\section{Introduction}
Let $S = K[x_1, \ldots, x_d]$ be a polynomial ring over a field $K$. 
We regard $S$ as a graded ring with $\deg x_i = 1$ for all $i$. 
The following is a well-known result. 

\begin{thm}[{c.f. \cite{EG}}]\label{flush point}
Let $M$ be a finitely generated graded $S$-module. 
For an integer $r$, the following conditions are equivalent. 
\begin{itemize}
\item[(1)] $H_{\mathfrak m}^i(M)_j = 0$ for all $i, j \in {\mathbb Z}$ with $i+j >r$. 
\item[(2)] The truncated module $M_{\geq r} := \bigoplus_{i \geq r} M_i$ 
has an $r$-linear free resolution. 
\end{itemize}
Here ${\mathfrak m} := (x_1, \ldots, x_d)$ is the irrelevant ideal of $S$, 
and $H^i_{\mathfrak m}(M)$ is the $i^{\rm th}$ local cohomology module. 
\end{thm}

If the conditions of Theorem~\ref{flush point} are satisfied, we say 
$M$ is {\it $r$-regular}. For a sufficiently large $r$, $M$ is $r$-regular.
We call $\operatorname{reg}(M) = \min \{ \,  r \mid  \text{$M$ is $r$-regular} \, \}$ 
the {\it Castelnuovo-Mumford regularity} of $M$. 
This is a very important invariant in commutative algebra.  

Let $A$ be a noetherian AS regular Koszul quiver algebra 
with the graded Jacobson radical ${\mathfrak m} := \bigoplus_{i\geq 1} A_i$.  
If $A$ is commutative, $A$ is essentially a polynomial ring. 
When $A$ is connected (i.e.,  $A_0 = K$), 
it is the coordinate ring of a 
``noncommutative projective space" in noncommutative 
algebraic geometry.  Let $\operatorname{gr} A$ be the category of finitely generated 
graded left $A$-modules and their degree preserving maps. 
(For a graded ring $B$, $\operatorname{gr} B$ means the similar category for $B$.) 
The local cohomology module $H_{\mathfrak m}^i(M)$ of 
$M \in \operatorname{gr} A$ behaves pretty much like in the commutative case. 
For example, we have ``Serre duality theorem" for the derived category 
$D^b(\operatorname{gr} A)$. See \cite{M98, Ye} and 
Theorem~\ref{Serre duality} below. By virtue of this duality, we can show 
that Theorem~\ref{flush point} also holds for {\it bounded complexes} in 
$\operatorname{gr} A$. 

\begin{thm}\label{2nd thm}
For a complex $M^\bullet \in D^b(\operatorname{gr} A)$ and an integer $r$, 
the following conditions are equivalent. 
\begin{itemize}
\item[(1)] $H_{\mathfrak m}^i(M^\bullet)_j = 0$ for all $i, j \in {\mathbb Z}$ with $i+j >r$. 
\item[(2)] The truncated complex $(M^\bullet)_{\geq r}$ has an $r$-linear 
projective resolution. 
\end{itemize}
Here $(M^\bullet)_{\geq r}$ is the subcomplex of $M^\bullet$ whose $i^{\rm th}$ term is 
$(M^i)_{\geq (r-i)}$. 
\end{thm}

For a sufficiently large $r$, the conditions of the above theorem are 
satisfied. The regularity $\operatorname{reg}(M^\bullet)$ of $M^\bullet$ 
is defined in the natural way. When $A$ is connected, J\o rgensen 
(\cite{Jor04}) has studied the regularity of complexes, and essentially proved 
the above result. See also \cite{Jor99, Mori}. 
(Even in the case when $A$ is a polynomial ring, it seems that 
nobody had considered Theorem~\ref{2nd thm} before \cite{Jor04}.) 
But his motivation and treatment are slightly different from ours. 

For $M^\bullet \in D^b(\operatorname{gr} A)$, set $\mathcal H(M^\bullet)$ to be a complex 
such that $\mathcal H(M^\bullet)^i = H^i(M)$ for all $i$ 
and the differential maps are zero. 
Then we have $\operatorname{reg}(\mathcal H(M^\bullet)) \geq \operatorname{reg}(M^\bullet)$. 
The difference $\operatorname{reg}(\mathcal H(M^\bullet)) - \operatorname{reg}(M^\bullet)$
is a theme of the last section of this paper. 

Let $A^!$ be the quadratic dual ring of $A$. 
For example, if $S=K[x_1, \ldots, x_d]$ is a polynomial ring, then 
$S^!$ is an exterior algebra $E= \bigwedge \< y_1, \ldots, y_d \>$. 
It is known that $A^!$ is always Koszul, finite dimensional, 
and selfinjective.
The Koszul duality functors 
${\mathcal F} : D^b(\operatorname{gr} {A^!}) \to D^b(\operatorname{gr} A)$ and 
${\mathcal G} : D^b(\operatorname{gr} A) \to D^b(\operatorname{gr} {A^!})$
give a category equivalence  $D^b(\operatorname{gr} {A^!}) \cong D^b(\operatorname{gr} A)$ 
(see \cite{BGS}). It is easy to check that 
$$\operatorname{reg}(M^\bullet) = \max \{ \, i \mid  H^i({\mathcal G}(M^\bullet)) \ne 0 \, \}$$
for $M^\bullet \in D^b(\operatorname{gr} A)$. 

Let $\operatorname{gr} {\Ao}$ be the category of finitely generated graded {\it right} 
$A$-modules. The above results on $\operatorname{gr} A$ also hold for 
$\operatorname{gr} {\Ao}$. Moreover, we have 
$$\operatorname{reg}( \, {\bf R} \underline{\Hom}_A(M^\bullet, {\mathcal D}^\bullet) \, ) = 
- \min \{ \, i \mid  H^i({\mathcal G}(M^\bullet)) \ne 0 \, \}$$
for $M^\bullet \in D^b(\operatorname{gr} A)$. 
Here ${\mathcal D}^\bullet$ is a balanced dualizing complex of $A$, which 
gives duality functors between $D^b(\operatorname{gr} A)$ and $D^b(\operatorname{gr} {\Ao})$. 

Let $B$ be a noetherian Koszul algebra. 
For $M \in \operatorname{gr} B$ and $i \in {\mathbb Z}$, $M_{\< i \>}$ denotes the 
submodule of $M$ generated by the degree $i$ component $M_i$ of $M$. 
We say $M$ is {\it weakly Koszul} if $M_{\< i \>}$ has a linear projective 
resolution for all $i$. This definition is different 
from the original one given in \cite{MZ}, but they are equivalent. 
(Weakly Koszul modules are also called ``componentwise linear modules" 
by some commutative algebraists.)  
 Martinez-Villa and Zacharia proved that if $N \in \operatorname{gr} {A^!}$ then 
the $i^{\rm th}$ syzygy $\Omega_i(N)$
of $N$ is weakly Koszul for $i \gg 0$. For $N \in \operatorname{gr} {A^!}$, set 
$$\operatorname{lpd}(N):= \min\{ \, i \in {\mathbb N}  \mid  \text{$\Omega_i(N)$ is 
weakly Koszul} \, \}.$$  

Let $N \in \operatorname{gr} {A^!}$ and  $N' :=\underline{\Hom}_{A^!}(N, A^!) \in \operatorname{gr} {(A^!)^{\sf op}}$ its dual. 
In Theorem~\ref{wKos}, we show that $N$ is weakly Koszul if and only if 
$\operatorname{reg}(\mathcal H \circ {\mathcal F}^{\sf op} (N')) = 0$, where 
${\mathcal F}^{\sf op} : D^b(\operatorname{gr} {(A^!)^{\sf op}}) \to D^b(\operatorname{gr} {\Ao})$ is the Koszul duality functor. 
(Since $\operatorname{reg}({\mathcal F}^{\sf op}(N')) = 0$, we have $\operatorname{reg}(\mathcal H \circ {\mathcal F}^{\sf op} (N')) 
\geq 0$ in general.) Moreover, we have 
$$\operatorname{lpd}(N) = \operatorname{reg}( \, \mathcal H \circ {\mathcal F}^{\sf op} (N') \,)$$
(Corollary~\ref{wKos=regH}). 
As an application of this formula, we refine a result of 
Herzog and R\"omer on monomial ideals of an exterior algebra. 
Among other things, in Proposition~\ref{d-2}, we show that 
if $J$ is a monomial ideal of an exterior algebra 
$E= \bigwedge \< y_1, \ldots, y_d \>$, $d \geq 3$, then 
$\operatorname{lpd}(E/J) \leq d-2$.

Finally, we remark that Herzog and Iyengar (\cite{HI}) studied the invariant 
$\operatorname{lpd}$ and related concepts over noetherian  commutative (graded) 
local rings. Among other things, they proved that 
$\operatorname{lpd}(N)$ is always finite over some ``nice" local 
rings (e.g., complete intersections whose associated graded rings 
are Koszul).   

\section{Preliminaries}
Let $K$ be a field. 
The ring $A$ treated in this paper is a (not necessarily commutative) 
$K$-algebra with some nice properties. More precisely, $A$ is 
a noetherian AS regular Koszul quiver algebra. 
If $A$ is commutative, it is essentially a polynomial ring. 
But even in this case, most results in \S4 and a few results in \S3 are new. 
(In the polynomial ring case, 
many results in \S3 were obtained in \cite{EFS}.) 
So one can read this paper assuming that $A$ is a polynomial ring. 

\medskip

We sketch the definition and basic properties of 
graded quiver algebras here. See \cite{GM} for further information.  

Let $Q$ be a finite quiver. That is, $Q = (Q_0, Q_1)$ is an oriented 
graph, where $Q_0$ is the set of vertices and $Q_1$ is the set of 
arrows. Here $Q_0$ and $Q_1$ are finite sets. 
The path algebra $KQ$ is a 
positively graded algebra with grading given by the lengths of paths. 
We denote the graded Jacobson radical of $KQ$ by $J$. That is, 
$J$ is the ideal generated by all arrows. 
If $I \subset J^2$ is a graded ideal, we say $A = KQ/I$ is a 
{\it graded quiver algebra}. Of course, $A = \bigoplus_{i \geq 0} A_i$ 
is a graded ring such that the degree $i$ component 
$A_i$ is a finite dimensional $K$-vector space for all $i$. 
The subalgebra $A_0$ is a product of copies of the field $K$, 
one copy for each element of $Q_0$.  
If $A_0 = K$ (i.e., $Q$ has only one vertex), we say $A$ is {\it connected}. 
Let  $R = \bigoplus_{i \geq 0} R_i$ be a graded algebra with 
$R_0 = K$ and $\dim_K R_1 =:n < \infty$. 
If $R$ is generated by $R_1$ as a $K$-algebra, 
then it can be regarded as a graded 
quiver algebra over a quiver with one vertex and $n$ loops. 
Let ${\mathfrak m} := \bigoplus_{i \geq 1} A_i$ be the graded Jacobson radical of $A$.  
Unless otherwise specified, we assume that $A$ is left and right noetherian 
throughout this paper. 


