1}) \, \mbox{ and } \chi(\mc{F})=\sum_i \chi(T_i(\mc{F})/T_{i-1}(\mc{F})).\]
Since $i \le m$, this implies $\mr{elw}_m(Y) \subset \mr{elw}_m^{ss}(Y)$.
Therefore, $\mr{elw}^{ss}_m(Y)=\mr{elw}_m(Y)$, for all $0 \le m \le n$. This proves the proposition.
\end{proof}

\section{Specialization properties of the index function}
We first observe (Example \ref{sp1}) that the Esnault-Levine-Wittenberg index does not vary in an algebraic way in flat families of projective
varieties (in particular, not semi-continuous). But  $\mr{elw}_i(-)$ is invariant under specialization (Theorem \ref{elw4}).

\begin{exa}\label{sp1} 
Denote by $Y:=\mb{A}_{\mb{R}}^1$ and $X$ the hypersurface in $\mb{P}^2 \times Y$ defined by the polynomial
$X^2+Y^2=t$, where $t$ is the coordinate corresponding to $\mb{A}^1$. Denote by $\pi:X \to Y$ the natural projection map.
Observe that $\pi$ is a flat morphism.

Denote by $I_\pi^i:Y \to \mb{Z}$ the function that associates to any $y \in Y$, the number $\mr{elw}_i(X_y)$.
We prove that $I_\pi^0$ is not semi-continuous. Observe that for $t>0$, there always exists rational point on the fiber $X_t$, hence $\mr{elw}_0(X_t)=1$. For $t \le 0$, there does not exist a rational point
but the resulting curve $X_t$ contains a $\mb{C}$-point. But $\mb{C}$ is an extension of degree $2$ over $\mb{R}$. Hence, $\mr{elw}_0(X_t)=2$ for $t \le 0$.
This shows that $I_\pi^0$ is not semi-continuous (in the Zariski topology).
\end{exa}

\begin{note}
 Let $R$ be a discrete valuation ring, $\pi:X_R \to \mr{Spec}(R)$ be a flat, projective morphism and $X_R$ is an integral scheme. Suppose that the dimension of the 
fibers is equal to $n$. Denote by $K$ (resp. $k$) the fraction field
(resp. residue field) of $R$. Denote by $X_k$ the special fiber and by $X_K$ the generic fiber. There is no assumption on the characteristic of $k$ and $K$.
\end{note}


\begin{prop}\label{elw3}
Suppose $\mc{F}_k$ is a semistable coherent sheaf on $X_k$. Then, there exists a semistable coherent sheaf $\mc{F}_R$ on $X_R$ such that $\mc{F}_R \otimes_R k \cong \mc{F}_k$. 
\end{prop}

\begin{proof}
As $\mc{F}_k$ is semistable there exists a free $\mo_{X_k}$-module $\mc{H}_k$ such that $\mc{F}_k$ is a quotient of $\mc{H}_k$. Let the resulting short exact sequence be of the form:
\[0 \to \mc{G}_k \to \mc{H}_k \to \mc{F}_k \to 0.\]
As $\mc{H}_k$ is a free $\mo_{X_k}$-module, there exists an unique free $\mo_{X_R}$-module $\mc{H}_R$ such that $\mc{H}_R \otimes_R k$. 
Denote by $\mc{G}_R$ the fiber product $\mc{H}_R \times_{\mc{H}_k} \mc{G}_k$ i.e., the set of 
sections of $\mc{H}_R$, which when restricted to $\mc{H}_k$ gives us $\mc{G}_k$. Denote by $\mc{F}_R$ the quotient of $\mc{H}_R$ by $\mc{G}_R$. As $\mc{G}_R\otimes_R k \cong \mc{G}_k$, the universal property of cokernel implies 
$\mc{F}_R \otimes_R k \cong \mc{F}_k$. Since $\mc{G}_R$ is a subsheaf of a free sheaf it is torsion free and as $R$ is principal ideal domain, $\mc{G}_R$ is $R$-flat (see \cite[Example III.$9.1.3$]{R1}).
Using \cite[Lemma $2.1.4$]{huy}, we have $\mc{F}_R$ is $R$-flat.
By the open nature of semi-stability \cite[Proposition $2.3.1$]{huy}, this implies $\mc{F}_R$ is semi-stable. This proves the proposition.
\end{proof}

We finally prove the main theorem of this article.

\begin{thm}\label{elw4}
 The following is true: for any $0 \le i  \le n$, $\mr{elw}_i(X_K)= \mr{elw}_i(X_k)$.
%Furthermore, if $R$ is an excellent, Henselian DVR and $k$ is algebraically closed  of characteristic $0$ then
%$\mr{elw}_i(X_K)=\mr{elw}_i(X_k)$. 
\end{thm}

\begin{proof}
By \cite[Proposition $4$]{koll} \[\mr{elw}_i(X_K)=(\chi(\mo_Z)|Z \subset X, \mbox{ of dimension } \le i).\] Given, any $Z$ on $X_K$, the scheme-theorectic closure $\bar{Z}$ is flat over $\Spec(R)$ (see \cite[Proposition III.$9.8$]{R1}). As Euler characteristic is invariant in flat families,
$\chi(\mo_Z)=\chi(\mo_{\bar{Z}_k})$, where $\bar{Z}_k$ is the special fiber for the flat morphism from $\bar{Z}$ to $\Spec(R)$. Hence,  for $0 \le i <n$, $\mr{elw}_i(X_K) \ge \mr{elw}_i(X_k)$.

Using Proposition \ref{elw3}, we conclude that $\mr{elw}^{ss}_i(X_K) \le \mr{elw}^{ss}_i(X_k)$, for all $0 \le i \le n$. By Proposition \ref{elw2}, we get
\[\mr{elw}_i(X_K)=\mr{elw}^{ss}_i(X_K) \le \mr{elw}^{ss}_i(X_k) = \mr{elw}_i(X_k).\] This implies  $\mr{elw}_i(X_K)=\mr{elw}_i(X_k)$. 

%In the case, $R$ is an excellent, Henselian DVR and $k$ is algebraically closed of characteristic $0$ then \cite[Theorem $39$]{koll}
%implies $\mr{elw}_0(X_K)=\mr{elw}_1(X_K)=...=\mr{elw}_n(X_K)$. As $\mr{elw}_i(X_K) \ge \mr{elw}_i(X_k)$ with equality for 
%$i=n$ and $\mr{elw}_n(X_k) \le \mr{elw}_{n-1}(X_k) \le ... \le \mr{elw}_0(X_k)$, we have $\mr{elw}_i(X_K)=\mr{elw}_i(X_k)$
%for all $0 \le i \le n$.
This completes the proof of the theorem.
\end{proof}


\bibliographystyle{alpha}
 \bibliography{researchbib}
 
 
 \vspace{2cm}
BCAM - Basque Centre for Applied Mathematics, Alameda de Mazarredo 14,
48009 Bilbao, Spain.\\
E-mail address: adan@bcamath.org\\
 
 
   Freie Universit\"{a}t Berlin, FB Mathematik und Informatik, Arnimallee 3, 14195 Berlin,
Germany.\\
E-mail address: kaur@math.fu-berlin.de

\end{document}




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\bibitem[HL10]{huy}
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\newblock Springer, 2010.

\bibitem[Kol13]{koll}
J{\'a}nos Koll{\'a}r.
\newblock Esnault-levine-wittenberg indices.
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\end{thebibliography}