Let ${\operatorname{Gr} A}$ (resp. $\operatorname{Gr} {\Ao}$) be the category of graded left 
(resp. right) $A$-modules and their degree preserving $A$-homomorphisms. 
Note that the degree $i$ component $M_i$ of $M \in {\operatorname{Gr} A}$ (or $M \in \operatorname{Gr} {\Ao}$) 
is an $A_0$-module for each $i$.  
Let $\operatorname{gr} A$ (resp. $\operatorname{gr} {\Ao}$) be the full subcategory of ${\operatorname{Gr} A}$ 
(resp. $\operatorname{Gr} {\Ao}$) consisting of finitely generated modules. 
Since we  assume that $A$ is noetherian, $\operatorname{gr} A$ and $\operatorname{gr} {\Ao}$ are 
abelian categories. 
In the sequel, we will define several concepts for ${\operatorname{Gr} A}$ and $\operatorname{gr} A$. 
But the corresponding concepts for $\operatorname{Gr} {\Ao}$ and $\operatorname{gr} {\Ao}$ can be defined 
in the same way. 

For $n \in {\mathbb Z}$ and $M \in {\operatorname{Gr} A}$, 
set $M_{\geq n} := \bigoplus_{i \geq n} M_i$ to be a 
submodule of $M$, and $M_{\leq n} := \bigoplus_{i \leq n} M_i$ 
to be a graded $K$-vector space. The $n^{\rm th}$ shift 
$M(n)$ of $M$ is defined by $M(n)_i = M_{n+i}$. 
Set $\sigma(M) := \sup \{ \, i \mid M_i \ne 0 \, \}$ and 
$\iota(M) := \inf \{ \, i \mid M_i \ne 0 \, \}$. 
If $M = 0$, we set $\sigma(M) = - \infty$ and $\iota(M) = + \infty$. 
Note that if $M \in \operatorname{gr} A$ then $\iota(M)>-\infty$. 
For a complex $M^\bullet$ in ${\operatorname{Gr} A}$, set 
$$\sigma(M^\bullet) := \sup \{ \, \sigma(H^i(M^\bullet)) + i \mid i \in {\mathbb Z} \, \}
\ \text{and} \ 
\iota(M^\bullet) := \inf \{ \, \iota(H^i(M^\bullet)) + i \mid i \in {\mathbb Z} \, \}.$$  

For $v \in Q_0$, we have the idempotent $e_v$ associated with $v$. 
Note that $1 = \sum_{v \in Q_0} e_v$. Set $P_v := A e_v $ and 
${}_v P :=  e_v A$. Then we have ${}_A A = \bigoplus_{v \in Q_0} P_v$ and 
$A_A = \bigoplus_{v \in Q_0} ({}_vP)$. 
Each $P_v$ and ${}_v P$ are indecomposable projectives. 
Conversely, any indecomposable projective in ${\operatorname{Gr} A}$ (resp. $\operatorname{Gr} {\Ao}$)
is isomorphic to  $P_v$ (resp. ${}_v P$) for some $v \in Q_0$ 
up to degree shifting. 
Set $K_v := P_v/ ({\mathfrak m} P_v )$ and ${}_v K := {}_v P / ({}_vP \, {\mathfrak m})$. 
Each $K_v$ and ${}_v K$ are simple. 
Conversely, any simple object in ${\operatorname{Gr} A}$ (resp. $\operatorname{Gr} {\Ao}$) is isomorphic to 
$K_v$ (resp. ${}_vK$) for some $v \in Q_0$ up to degree shifting.  

We say a graded left (or right) $A$-module $M$ 
is {\it locally finite} 
if $\dim_K M_i < \infty$ for all $i$. If $M \in \operatorname{gr} A$, then it is 
locally finite. 
Let $\operatorname{lf} A$ (resp. $\operatorname{lf} {\Ao}$) be the full subcategory of 
${\operatorname{Gr} A}$ (resp. $\operatorname{Gr} {\Ao}$) consisting of locally finite modules. 

Let $C^b({\operatorname{Gr} A})$ be the category of bounded cochain complexes in 
${\operatorname{Gr} A}$, and $D^b({\operatorname{Gr} A})$ its derived category. 
We have similar categories for $\operatorname{Gr} {\Ao}$, $\operatorname{lf} A$, $\operatorname{lf} {\Ao}$, $\operatorname{gr} A$ and $\operatorname{gr} {\Ao}$. 
For a complex $M^\bullet$ and an integer $p$, let $M^\bullet[p]$ be the $p^{\rm th}$ 
translation of $M^\bullet$. That is, $M^\bullet[p]$ is a complex with $M^i[p] = M^{i+p}$. 
Since $D^b(\operatorname{gr} A) \cong D^b_{\operatorname{gr} A}({\operatorname{Gr} A}) \cong D^b_{\operatorname{gr} A}(\operatorname{lf} A)$, 
we freely identify these categories. 
A module $M$ can be regarded as a complex 
$\cdots \to 0 \to M \to 0 \to \cdots $ 
with $M$ at the $0^{\rm th}$ term. We can regard ${\operatorname{Gr} A}$ as a full 
subcategory of $C^b({\operatorname{Gr} A})$ and $D^b({\operatorname{Gr} A})$ in this way. 


For $M,N \in {\operatorname{Gr} A}$, set $\underline{\Hom}_A(M,N):= \bigoplus_{i \in {\mathbb Z}} 
\operatorname{Hom}_{{\operatorname{Gr} A}}(M,N(i))$ to be a graded $K$-vector space with 
$\underline{\Hom}_A(M,N)_i = \operatorname{Hom}_{{\operatorname{Gr} A}}(M,N(i))$. Similarly, we can also define 
$\underline{\Hom}_A^\bullet(M^\bullet,N^\bullet)$, ${\bf R}\underline{\Hom}_A(M^\bullet,N^\bullet)$, and $\underline{\Ext}^i_A(M^\bullet,N^\bullet)$ 
for $M^\bullet, N^\bullet \in D^b({\operatorname{Gr} A})$. 

If $V$ is a $K$-vector space, $V^*$ denotes the dual vector 
space $\operatorname{Hom}_K(V,K)$. For $M \in {\operatorname{Gr} A}$ (resp. $M \in \operatorname{Gr} {\Ao}$), 
$M^\vee := \bigoplus_{i \in {\mathbb Z}} (M_i)^*$ 
has a graded {\it right} (resp. {\it left}) $A$-module 
structure given by $(fa)(x)= f(ax)$ (resp. $(af)(x)= f(xa)$) and 
$(M^\vee)_i = (M_{-i})^*$. 
If $M \in \operatorname{lf} A$, then $M^\vee \in \operatorname{lf} {\Ao}$ and $M^{\vee \vee} \cong M$. 
In other words, $(-)^\vee$ gives exact duality functors between $\operatorname{lf} A$ and 
$\operatorname{lf} {\Ao}$, which can be extended to duality functors between 
$C^b(\operatorname{lf} A)$ and $C^b(\operatorname{lf} {\Ao})$, or 
between $D^b(\operatorname{lf} A)$ and $D^b(\operatorname{lf} {\Ao})$. 
In this paper, when we say $W$ is an $A$-$A$ bimodule, 
we always assume that $(aw)a' = a(wa')$ for all 
$w \in W$ and $a, a' \in A$. If $W$ is a graded $A$-$A$ bimodule, 
then so is $W^\vee$. 

It is easy to see that  $I_v :=(_vP)^\vee$ (resp. 
${}_vI := (P_v)^\vee$) is injective in ${\operatorname{Gr} A}$ (resp. $\operatorname{Gr} {\Ao}$).  
Moreover, $I_v$ and ${}_v I$ are graded injective hulls of $K_v$ and 
${}_v K$ respectively.  
In particular, the $A$-$A$ bimodule 
$A^\vee$ is injective both in ${\operatorname{Gr} A}$ and in $\operatorname{Gr} {\Ao}$. 

Let $W$ be a graded $A$-$A$-bimodule. 
For $M \in {\operatorname{Gr} A}$, we can regard $\underline{\Hom}_A(M,W)$ 
as a graded {\it right} $A$-module by $(fa)(x)= f(x)a$. 
We can also define ${\bf R}\underline{\Hom}_A(M^\bullet,W) \in D^b(\operatorname{Gr} {\Ao})$ and 
$\underline{\Ext}_A^i(M^\bullet,W) \in \operatorname{Gr} {\Ao}$ for $M^\bullet \in D^b({\operatorname{Gr} A})$ in this way. 
Similarly,  for $M^\bullet \in D^b(\operatorname{Gr} {\Ao})$, 
we can make  ${\bf R}\underline{\Hom}_{A^{\sf op}}(M^\bullet,W)$ and 
$\underline{\Ext}_{A^{\sf op}}^i(M^\bullet,W)$  (bounded complex of) graded left $A$-modules. 
For $M \in {\operatorname{Gr} A}$, we can regard $\underline{\Hom}_A(W,M)$ 
as a graded {\it left} $A$-module by $(af)(x)= f(xa)$. 



For the functor $\underline{\Hom}_A(-,W)$  , 
we mainly consider the case when $W =A$ or $W = A^\vee$. 
But, we have $\underline{\Hom}_A(-,A^\vee) \cong (-)^\vee$. To see this, note that 
\begin{eqnarray*}
(M^\vee)_i = \operatorname{Hom}_K(M_{-i},K) &=& \bigoplus_{v \in Q_0} \operatorname{Hom}_K(e_vM_{-i},K) \\
&\cong& \bigoplus_{v \in Q_0} \operatorname{Hom}_K (e_vM_{-i},K_v)\\
&\cong& \operatorname{Hom}_{A_0} (M_{-i}, A_0).  
\end{eqnarray*}
Via the identification $(A^\vee)_0 \cong (A_0)^* \cong A_0$, 
$f \in (M^\vee)_i \cong \operatorname{Hom}_{A_0}(M_{-i}, A_0)$ gives a morphism 
$f': M_{\geq -i} \to A^\vee(i)$ in ${\operatorname{Gr} A}$. Since 
$\operatorname{Hom}_{\operatorname{Gr} A} (M/M_{\geq -i}, A^\vee(i)) = 0$ and $A^\vee$ is injective, 
the short exact sequence $0 \to M_{\geq -i} \to M \to M/M_{\geq -i} \to 0$ 
induces a unique extension $f'' : M \to A^\vee(i)$ of $f'$. 
From this correspondence, we have $\underline{\Hom}_A(M,A^\vee) \cong M^\vee$.


Let $P^\bullet$ be a right bounded complex in $\operatorname{gr} A$ such that each $P^i$ 
is projective. We say $P^\bullet$ is {\it minimal} 
if $d(P^i) \subset {\mathfrak m} P^{i+1}$ for all $i$. Here $d$ is the 
differential map. Any complex $M^\bullet \in C^b(\operatorname{gr} A)$ 
has a minimal projective resolution, that is, 
we have a minimal complex $P^\bullet$ of 
projective objects and a graded quasi-isomorphism $P^\bullet \to M^\bullet$. 
A minimal projective resolution of $M^\bullet$ is unique up to isomorphism.
We denote a graded module $A/{\mathfrak m}$ by $A_0$.  
Set $\beta^{i,j}(M^\bullet) := \dim_K \underline{\Ext}_A^{-i}(M^\bullet,A_0)_{-j}$. 
Let $P^\bullet$ be a minimal projective resolution of $M^\bullet$, 
and $P^i := \bigoplus_{l=1}^m T^{i, \, l}$ an indecomposable decomposition. 
Then we have 
$$\beta^{i,j}(M^\bullet) = \# \{ \, l \mid \text{$T^{i, \, l}(j) 
\cong P_v$ for some $v$}\, \}.$$
We can also define $\beta^{i,j}(M^\bullet)$ as the dimension of 
$\operatorname{Tor}^A_{-i}(A_0, M^\bullet)_j$. This definition must be much more familiar to 
commutative algebraists. Note that $\beta^{i,j}(-)$ is an invariant of 
isomorphism classes of the derived category $D^b(\operatorname{gr} A)$. Note that 
these facts on minimal projective resolutions also hold over any 
noetherian graded algebra. 

\begin{dfn}\label{AS regular} 
Let $A$ be a (not necessarily noetherian) graded quiver algebra. 
We say $A$ is {\it Artin-Schelter regular} 
(AS-regular, for short), if 
\begin{itemize}
\item $A$ has finite global dimension $d$. 
\item $\underline{\Ext}_A^i(K_v, A) = \underline{\Ext}_{A^{\sf op}}^i({}_v K,A)= 0$ for all $i \ne d$ 
and all $v \in Q_0$.  
\item  There are a permutation $\delta$ on $Q_0$ and 
an integer $n_v$ for each $v \in Q_0$ such that  
$\underline{\Ext}_A^d(K_v, A) \cong {}_{\delta(v)} K(n_v)$ (equivalently, 
$\underline{\Ext}_{A^{\sf op}}^d({}_v K, A) \cong K_{\delta^{-1}(v)}(n_v)$ ) 
for all $v$.   
\end{itemize}
\end{dfn}

\begin{rem}\label{original definition}
The AS regularity is a very important concept in 
non-commutative algebraic geometry. 
In the original definition,  it is assumed that 
an AS regular algebra $A$ is connected and  
there is a positive real number $\gamma$ such that 
$\dim_K A_n < n^\gamma$ for $n \gg 0$, while some authors do not require 
the latter condition. We also remark that 
Martinez-Villa and coworkers called rings satisfying 
the conditions of Definition~\ref{AS regular} {\it generalized Auslander 
regular algebras} in \cite{GMT, M98}. 
\end{rem}

\begin{dfn}
For an integer $l \in {\mathbb Z}$, 
we say $M^\bullet \in \operatorname{gr} A$ has an {\it $l$-linear (projective) resolution}, 
if $$\beta^{i,j}(M^\bullet) \ne 0 \Rightarrow i+j=l.$$
If $M^\bullet$ has an $l$-linear resolution for some $l$, 
we say $M^\bullet$ has a {\it linear resolution}. 
\end{dfn}


\begin{dfn}
We say $A$ is {\it Koszul}, if the graded left 
$A$-module $A_0$ has a linear resolution. 
\end{dfn}

In the definition of the Koszul property, we can regard $A_0$ as a 
right $A$-module. (We get the equivalent definition.)  
That is, $A$ is Koszul if and only if any simple   
graded left (or, right) $A$-module has a linear resolution. 

\begin{lem}
If $A$ is noetherian, AS-regular, Koszul, and has global dimension $d$, 
then $\underline{\Ext}_A^d(K_v, A) \cong {}_{\delta(v)} K (d)$ and 
$\underline{\Ext}_{A^{\sf op}}^d({}_v K, A) \cong K_{\delta^{-1} (v)}(d)$ for all $v$.  
Here $\delta$ is the permutation of $Q_0$ given in 
Definition~\ref{AS regular}. 
\end{lem}

\begin{proof}
Since $A$ is Koszul, $P^{-d}$ of 
a minimal projective resolution $P^\bullet : 0 \to P^{-d} \to \cdots \to P^0 \to 0$ 
of $K_v$ is generated by its degree $d$-part $(P^{-d})_d$ 
(more precisely, $P^{-d} = P_{\delta(v)}(-d)$). 
\end{proof}


{\it In the rest of this paper, $A$ is always a noetherian AS-regular 
Koszul quiver algebra of global dimension $d$.}


\begin{exmp}
(1) A polynomial ring $K[x_, \ldots, x_d]$ is clearly 
a noetherian AS-regular Koszul (quiver) algebra of global dimension $d$.
Conversely, if a regular noetherian graded algebra 
is connected and commutative, it is a polynomial ring. 

(2) Let $K\< x_1, \ldots, x_d\>$ be the free associative algebra, and 
$(q_{i,j})$ a $d \times d$ matrix with entries in $K$ satisfying 
$q_{i,j} q_{j,i} = q_{i,i} = 1$ for all $i, j$. Then the quotient ring 
$A = K\< x_1, \ldots, x_n\>/\<x_j x_i -q_{i,j} x_i x_j \mid 
1 \leq i, j \leq d\>$ is a noetherian AS-regular Koszul algebra with 
global dimension $d$.  This fact must be well-known to specialists, 
but we will sketch a proof here for the reader's convenience. 
Since $x_1, ..., x_d \in A_1$ form a regular 
normalizing sequence with the quotient ring $K = A/(x_1, \ldots, x_d)$, 
$A$ is a noetherian ring with a balanced dualizing complex 
by \cite[Lemma 7.3]{Mori}.  
It is not difficult to construct a minimal free resolution of the module 
$K = A/{\mathfrak m}$, which is a ``$q$-analog" of the Koszul complex of 
a polynomial ring $K[x_1, \ldots, x_d]$. So $A$ is Koszul 
and has global dimension $d$.  
Since $A$ has finite global dimension and admits 
a balanced dualizing complex,  
it is AS-regular (c.f. \cite[Remark 3.6 (3)]{Mori}).  


Artin, Tate and Van den Bergh (\cite{ATV}) classified connected 
AS regular algebras of global dimension 3.  
(Their definition of AS regularity is stronger than ours. 
See Remark~\ref{original definition}.) All of the algebras they listed   
are noetherian (\cite[Theroem~8.1]{ATV}). 
But some are Koszul and some are not. 

 


(3) A {\it preprojective algebra} is an important example of 
non-connected AS regular algebras. 
See \cite{GMT} and the references cited there for the definition 
of this algebra and further information. 
The preprojective algebra $A$ of a finite quiver $Q$ is a graded quiver 
algebra over the {\it inverse completion} $\overline{Q}$ of $Q$. 
If the quiver $Q$ is connected (of course, it does not mean $A$ is connected), 
then $A$ is (almost) always an AS regular algebra of global dimension 2, 
but it is not Koszul in some cases, and not noetherian in many cases. 
Let $G$ be the bipartite graph of $Q$ in the sense of 
\cite[\S3]{GMT}. If $G$ is Euclidean, then $A$ is a noetherian 
AS-regular Koszul algebra of global dimension 2.  
\end{exmp}

 For $M \in {\operatorname{Gr} A}$, set 
 $$\Gamma_{\mathfrak m}(M) = \lim\limits_{ \to} \underline{\Hom}_A(A/{\mathfrak m}^n, M) 
 = \{ \, x \in M \mid \text{$A_{n} \, x =0$ for $n \gg 0$} \, \}\in {\operatorname{Gr} A}.$$  
Then $\Gamma_{\mathfrak m}(-)$ gives a left exact functor from ${\operatorname{Gr} A}$ to itself.
So we have a right derived functor ${\bf R} \Gamma_{\mathfrak m} : D^b({\operatorname{Gr} A}) \to 
D^b({\operatorname{Gr} A})$.  For $M^\bullet \in D^b({\operatorname{Gr} A})$, $H_{\mathfrak m}^i(M^\bullet)$ denotes the $i^{\rm th}$ 
cohomology of ${\bf R} \Gamma_{\mathfrak m}(M^\bullet)$, and we call it the $i^{\rm th}$  
{\it local cohomology} of $M^\bullet$. 
It is easy to see that $H_{\mathfrak m}^i(M^\bullet) =  \lim\limits_{ \to} \underline{\Ext}^i_A(A/{\mathfrak m}^n, M^\bullet)$. 
Similarly, we can define ${\bf R} \Gamma_{{\mathfrak m}^{\sf op}}: D^b(\operatorname{Gr} {\Ao}) \to 
D^b(\operatorname{Gr} {\Ao})$ and $H^i_{{\mathfrak m}^{\sf op}}: D^b(\operatorname{Gr} {\Ao}) \to \operatorname{Gr} {\Ao}$ in the same way. 
If $M$ is an $A$-$A$ bimodule, $H_{\mathfrak m}^i(M)$ and $H_{{\mathfrak m}^{\sf op}}^i(M)$ 
are also. 

Let $I \in {\operatorname{Gr} A}$ be an indecomposable injective.  
Then $\Gamma_{\mathfrak m}(I) \ne 0$, if and only if $I \cong I_v(n)$ 
for some $v \in Q_0$ and $n \in {\mathbb Z}$, if and only if 
$\Gamma_{\mathfrak m}(I) = I$.  Similarly, $\underline{\Hom}_A(A_0, I) \ne 0$ if and 
only if  $I \cong I_v(n)$ for some $v \in Q_0$ and $n \in {\mathbb Z}$. In this case, 
$\underline{\Hom}_A(A_0, I) = K_v(n)$.  
The same is true for an indecomposable injective $I \in \operatorname{Gr} {\Ao}$. 


Let $I^\bullet$ be a minimal injective resolution of $A$ in $\operatorname{gr} A$. 
Since $A$ is AS regular, $I^i = 0$ for all $i>d$, $\Gamma_{\mathfrak m}(I^i) = 0$ 
for all $i < d$, and $\Gamma_{\mathfrak m}(I^d) = A^\vee(d)$. 
Hence ${\bf R}\Gamma_{\mathfrak m}(A) \cong A^\vee(d)[-d]$ in $D^b(\operatorname{gr} A)$. 
By the same argument as \cite[Proposition~4.4]{Ye}, we also have 
${\bf R}\Gamma_{\mathfrak m}(A) \cong A^\vee(d)[-d]$ in $D^b(\operatorname{gr} {\Ao})$. It does not mean that 
$H_{\mathfrak m}^d(A) \cong A^\vee(d)$ as $A$-$A$ bimodules. 
But there is an $A$-$A$ bimodule $L$ such that 
$L \otimes_A H_{\mathfrak m}^d(A)  \cong A^\vee(d)$ as $A$-$A$ bimodules. 
Here the underlying additive group of $L$ is $A$, 
but the bimodule structure is give by 
$A \times L \times A \ni (a, l,  b) \mapsto \phi(a) l b \in A = L$ 
for a (fixed) $K$-algebra automorphism $\phi$ of $A$. 
In particular, $L \cong A$ as 
left $A$-modules and as right $A$-modules (separately). 
Note that $\phi(e_v) = e_{\delta(v)}$ for all $v \in Q_0$, where 
$\delta$ is the permutation on $Q_0$ appeared in 
Definition~\ref{AS regular}. If $A$ is commutative, then 
$\phi$ is the identity map. 

We give a new $A$-$A$ bimodule structure $L'$ to the additive group $A$ by  
$A \times L' \times A \ni (a, l,  b) \mapsto a l \phi(b) \in A = L'$. 
Then $L' \cong \underline{\Hom}_A(L,A)$.  Set ${\mathcal D}^\bullet := L'(-d)[d]$. 
Note that ${\mathcal D}^\bullet$ belongs both $D^b(\operatorname{gr} A)$ and $D^b(\operatorname{gr} {\Ao})$.  We have 
$H_{\mathfrak m}^i({\mathcal D}^\bullet) = H_{{\mathfrak m}^{\sf op}}^i({\mathcal D}^\bullet) = 0$ for all $i \ne 0$ and 
$H^0_{\mathfrak m} ({\mathcal D}^\bullet) \cong H_{m^{\sf op}}^0({\mathcal D}^\bullet) \cong A^\vee$ as $A$-$A$ 
bimodules by the same argument as \cite[\S4]{Ye}. 
Thus (an injective resolution of) ${\mathcal D}^\bullet$ is a 
{\it balanced dualizing complex} of $A$ in the sense of \cite{Ye} 
(the paper only concerns connect rings, but the definition 
can be generalized in the obvious way).

Easy computation shows that 
$\underline{\Hom}_A(P_v, L') \cong {}_{\delta^{-1}(v)} P$ and $\underline{\Hom}_{A^{\sf op}}
( {}_vP, L') \cong P_{\delta(v)}$ for all $v \in Q_0$. Since   
${\bf R} \underline{\Hom}_A(M^\bullet, {\mathcal D}^\bullet)$ (resp. ${\bf R} \underline{\Hom}_{A^{\sf op}}(M^\bullet, {\mathcal D}^\bullet)$) for 
$M^\bullet \in \operatorname{gr} A$ (resp. $M^\bullet \in \operatorname{gr} {\Ao}$) can be computed by a projective 
resolution of $M^\bullet$, ${\bf R} \underline{\Hom}_A(-, {\mathcal D}^\bullet)$ and ${\bf R} \underline{\Hom}_{A^{\sf op}}(-, {\mathcal D}^\bullet)$ 
give duality functors between $D^b(\operatorname{gr} A)$ and $D^b(\operatorname{gr} {\Ao})$.  
(Of course, we can also prove this by 
the same argument as \cite[Proposition~3.4]{Ye}.)


\begin{thm}[{Yekutieli \cite[Theorem 4.18]{Ye}, Martinez-Villa 
\cite[Proposition~4.6]{M98}}]\label{Serre duality}
For $M^\bullet \in D^b(\operatorname{gr} A)$, we have 
$${\bf R} \Gamma_{\mathfrak m}(M^\bullet)^\vee \cong {\bf R}\underline{\Hom}_A(M^\bullet, {\mathcal D}^\bullet).$$
In particular, $$(H^i_{\mathfrak m}(M^\bullet)_j)^* \cong \underline{\Ext}_A^{-i}(M^\bullet, {\mathcal D}^\bullet)_{-j}.$$
\end{thm}

\begin{proof}
The above result was proved by Yekutieli in the connected case. 
(In some sense, Martinez-Villa proved a more general result than ours, 
but he did not concern complexes.) But, the proof of  
\cite[Theorem 4.18]{Ye} only uses formal properties such as  
$A$ is noetherian, ${\bf R}\underline{\Hom}_{A^{\sf op}} ({\bf R}\underline{\Hom}_A(-, {\mathcal D}^\bullet), \, {\mathcal D}^\bullet) \cong \operatorname{Id}$, 
and ${\bf R}\Gamma_{\mathfrak m} {\mathcal D}^\bullet \cong A^\vee$.  
So the proof also works in our case. 
\end{proof}


\begin{dfn}[J\o rgensen, \cite{Jor04}]
For $M^\bullet \in D^b(\operatorname{gr} A)$, we say  
$$\operatorname{reg}(M^\bullet) := \sigma({\bf R} \Gamma_{\mathfrak m}(M^\bullet))
= \sup \{ \, i+j \mid H_{\mathfrak m}^i(M^\bullet)_j \ne 0 \, \}$$ 
is the {\it Castelnuovo-Mumford regularity} of $M^\bullet$. 
\end{dfn}

By Theorem~\ref{Serre duality} and the fact that 
${\bf R} \underline{\Hom}_A(M^\bullet, {\mathcal D}^\bullet) \in D^b(\operatorname{gr} {\Ao})$, 
we have $\operatorname{reg}(M^\bullet) < \infty$ for all $M^\bullet \in D^b(\operatorname{gr} A)$. 

\begin{thm}[J\o rgensen, \cite{Jor04}]\label{CM reg = Ext reg} 
If $M^\bullet \in C^b(\operatorname{gr} A)$, then 
\begin{equation}\label{reg}
\operatorname{reg}(M^\bullet) = \max \{ \, i+j \mid \beta^{i,j}(M^\bullet) \ne 0 \, \}.
\end{equation}
\end{thm}

When $A$ is a polynomial ring and $M^\bullet$ is a module, the above theorem 
is a fundamental result obtained by Eisenbud and Goto \cite{EG}. 
In the non-commutative case, 
under the assumption that $A$ is connected but not necessarily regular, 
this has been proved by J\o rgensen \cite[Corollary~2.8]{Jor04}. 
(If $A$ is not regular, we have $\operatorname{reg}(A) > 0$ in many cases.  
So one has to assume that $\operatorname{reg} A = 0$ there.) 
In our case (i.e., $A$ is AS-regular), we have a much simpler 
proof. So we will give it here. This proof is also different from one given in 
\cite{EG}. 

\begin{proof}
Set $Q^\bullet := \underline{\Hom}_A^\bullet(P^\bullet, L'(-d)[d] \, )$. Here 
$P^\bullet$ is a minimal projective resolution of $M^\bullet$, and $L'$ is the $A$-$A$ 
bimodule given in the construction of the dualizing complex ${\mathcal D}^\bullet$. 
Recall that $\underline{\Hom}_A(P_v, L') \cong {}_{\delta^{-1}(v)} P$ for all 
$v \in Q_0$. Let $s$ be the right hand side of \eqref{reg}, and 
$l$ the minimal integer 
with the property that $\beta^{l, s-l}(M^\bullet) \ne 0$. 
Then $\iota(Q^{-d-l}) = l-s+d$, and $(Q^{-d-l+1})_{\leq (l-s+d-1)} = 0$ 
(Note that $\beta^{l-1,m}(M^\bullet) = 0$ for all $m \geq s -l +1$). 
Since $Q^\bullet$ is a minimal complex, we have 
$$0 \ne H^{-d-l}(Q^\bullet)_{l-s+d} = \underline{\Ext}_A^{-d-l}(M^\bullet,{\mathcal D}^\bullet)_{l-s+d} 
= (H^{d+l}_{{\mathfrak m}}(M^\bullet)_{-l+s-d})^*.$$ 
Thus $\operatorname{reg}(M^\bullet) \geq \max \{ \, i+j \mid \beta^{i,j}(M^\bullet) \ne 0 \, \}$. 

On the other hand,  if $H^{d+l}_{{\mathfrak m}}(M^\bullet)_{-l+r-d} \ne 0$, 
we have that $\beta^{l, t-l}(M^\bullet) \ne 0$ for some $t \geq r$ by an argument 
similar to the above. Hence $\operatorname{reg}(M^\bullet) \leq 
\max \{ \, i+j \mid \beta^{i,j}(M^\bullet) \ne 0 \, \}$, and we are done. 
\end{proof}

For $M^\bullet \in D^b(\operatorname{gr} A)$, set $\mathcal H(M^\bullet)$ to be the complex 
such that $\mathcal H(M^\bullet)^i = H^i(M)$ for all $i$ 
and all differential maps are zero. 

\begin{lem}\label{degenerate} 
We have $\beta^{i,j}(\mathcal H(M^\bullet)) \geq \beta^{i,j}(M^\bullet)$ 
for all $M^\bullet \in D^b(\operatorname{gr} A)$ and 
all $i,j \in {\mathbb Z}$. In particular, $\operatorname{reg}(\mathcal H(M^\bullet)) \geq \operatorname{reg}(M^\bullet)$. 
\end{lem}

The difference between $\operatorname{reg}(M^\bullet)$ and $\operatorname{reg}(\mathcal H(M^\bullet))$ can be 
arbitrary large. In the last section, we will study 
the relation between this difference and a work of Martinez-Villa 
and Zacharia~\cite{MZ}.  

\begin{proof}
The assertion easily follows from the spectral sequence 
$$E_2^{p,q} = \underline{\Ext}_A^p(H^{-q}(N^\bullet),A_0) {\longrightarrow} \underline{\Ext}^{p+q}_A(N^\bullet,A_0).$$
\end{proof}

For a complex $M^\bullet \in C^b(\operatorname{gr} A)$ and an integer $r$, $(M^\bullet)_{\geq r}$ 
denotes the subcomplex of $M^\bullet$ whose $i^{\rm th}$ term is 
$(M^i)_{\geq (r-i)}$. 
Even if $M^\bullet \cong N^\bullet$ in $D^b(\operatorname{gr} A)$, we have $(M^\bullet)_{\geq r} \not \cong 
(N^\bullet)_{\geq r}$ in general. 

In the module case, the following is a well-known property of 
Castelnuovo-Mumford regularity. 

\begin{prop}\label{truncate}
Let $M^\bullet \in C^b(\operatorname{gr} A)$. Then $(M^\bullet)_{\geq r}$ has an 
$r$-linear resolution if and only if $r \geq \operatorname{reg}(M^\bullet)$.   
\end{prop}

To prove the proposition, we need the following lemma. 

\begin{lem}\label{artinian}
For a module $M \in \operatorname{gr} A$ with $\dim_K M < \infty$, we have 
$H^0_{\mathfrak m}(M)=M$ and $H^i_{\mathfrak m}(M)=0$ for all $i \ne 0$. 
In particular, $\operatorname{reg}(M) = \sigma(M)$ in this case. 
\end{lem} 

\begin{proof}
If $P^\bullet$ is a minimal projective resolution of $M^\vee \in \operatorname{gr} {\Ao}$, then  
$I^\bullet := (P^\bullet)^\vee$ is a minimal injective resolution of $M$. 
Since each indecomposable summand of $I^i$ is isomorphic to $I_v(n)$ for 
some $v \in Q_0$ and $n \in {\mathbb Z}$, we have $\Gamma_{\mathfrak m}(I^\bullet) = I^\bullet$.   
\end{proof}


\noindent{\it Proof of Proposition~\ref{truncate}.}
For a complex $T^\bullet \in D^b(\operatorname{gr} A)$, it is easy to see that 
$\iota(T^\bullet) = \min \{ \, i+j \mid \beta^{i,j}(T^\bullet) \ne 0 \, \}$. 
In particular, $\iota(T^\bullet) \leq \operatorname{reg}(T^\bullet)$. Hence $T^\bullet$ has an $l$-linear 
projective resolution if and only if $\iota(T^\bullet) = \operatorname{reg}(T^\bullet) = l$. 

Consider the short exact sequence of complexes 
\begin{equation}\label{find the way}
0 \to (M^\bullet)_{\geq r} \to M^\bullet \to M^\bullet/(M^\bullet)_{\geq r} \to 0, 
\end{equation} and set $N^\bullet:= M^\bullet/(M^\bullet)_{\geq r}$.  Note that 
$\dim_K H^i(N) < \infty$ for all $i$.  
By Lemmas~\ref{artinian} and \ref{degenerate}, 
we have $$r > \sigma(N^\bullet) = \max\{ \, \operatorname{reg}(H^i(N^\bullet))+i \mid i \in {\mathbb Z}  \, \} 
= \operatorname{reg}(\mathcal H(N^\bullet)) \geq \operatorname{reg}(N^\bullet).$$ 
By the long exact sequence of $\underline{\Ext}_A^\bullet(-, A_0)$ induced by 
\eqref{find the way}, we have 
\begin{eqnarray*}
r \leq \iota( (M^\bullet)_{\geq r} ) 
\leq \operatorname{reg}((M^\bullet)_{\geq r}) 
&\leq&  \max \{ \, \operatorname{reg}(N^\bullet)+1, \, \operatorname{reg}(M^\bullet) \, \} \\ 
&\leq& \max \{ \, r, \, \operatorname{reg}(M^\bullet) \, \}.
\end{eqnarray*} 
Moreover, if $r < \operatorname{reg}(M^\bullet)$ then we have $\operatorname{reg} (N^\bullet)+1 < \operatorname{reg}(M^\bullet)$ and 
$\operatorname{reg}((M^\bullet)_{\geq r}) = \operatorname{reg}(M^\bullet) > r$. 
Hence $(M^\bullet)_{\geq r}$ has an $r$-linear resolution if and only if 
$r \geq \operatorname{reg}(M^\bullet)$.  
\qed

\medskip

The following is one of the most basic 
results on Castelnuovo-Mumford regularity (see \cite{EG}). 
J\o rgensen~\cite{Jor99} proved the same result for $M \in \operatorname{gr} A$. 

\medskip

{\it Let $S = K[x_1, \ldots, x_d]$ be a polynomial ring. 
If $M \in \operatorname{gr} S$ satisfies $H_{\mathfrak m}^0(M)_{\geq r+1} = 0$ and 
$H_{\mathfrak m}^i(M)_{r+1-i}=0$ for all $i \geq 1$, then $r \geq \operatorname{reg}(M)$ 
(i.e., $H_{\mathfrak m}^i(M)_{\geq r+1-i}=0$ for all $i \geq 1$). }

\medskip

The similar result also holds for $M^\bullet \in D^b(\operatorname{gr} A)$. 
Since a minor adaptation of the proof of 
\cite[Theorem~2.4]{Jor99} also works for complexes, 
we leave the proof to the reader. 

\begin{prop} 
If $M^\bullet \in D^b(\operatorname{gr} A)$ with 
$t := \max\{ \, i \mid H^i(M^\bullet) \ne 0  \, \}$ satisfies 
\begin{itemize}
\item $H_{\mathfrak m}^i(M^\bullet)_{\geq r+1-i} = 0$ for all $i \leq t$ 
\item $H_{\mathfrak m}^i(M^\bullet)_{r+1-i} = 0$ for all $i > t$, 
\end{itemize}
then $r \geq \operatorname{reg}(M^\bullet)$ (i.e., $H_{\mathfrak m}^i(M^\bullet)_{\geq r+1-i} = 0$ for all $i > t$).
\end{prop}


\section{Koszul duality}
In this section, we study the relation between the Castelnuovo-Mumford 
regularity of complexes and the Koszul duality. 
For precise information of this duality, see \cite[\S2]{BGS}.
There, the symbol $A$ (resp. $A^!$) basically 
means a finite dimensional (resp. noetherian) Koszul algebra. 
This convention is opposite to ours. So the reader should be careful. 

\medskip

Recall that $A = KQ/I$ is a graded quiver algebra over a finite quiver $Q$. 
Let $Q^{\sf op}$ be the {\it opposite quiver} of $Q$. That is, $Q_0^{\sf op} = Q_0$ and 
there is a bijection from $Q_1$ to $Q^{\sf op}_1$ which sends an arrow 
$\alpha: v \to u$ in $Q_1$ to the arrow 
$\alpha^{\sf op} : u \to v$ in $Q_1^{\sf op}$. 
Consider the bilinear form $\< -, - \> : (KQ)_2 \times (KQ^{\sf op})_2 
\to K$ defined by 
$$
\< \alpha \beta, \gamma^{\sf op} \delta^{\sf op} \> = \begin{cases}
1 & \text{if $\alpha = \delta$ and $\beta = \gamma$,} \\
0 & \text{otherwise}
\end{cases}
$$
for all $\alpha, \beta, \gamma, \delta \in Q_1$. 
Let $I^\bot \subset KQ^{\sf op}$ be the ideal generated by 
$$\{ \, y \in (KQ^{\sf op})_2 \mid 
\text{$\<x, y\> = 0$ for all $x \in I_2$} \, \}.$$ 
We say $KQ^{\sf op}/I^\bot$ is the  {\it quadratic dual ring} of $A$, 
and denote it by $A^!$.  Clearly, $(A^!)_0 = A_0$.  
Since $A$ is Koszul, so is $A^!$ and 
$(A^!)^! \cong A$. Since $A$ is AS regular, $A^!$ is a finite dimensional 
selfinjective algebra with $A = \bigoplus_{i=0}^d A_i$ 
by \cite[Theorem~5.1]{M99}. If $A$ is a polynomial ring, then 
$A^!$ is the exterior algebra $\bigwedge (A_1)^*$.  

Since $A^!$ is selfinjective, 
${\bf D}_{A^!} := \underline{\Hom}_{A^!}(-, A^!)$ and ${\bf D}_{{(A^!)^{\sf op}}} := \underline{\Hom}_{(A^!)^{\sf op}}(-, A^!)$
give exact duality functors between $\operatorname{gr} {A^!}$ and $\operatorname{gr} {(A^!)^{\sf op}}$. 
They induce duality functors between  
$D^b(\operatorname{gr} {A^!})$ and $D^b(\operatorname{gr} {(A^!)^{\sf op}})$, which are also denoted by ${\bf D}_{A^!}$and ${\bf D}_{{(A^!)^{\sf op}}}$. 
It is easy to see that ${\bf D}_{A^!}(N) \cong \operatorname{Hom}_K(N,K)(-d)$.   


We say a complex $F^\bullet \in C(\operatorname{gr} {A^!})$ is a projective 
(resp. injective) resolution of a complex $N^\bullet \in C^b(\operatorname{gr} {A^!})$, 
if each term $F^i$ is projective ($=$ injective), 
$F^\bullet$ is right (resp. left) bounded, and there is a graded quasi isomorphism 
$F^\bullet \to N^\bullet$ (resp. $N^\bullet \to F^\bullet$). 
We say a projective (or, injective) resolution $F^\bullet \in C^b(\operatorname{gr} {A^!})$ is 
{\it minimal} if $d^i(F^i) \subset {\mathfrak n} F^{i+1}$ for all $i$, 
where ${\mathfrak n}$ is the graded Jacobson radical of $A^!$. 
(The usual definition of a minimal injective resolution is different 
from the above one. But they coincide in our case.) 
A bounded complex $N^\bullet \in C^b(\operatorname{gr} {A^!})$ has a minimal projective 
resolution and a minimal injective resolution, and they are unique 
up to isomorphism. If $F^\bullet$ is a minimal projective (resp. injective) 
resolution of $N^\bullet$ then ${\bf D}_{A^!}(F^\bullet)$ is a  minimal injective (resp. projective) 
resolution of ${\bf D}_{A^!}(N^\bullet)$.  

For $N^\bullet \in D^b(\operatorname{gr} {A^!})$, set 
$$\mu^{i,j}(N^\bullet):= \dim_K \underline{\Ext}_{A^!}^i(A_0, N^\bullet)_j.$$ 
Then $\mu^{i,j}(N^\bullet)$  measures the size of 
a minimal injective resolution of $N^\bullet$. More precisely, 
if $F^\bullet$ is a minimal injective resolution of $N^\bullet$, 
and $F^i := \bigoplus_{l =1}^m T^{i,\, l}$ is an indecomposable 
decomposition, then we have 
\begin{eqnarray*}
\mu^{i,j}(N^\bullet) &=& \# \{ \, l \mid \text{
$\operatorname{soc}(T^{i,\, l}) = (T^{i,\, l})_j$ }\, \}\\
&=& \# \{ \, l \mid  \, \text{$T^{i,\, l}(j)$ 
is isomorphic to a direct summand of $A^!(d)$} \, \}.
\end{eqnarray*} 

\medskip

Let $V$ be a finitely generated left $A_0$-module. Then  
$\operatorname{Hom}_{A_0}(A^!, V)$ is a graded  {\it left}  $A^!$-module with 
$(af)(a') = f(a'a)$ and $\operatorname{Hom}_{A_0}(A^!, V)_i = 
\operatorname{Hom}_{A_0} ((A^!)_{-i}, V)$. 
Since $A^!$ is selfinjective, we have 
$\operatorname{Hom}_{A_0}(A^!, A_0) \cong A^!(d)$. Hence $\operatorname{Hom}_{A_0}(A^!,V)$ is 
a projective (and injective) left $A^!$-module for all $V$. 
If $V$ has degree $i$ (e.g.,  $V = M_i$ for some $M \in \operatorname{gr} A$), then 
we set  $\operatorname{Hom}_{A_0}(A^!, V)_j = \operatorname{Hom}_{A_0}(A^!_{-j-i}, V)$. 

For $M^\bullet \in C^b(\operatorname{gr} A)$, let 
${\mathcal G}(M^\bullet):= \operatorname{Hom}_{A_0}(A^!,M^\bullet) \in C^b(\operatorname{gr} {A^!})$ be  
the total complex of the double complex with 
${\mathcal G}(M^\bullet)^{i,j} =  \operatorname{Hom}_{A_0}(A^!, M^i_j)$ whose vertical and horizontal 
differentials $d'$ and $d''$ are defined by 
$$d'(f)(x)= \sum_{\alpha \in Q_1} \alpha f(\alpha^{\sf op} x),  
\qquad \quad d''(f)(x) = \partial_{M^\bullet}(f(x))$$
for $f \in \operatorname{Hom}_{A_0}(A^!, M^i_j)$ and $x \in A^!$.  
The gradings of ${\mathcal G}(M^\bullet)$ is given by 
$${\mathcal G}(M^\bullet)^p_q := \bigoplus_{p=i+j, \, q = -l-j} 
\operatorname{Hom}_{A_0}((A^!)_l, M^i_j).$$ 

Each term of ${\mathcal G}(M^\bullet)$ is injective. 
For a module $M \in \operatorname{gr} A$, ${\mathcal G}(M)$ is a minimal complex. 
Thus we have 
\begin{equation}
\mu^{i,j}({\mathcal G}(M)) = \begin{cases} 
\dim_K M_i & \text{if $i+j = 0$,}\\
0 & \text{otherwise.}
\end{cases}
\end{equation}



Similarly, for a complex $N^\bullet \in C^b(\operatorname{gr} {A^!})$, we can define a new complex 
${\mathcal F}(N^\bullet):= A \otimes_{A_0} N^\bullet \in C^b(\operatorname{gr} A)$ as the total complex 
of the double complex with 
${\mathcal F}(N^\bullet)^{i,j} =  A \otimes_{A_0} N^i_j$ whose vertical and horizontal 
differentials $d'$ and $d''$ are defined by 
$$d'(a \otimes x)= \sum_{\alpha \in Q_1} a \alpha \otimes  \alpha^{\sf op} x, 
\qquad d''(a \otimes x)=  a \otimes \partial_{N^\bullet}(x) $$
for $a \otimes x \in A \otimes_{A_0} N^i$. The gradings of ${\mathcal F}(N^\bullet)$
is given by 
$${\mathcal F}(N^\bullet)^p_q:= \bigoplus_{p=i+j, \, q = l-j} A_l \otimes_{A_0} N^i_j.$$
Clearly, each term of ${\mathcal F}(N^\bullet)$ is a projective $A$-module.  
For a module 
$N \in \operatorname{gr} {A^!}$,  ${\mathcal F}(N)$ is a minimal complex. Hence we have 
\begin{equation}\label{betti of DF}
\beta^{i,j}({\mathcal F}(N)) = \begin{cases} 
\dim_K N_i & \text{if $i+j = 0$,}\\
0 & \text{otherwise.}
\end{cases}
\end{equation}


It is well-known that the operations ${\mathcal F}$ and ${\mathcal G}$ 
define functors ${\mathcal F} : D^b(\operatorname{gr} {A^!}) \to D^b(\operatorname{gr} A)$ and 
${\mathcal G}:  D^b(\operatorname{gr} A) \to D^b(\operatorname{gr} {A^!})$, and they give an equivalence 
$D^b(\operatorname{gr} A) \cong D^b(\operatorname{gr} {A^!})$ of triangulated categories. 
This equivalence is called the {\it Koszul duality}. 
When $A$ is a polynomial ring, 
this equivalence is called {\it Bernstein-Gel'fand-Gel'fand 
correspondence}. See, for example, \cite{EFS}. 

Note that $(A^{\sf op})^! \cong (A^!)^{\sf op}$. 
We have the functors ${\mathcal F}^{\sf op} : D^b(\operatorname{gr} {(A^!)^{\sf op}}) \to D^b(\operatorname{gr} {\Ao})$ 
and ${\mathcal G}^{\sf op}: D^b(\operatorname{gr} {\Ao}) \to D^b(\operatorname{gr} {(A^!)^{\sf op}})$ giving  
$D^b(\operatorname{gr} {\Ao}) \cong D^b(\operatorname{gr} {(A^!)^{\sf op}})$. 


\begin{prop}[{c.f. \cite[Proposition~2.3]{EFS}}]
\label{generalization of EFS} In the above situation, we have 
$$\beta^{i,j}(M^\bullet) = \dim_K H^{i+j}({\mathcal G}(M^\bullet))_{-j} \quad  \text{and} \quad  
\mu^{i,j}(N^\bullet) = \dim_K H^{i+j}({\mathcal F}(N^\bullet))_{-j}.$$
\end{prop}

\begin{proof} 
While the assertion follows from Proposition~\ref{strand} below, 
we give a direct proof here. We have 
\begin{eqnarray*}
\underline{\Ext}_{A^!}^i(A_0, N^\bullet)_j &\cong& 
\operatorname{Hom}_{D^b(\operatorname{gr} {A^!})}(A_0, N^\bullet[i](j))\\
&\cong& \operatorname{Hom}_{D^b(\operatorname{gr} A)}( \, {\mathcal F}(A_0), \, {\mathcal F}(N^\bullet[i](j)) \, )\\ 
&\cong& \operatorname{Hom}_{D^b(\operatorname{gr} A)}( \, A, \, {\mathcal F}(N^\bullet)[i+j](-j) \, )\\
&\cong& H^{i+j}({\mathcal F}(N^\bullet))_{-j}. 
\end{eqnarray*}
Since $\mu^{i,j}(N^\bullet)= \dim_K \underline{\Ext}_{A^!}^i(A_0, N^\bullet)_j$,  
the second equation of the proposition follows. 
We can prove the first equation by a similar argument. 
But in this time we use the contravariant functor 
${\bf D}_{A^!} \circ {\mathcal G} : D^b(\operatorname{gr} A) \to D^b(\operatorname{gr} {(A^!)^{\sf op}})$ 
and the fact that ${\bf D}_{A^!} \circ {\mathcal G}(A_0) \cong 
{\bf D}_{A^!}(A^!(d)) \cong A^!(-d)$. 
\end{proof}

\begin{cor}\label{reg & DG}
$\operatorname{reg}(M^\bullet) = \max \{ \, i \mid  H^i({\mathcal G}(M^\bullet)) \ne 0 \, \}$.
\end{cor}

\begin{proof}
Follows Theorem~\ref{CM reg = Ext reg} 
and Proposition~\ref{generalization of EFS}.
\end{proof}

Recall that ${\bf D}_A := {\bf R} \underline{\Hom}_A(-,{\mathcal D}^\bullet)$ is a duality functor 
from $D^b(\operatorname{gr} A)$ to $D^b(\operatorname{gr} {\Ao})$. 

\begin{prop}\label{reg of dual}
$\operatorname{reg}( \, {\bf D}_A(M^\bullet) \, ) = - \min \{ \, i \mid  H^i({\mathcal G}(M^\bullet)) \ne 0 \, \}$. 
\end{prop}

\begin{proof}
Let $L'$ be the $A$-$A$ 
bimodule given in the construction of the dualizing complex ${\mathcal D}^\bullet$. 
Note that ${\bf D}_A(M^\bullet) \cong \underline{\Hom}_A^\bullet(P^\bullet,L'(-d)[d]) =:Q^\bullet$ 
for a projective resolution $P^\bullet$ of $M^\bullet$. 
Since ${\bf D}_A(P_v) = {}_{\delta^{-1}(v)} P(-d)[d]$, $Q^\bullet$ 
is a complex of projectives. And $Q^\bullet$ 
is a minimal complex if and only if $P^\bullet$ is. Hence 
$\beta^{-i-d, -j+d}({\bf D}_A(M^\bullet)) = \beta^{i,j}(M^\bullet)$. Therefore, the assertion 
follows from Proposition~\ref{generalization of EFS}. 
\end{proof}

We can refine Proposition~\ref{generalization of EFS} using 
the notion of {\it linear strands} of projective (or injective) resolutions, 
which was introduced by Eisenbud et. al. (See \cite[\S3]{EFS}.) 
First, we will generalize this notion to our rings. 
Let $B$ be a noetherian Koszul algebra (e.g. $B= A$ or $A^!$) 
with the graded Jacobson radical ${\mathfrak m}$, 
and $P^\bullet$ a {\it minimal} projective resolution of a bounded complex 
$M^\bullet \in D^b(\operatorname{gr} B)$.  
Consider the decomposition $P^i := \bigoplus_{j \in {\mathbb Z}} P^{i,j}$ 
such that any indecomposable summand of $P^{i,j}$ 
is isomorphic to a summand of $B(-j)$. 
For an integer $l$, we define the {\it $l$-linear strand} 
$\operatorname{proj.lin}_l(M^\bullet)$ of a projective resolution of 
$M^\bullet$ as follows: The term $\operatorname{proj.lin}_l(M^\bullet)^i$ of cohomological degree $i$  
is $P^{i, l-i}$ and the differential 
$P^{i, l-i} \to P^{i+1, l-i-1}$ 
is the corresponding component of the differential 
$P^i \to P^{i+1}$ of $P^\bullet$.  So the differential of $\operatorname{proj.lin}_l(M^\bullet)$ 
is  represented by a matrix whose entries are elements in $A_1$. 
Set $\operatorname{proj.lin}(M^\bullet) := \bigoplus_{l \in {\mathbb Z}} \operatorname{proj.lin}_l(M^\bullet)$. 
It is obvious that $\beta^{i,j}(M^\bullet) = \beta^{i,j}(\operatorname{proj.lin}(M^\bullet))$ 
for all $i,j$. 

Using spectral sequence argument, we can construct $\operatorname{proj.lin} (M^\bullet)$ from  
a (not necessarily minimal) projective resolution $Q^\bullet$ of $M^\bullet$. 
Consider the ${\mathfrak m}$-adic filtration 
$Q^\bullet = F_0Q^\bullet \supset F_1 Q^\bullet \supset \cdots$ of 
$Q^\bullet$ with $F_p Q^i = {\mathfrak m}^p Q^i$ and the associated spectral 
sequence $\{E_r^{*,*},d_r \}$. The associated graded object 
$\operatorname{gr}_\m M := \bigoplus_{p \geq 0} {\mathfrak m}^p 
M/{\mathfrak m}^{p+1}M$ of $M \in \operatorname{gr} B$ is a module over 
$\operatorname{gr}_\m B = \bigoplus_{p \geq 0} {\mathfrak m}^p/{\mathfrak m}^{p+1} \cong B$. 
Since ${\mathfrak m}^p M$ is a graded submodule of $M$, we can make $\operatorname{gr}_\m M$ 
a graded module using the original grading of $M$ 
(so $(\operatorname{gr}_\m M)_i$ is {\it not} ${\mathfrak m}^i M /{\mathfrak m}^{i+1} M$ here). 
Under the identification $\operatorname{gr}_\m B$ with $B$,  
we have $\operatorname{gr}_\m M \not \cong M$ in general. 
But if each indecomposable summand $N$ of $M$ is generated by 
$N_{\iota(N)}$ then $\operatorname{gr}_\m M \cong M$. 
Since $Q^t$ is a projective $B$-module, $Q_0^t := \bigoplus_{p+q=t}
E_0^{p,q} = \bigoplus_{p \geq 0} {\mathfrak m}^p Q^t / {\mathfrak m}^{p+1} Q^t = \operatorname{gr}_\m Q^t$ 
is isomorphic to $Q^t$. 
The maps $d_0^{p,q} : E_0^{p,q} \to E_0^{p,q+1}$ make $Q^\bullet_0$ 
a cochain complex of projective $\operatorname{gr}_\m B$-modules.  
Consider the decomposition $Q^\bullet = P^\bullet \oplus C^\bullet$, where $P^\bullet$ is minimal 
and $C^\bullet$ is exact. (We always have such a decomposition.) 
If we identify $Q_0^t$ with 
$Q^t = P^t \oplus C^t$, the differential $d_0$ of $Q^\bullet_0$ is given by 
$(0, d_{C^\bullet})$. 
Hence we have $Q_1^t = \bigoplus_{p+q=t}E_1^{p,q} \cong P^t$. 
The maps $d_1^{p,q}: E_1^{p,q} = {\mathfrak m}^p P^t/{\mathfrak m}^{p+1} P^t  
\to E_1^{p+1,q} = {\mathfrak m}^{p+1} P^{t+1}/{\mathfrak m}^{p+2} P^{t+1}$ make 
$Q^\bullet_1$ a cochain complex of projective $\operatorname{gr}_\m B (\cong B)$-modules 
whose differential is the ``linear component" of the differential 
$d_{P^\bullet}$ of $P^\bullet$. Thus the complex 
$(Q_1^\bullet, d_1)$ is isomorphic to  $\operatorname{proj.lin} (M^\bullet)$. 

\medskip

Since $A^!$ is selfinjective, we can consider the linear strands 
of an injective resolution. More precisely, 
starting from a minimal injective resolution of $N^\bullet \in D^b(\operatorname{gr} {A^!})$, 
we can construct its $l$-linear strand 
$\operatorname{inj.lin}_l(N^\bullet)$  in a similar way. Here, if $I^i$ 
is the cohomological degree $i^{\rm th}$ term of $\operatorname{inj.lin}_l(N^\bullet)$, 
then the socle of $I^i$ coincides with $(I^i)_{l-i}$. In other words, 
any indecomposable summand of $I^i$ 
is isomorphic to a summand of $A^!(i-l+d)$.  
Set $\operatorname{inj.lin} (N^\bullet) = \bigoplus_{l \in {\mathbb Z}} \operatorname{inj.lin}_l(N^\bullet)$. 
This complex can also be constructed using spectral sequences. 

We have that  ${\bf D}_{A^!}(\operatorname{inj.lin}(N^\bullet)) \cong \operatorname{proj.lin}({\bf D}_{A^!}(N^\bullet))$ 
and ${\bf D}_{A^!}(\operatorname{proj.lin}(N^\bullet)) \cong \operatorname{inj.lin}({\bf D}_{A^!}(N^\bullet))$. 

\begin{prop}[{c.f. \cite[Corollary~3.6]{EFS}}]\label{strand} 
For  $M^\bullet \in D^b(\operatorname{gr} A)$ and $N^\bullet \in D^b(\operatorname{gr} {A^!})$, 
we have $$\operatorname{proj.lin} ({\mathcal F}(N^\bullet)) = {\mathcal F}(\mathcal H(N^\bullet)) \quad  \text{and} \quad   
\operatorname{inj.lin} ({\mathcal G}(M^\bullet)) = {\mathcal G}(\mathcal H(M^\bullet)).$$ More precisely, $$\operatorname{proj.lin}_l ({\mathcal F}(N^\bullet)) = 
{\mathcal F}(H^l(N^\bullet))[-l] \quad  \text{and} \quad  
\operatorname{inj.lin}_l({\mathcal G}(M^\bullet)) = {\mathcal G}(H^l(M^\bullet))[-l].$$ 
\end{prop}

\begin{proof}
Set $Q^\bullet = {\mathcal F}(N^\bullet)$. Note that $Q^\bullet$ is a (non minimal) complex 
of projective modules. We use the above spectral sequence argument 
(and the notation there). Then the differential  
$d_0^t : Q_0^t \cong {\mathcal F}^t(N^\bullet) \to Q_0^{t+1} \cong 
{\mathcal F}^{t+1}(N^\bullet)$ is given by $\pm \partial_{N^\bullet}$. Thus  
$$Q_1^t \cong \bigoplus_{t = i+j} A \otimes_{A_0} H^i(N^\bullet)_j 
= \bigoplus_{t = i+j}{\mathcal F}^j(H^i(N^\bullet)),$$ 
and the differential of $Q^\bullet_1$ is induced by that of 
${\mathcal F}(N^i)$. Hence we can easily check that $Q^\bullet_1$, 
which can be identified with $\operatorname{proj.lin} ({\mathcal F}(N^\bullet))$, is isomorphic to 
${\mathcal F}(\mathcal H(N^\bullet)) \cong \bigoplus_{i \in {\mathbb Z}} {\mathcal F}(H^i(N^\bullet))[-i]$. 
We can prove the statement for $\operatorname{inj.lin} ({\mathcal G}(M^\bullet))$ in the same way.  
\end{proof}




\section{Weakly Koszul Modules}

Let $B$ be a noetherian Koszul algebra (e.g. $B= A$ or $A^!$) 
with the graded Jacobson radical ${\mathfrak m}$. For $M \in \operatorname{gr} B$ and 
an integer $i$, $M_{\< i \>}$ denotes the submodule of $M$ generated by 
its degree $i$ component $M_i$.


\begin{prop}\label{my old result}
In the above situation, the following are equivalent. 
\begin{itemize}
\item[(1)] $M_{\< i \>}$ has a linear projective resolution for all $i$. 
\item[(2)] $H^i(\operatorname{proj.lin}(M)) = 0$ for all $i \ne 0$. 
\item[(3)] All indecomposable summands of 
$\operatorname{gr}_\m M$ have linear resolutions as $B \, (\cong \operatorname{gr}_\m B)$-modules.  
\end{itemize}
\end{prop}


\begin{proof}
This result was proved in \cite[Proposition~4.9]{Y} under the assumption 
that $B$ is a polynomial ring. (R\"omer also proved this for a 
commutative  Koszul algebra. See \cite[Theorem~3.2.8]{R02}.)
In this proof, only the Koszul property of a polynomial ring 
is essential, and the proof also works in our case. But, to refer this, 
the reader should be careful with the following points. 

\smallskip

(a) In \cite{Y}, the grading of $\operatorname{gr}_\m M$ is given by a different way. 
There, $(\operatorname{gr}_\m M)_i = {\mathfrak m}^i M /{\mathfrak m}^{i+1} M$. It is easy to see that $\operatorname{gr}_\m M$ 
has a linear resolution in this grading if and only if 
the condition (3) of the proposition is satisfied in our grading.  

\smallskip

(b) In the proof of \cite[Proposition~4.9]{Y},  
the regularity $\operatorname{reg}(N)$ of $N \in \operatorname{gr} B$ is  an important tool. 
Unless $B$ is AS regular, one cannot define $\operatorname{reg}(N)$ 
using the local cohomologies of $N$. But if we set 
$\operatorname{reg}(N) := \sup \{ \, i+j \mid  \beta^{i,j} (N) \ne 0 \, \},$ 
then everything works well. It is not clear whether 
$\operatorname{reg}(N) < \infty$ for all $N \in \operatorname{gr} B$ (c.f. \cite{Jor04}). 
But modules appearing in the argument similar to 
the proof of \cite[Proposition~4.9]{Y} have finite regularities. 

\smallskip

(c) In the proof of \cite[Proposition~4.9]{Y}, a few basic properties 
of the Castelnuovo-Mumford regularity (over a polynomial ring) are used. 
But $\operatorname{reg}(N)$ of $N \in \operatorname{gr} B$ also has these properties, if 
we define $\operatorname{reg}(N)$ as (b). 
For example, if $N \in \operatorname{gr} B$ satisfies $\dim_K N < \infty$, then 
$\operatorname{reg}(N) = \sigma(N)$. This can be proved by induction on 
$\dim_K N$. Using the short exact sequence $0 \to N_{\geq r} \to 
N \to N/N_{\geq r} \to 0$, we can also prove that $N_{\geq r}$ 
has an $r$ linear resolution if and only if $r \geq \operatorname{reg}(N)$ 
(see also Proposition~\ref{truncate}).  

\smallskip

(d) For the implication $(2) \Rightarrow (3)$, \cite{Y} refers another paper. 
But this implication can be proved by spectral sequence argument, 
since $\operatorname{proj.lin}(M)$ can be constructed using a spectral sequence 
as we have seen in the previous section. 
\end{proof}

\begin{dfn}[{\cite{GM, MZ}}]
In the above situation, we say $M \in \operatorname{gr} B$ is {\it weakly Koszul}, 
if it satisfies the equivalent conditions of Proposition~\ref{my old result}. 
\end{dfn}


\begin{rem}\label{wKos remarks}
(1) If $M \in \operatorname{gr} B$ has a linear resolution, then it is weakly Koszul. 

(2) The notion of weakly Koszul modules was first introduced by 
Green and Martinez-Villa \cite{GM}. But they used the 
name ``strongly quasi Koszul modules". 
Weakly Koszul modules are also called ``componentwise 
linear modules" by some commutative algebraists (see \cite{HH}). 
\end{rem}

\begin{thm}\label{wKos} 
Let $0 \ne N \in \operatorname{gr} {A^!}$ and set $N' := {\bf D}_{A^!}(N)$.  
Then the following are equivalent. 
\begin{itemize}
\item[(1)] $N$ is weakly Koszul.  
\item[(2)] $H^i({\mathcal F}^{\sf op} (N'))$ has a $(-i)$-linear 
projective resolution for all $i$. 
\item[(3)] $\operatorname{reg}( \, \mathcal H \circ {\mathcal F}^{\sf op} (N') \,) = 0$. 
\item[(4)] $\operatorname{reg}( \, \mathcal H \circ {\mathcal F}^{\sf op} (N') \,) \leq 0$.
\end{itemize}
\end{thm}


\begin{proof}
Since $\iota(\mathcal H \circ {\mathcal F}^{\sf op} (N'))) \geq 0$ (i.e., 
$\iota(H^i( {\mathcal F}^{\sf op} (N'))) \geq -i$ for all $i$), 
the equivalence among (2), (3) and (4) follows from 
Proposition~\ref{truncate}. So it suffices to prove 
$(1) \Leftrightarrow (4)$. Since ${\bf D}_{(A^!)^{\sf op}}(\operatorname{inj.lin}(N')) \cong \operatorname{proj.lin}(N)$, 
$N$ is weakly Koszul if and only if 
$H^i(\operatorname{inj.lin}( N'))=0$ for all $i > 0$. 
By Proposition~\ref{strand}, we have 
$$\operatorname{inj.lin}(N') = \operatorname{inj.lin}({\mathcal G}^{\sf op} \circ {\mathcal F}^{\sf op} (N')) 
= {\mathcal G}^{\sf op} \circ \mathcal H \circ  {\mathcal F}^{\sf op} (N').$$
Therefore, by Corollary~\ref{reg & DG},
$H^i(\operatorname{inj.lin}( N'))=0$ for all $i > 0$ 
if and only if the condition (4) holds.  
\end{proof}

\begin{rem}
Martinez-Villa and Zacharia proved that if $N$ is weakly Koszul then 
there is a filtration 
$$U_0 \subset U_1 \subset \cdots \subset U_p = N$$ 
such that $U_{i+1}/U_i$ has a linear resolution for each $i$ 
(see \cite[pp. 676--677]{MZ}). 
We can interpret this fact using Theorem~\ref{wKos} in our case. 

Let $N \in \operatorname{gr} {A^!}$ be a weakly Koszul module. Set $N' := {\bf D}_{A^!}(N)$ and 
$T^\bullet := {\mathcal F}^{\sf op}(N')$.  Assume that $N$ does not have a linear 
resolution. Then $H^i(T^\bullet) \ne 0$ for several $i$. 
Set $n = \min \{ \, i \mid H^i(T^\bullet) \ne 0 \, \}$. 
Consider the truncation
$$\sigma_{> n} T^\bullet : \cdots {\longrightarrow} 0 {\longrightarrow} \operatorname{im}d^n 
{\longrightarrow} T^{n+1} {\longrightarrow} T^{n+2} {\longrightarrow} \cdots$$ of $T^\bullet$. 
Then we have $H^i(T^\bullet) = H^i(\sigma_{> n} T^\bullet )$ for all $i > n$ and 
$H^i(\sigma_{> n} T^\bullet ) = 0$ for all $i \leq n$. 
We have a triangle 
\begin{equation}\label{triangle}
H^n(T^\bullet)[-n] \to T^\bullet \to \sigma_{> n} T^\bullet \to H^n(T^\bullet)[-n+1].
\end{equation} 
By Theorem~\ref{wKos}, $H^n(T^\bullet)[-n]$ has a 0-linear resolution.  
On the other hand, $$0 = \operatorname{reg} ( \, \mathcal H(\sigma_{> n} T^\bullet) \, ) \geq  
\operatorname{reg} (\sigma_{> n} T^\bullet) \geq \iota(\sigma_{> n} T^\bullet) \geq 0.$$ 
Hence $\sigma_{> n} T^\bullet$ also has a 0-linear resolution.  Therefore, both 
${\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} (\sigma_{> n} T^\bullet)$ and 
${\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} ( H^n(T^\bullet)[-n]  )$ are acyclic complexes 
(that is, the $i^{\rm th}$ cohomology vanishes for all $i \ne 0$).  
Set $$U := H^0( \, {\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} (\sigma_{> n} T^\bullet) \, ) 
\quad  \text{and} \quad 
V := H^0( \, {\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} ( H^n(T^\bullet)[-n]  ) \, ).$$  
Since $N = {\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} (T^\bullet)$, the triangle 
\eqref{triangle} induces a short exact sequence 
$0 \to U \to N \to V \to 0$ in $\operatorname{gr} {A^!}$. 
It is easy to see that $V$ has a linear resolution.  
Since $\mathcal H \circ {\mathcal F}^{\sf op} \circ {\bf D}_{A^!} (U) = 
\mathcal H(\sigma_{> n} T^\bullet)$, $U$ is weakly Koszul by 
Theorem~\ref{wKos}. Repeating this procedure, we can get the expected 
filtration.  
\end{rem}

Let $N \in \operatorname{gr} {A^!}$ and $\cdots \stackrel{f_2}{{\longrightarrow}} 
P^{-1} \stackrel{f_1}{{\longrightarrow}} P^0 \stackrel{f_0}{{\longrightarrow}} N \to 0$ 
its minimal projective resolution. For $i \geq 1$, 
we call $\Omega_i(N) := \ker (f_{i-1})$ 
the $i^{\rm th}$ {\it syzygy} of $N$. 
Note that $\Omega_i(N) = \operatorname{im}(f_i) = \operatorname{coker}(f_{i+1})$. 

By the original definition of a weakly Koszul module given in \cite{GM, MZ}, 
if $N \in \operatorname{gr} {A^!}$ is weakly Koszul then so is $\Omega_i(N)$ for all 
$i \geq 1$. 

\begin{dfn}[Herzog-R\"omer, \cite{R02}]
For $0 \ne N \in \operatorname{gr} {A^!}$, set
$$\operatorname{lpd}(N):= \inf\{ \, i \in {\mathbb N}  \mid  \text{$\Omega_i(N)$ is 
weakly Koszul} \, \},$$
and call it the {\it linear part dominates} of $N$. 
\end{dfn}

Since $A \, (\cong (A^!)^!)$ 
is a noetherian ring of finite global dimension,   
$\operatorname{lpd}(N)$ is finite for all $N \in \operatorname{gr} {A^!}$ by \cite[Theorem~4.5]{MZ}.  

\begin{thm}\label{wKos=regH}
Let $N \in \operatorname{gr} {A^!}$ and set $N' := {\bf D}_{A^!}(N)$.   Then we have 
\begin{eqnarray*}
\operatorname{lpd}(N) &=& \operatorname{reg}( \, \mathcal H \circ {\mathcal F}^{\sf op} (N') \,)\\
&=& \max 
\{ \, \operatorname{reg}(H^i({\mathcal F}^{\sf op}(N') )) +i \mid i \in {\mathbb Z} \, \}.  
\end{eqnarray*}
\end{thm}

\begin{proof}
Note that $P^\bullet := {\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} \circ {\mathcal F}^{\sf op}(N')$ 
is a projective resolution of $N$, and 
$Q^\bullet := {\bf D}_{(A^!)^{\sf op}} \circ {\mathcal G}^{\sf op} ({\mathcal F}^{\sf op}(N')_{\geq i})$ is the truncation 
$\cdots \to P^{-i-1} \to P^{-i} \to 0 \to \cdots $ 
of $P^\bullet$ for each $i \geq 1$. 
Hence we have $H^{j}(Q^\bullet) = 0$ for all $j \ne -i$ and 
there is a projective module $P$ such that 
$H^{-i}(Q^\bullet) \cong \Omega_i(N) \oplus P$. 
Since $P$ is weakly Koszul, $\Omega_i(N)$  is weakly 
Koszul if and only if so is $Q := H^{-i}(Q^\bullet)$. We have 
$$\operatorname{proj.lin} (Q)[i] \cong {\bf D}_{{(A^!)^{\sf op}}} \circ {\mathcal G}^{\sf op} \circ \mathcal H 
({\mathcal F}^{\sf op}(N')_{\geq i}).$$
By Theorem~\ref{wKos}, $Q$ is weakly Koszul if and only if 
$\mathcal H ({\mathcal F}^{\sf op}(N')_{\geq i})$ has an $i$-linear resolution, that is, 
$H^j ({\mathcal F}^{\sf op}(N')_{\geq i})$ has an 
$(i-j)$-linear resolution for all $j$. But there is 
some $L \in \operatorname{gr} {(A^!)^{\sf op}}$ such that $L = L_{i-j}$ and 
$H^j ({\mathcal F}^{\sf op}(N')_{\geq i}) \cong H^j({\mathcal F}^{\sf op}(N'))_{\geq i-j} \oplus L$. 
Note that $L$ has an $(i-j)$-linear resolution. 
Therefore, $H^j ({\mathcal F}^{\sf op}(N')_{\geq i})$ has an 
$(i-j)$-linear resolution if and only if so does 
$H^j({\mathcal F}^{\sf op}(N'))_{\geq i-j}$. Summing up the above facts, we have that 
$\Omega_i(N)$ is weakly Koszul if and only if 
$(\mathcal H \circ {\mathcal F}^{\sf op}(N'))_{\geq i}$ has an $i$-linear resolution.   
By Proposition~\ref{truncate}, the last condition is 
equivalent to the condition that $i \geq \operatorname{reg} (\mathcal H \circ {\mathcal F}^{\sf op}(N'))$. 
\end{proof}

\begin{rem}
Assume that $A$ is noetherian, Koszul, and has finite global dimension, 
but not necessarily AS regular. Then $A^!$ is a finite dimensional Koszul  
algebra, but not necessarily selfinjective. 
Even in this case, ${\mathcal G}(M^\bullet)$ for $M^\bullet \in D^b(\operatorname{gr} A)$ 
is a complex of injective $A^!$-modules, 
and the results in \S3 and Theorem~\ref{wKos=regH} also hold. 
But now we should set 
$\operatorname{reg}(M^\bullet) := \sup \{ \, i+j \mid  \beta^{i,j} (M^\bullet) \ne 0 \, \}$  
for $M^\bullet \in D^b(\operatorname{gr} A)$ (local cohomology is not 
helpful to define the regularity). 
Since  $A$ is noetherian and has finite global 
dimension, we have $\operatorname{reg}(M^\bullet) < \infty$ for all $M^\bullet$.
In particular, we have $\operatorname{lpd}(N) < \infty$ for all $N \in \operatorname{gr} {A^!}$ 
(if $A$ is {\it right} noetherian) as proved in \cite[Theorem~4.5]{MZ}. 
\end{rem}

If $\operatorname{lpd}(N) \geq 1$ for some $N \in \operatorname{gr} {A^!}$, then 
$\sup \{ \, \operatorname{lpd}(T) \mid  T \in \operatorname{gr} {A^!} \, \} = \infty$. 
In fact, if $\Omega_{-i}(N)$ is the $i^{\rm th}$ {\it cosyzygy} of $N$ 
(since $A^!$ is selfinjective, we can consider cosyzygies), 
then $\operatorname{lpd}(\Omega_{-i}(N)) > i$. 
But Herzog and R\"omer proved that if $J$ is a monomial ideal 
of an exterior algebra $E =  \bigwedge \< y_1, \ldots, y_d \>$ 
then $\operatorname{lpd} (E/J) \leq d-1$ (c.f. \cite[\S3.3]{R02}).  
We will refine their results using Theorem~\ref{wKos=regH}. 

\medskip

In the sequel, we regard the polynomial ring 
$S=K[x_1, \ldots, x_d]$, $d \geq 1$, as an ${\mathbb N}^d$-graded ring 
with $\deg x_i = (0, \ldots, 0,1,0, \ldots, 0)$ where 1 is at the $i^{\rm th}$ 
position. Similarly, the exterior algebra $E = S^!= \bigwedge
\<y_1, \ldots, y_d \>$ is also an  ${\mathbb N}^d$-graded ring. 
Let $\operatorname{*Gr} S$ be the category of ${\mathbb Z}^d$-graded $S$-modules and their 
degree preserving $S$-homomorphisms, 
and $\operatorname{*gr} S$ its full subcategory consisting 
of finitely generated modules. We have a similar category $\operatorname{*gr} E$ for $E$. 
For $S$-modules and graded $E$-modules, we do not have to 
distinguish left modules from right modules. Since ${\mathbb Z}^d$-graded modules 
can be regarded as ${\mathbb Z}$-graded modules in the natural way, 
we can discuss $\operatorname{reg}(M^\bullet)$ for $M^\bullet \in D^b(\operatorname{*gr} S)$ and 
$\operatorname{lpd}(N)$ for $N \in \operatorname{*gr} E$. 

Note that ${\bf D}_E(-) = \bigoplus_{{\bf a} \in {\mathbb Z}^d} 
\operatorname{Hom}_{\operatorname{*gr} E}(-,E({\bf a}))$ gives an exact duality functor 
from $\operatorname{*gr} E$ to itself. Sometimes, we simply denote ${\bf D}_E(N)$ by $N'$. 
Set ${\mathbf 1} := (1, 1, \ldots, 1) \in {\mathbb Z}^d$. Then 
${\mathcal D}^\bullet_S := S(-{\mathbf 1})[d] \in D^b(\operatorname{*gr} S)$ is a ${\mathbb Z}^d$-graded 
normalized dualizing complex and ${\bf D}_S(-):= {\bf R} \underline{\Hom}_S(-,{\mathcal D}^\bullet_S) 
= \bigoplus_{{\bf a} \in {\mathbb Z}^d} {\bf R} \operatorname{Hom}_{\operatorname{*Gr} S} (-, {\mathcal D}^\bullet_S({\bf a}))$ 
gives a duality functor from $D^b(\operatorname{*gr} S)$ to itself. 
As shown in \cite[Theorem~4.1]{Y5}, we have the ${\mathbb Z}^d$-graded 
Koszul duality functors ${\mathcal F}^*$ and ${\mathcal G}^*$ giving an equivalence 
$D^b(\operatorname{*gr} S) \cong D^b(\operatorname{*gr} E)$. These functors are defined in the same 
way as in the ${\mathbb Z}$-graded case. 

For ${\bf a} = (a_1, \ldots, a_d)\in {\mathbb Z}^d$, 
set $\operatorname{supp} ({\bf a}) := \{i  \mid a_i > 0\} \subset [d] := \{1, \ldots, d \}$. 
We say  ${\bf a} \in {\mathbb Z}^d$ is {\it squarefree} if $a_i= 0,1$ for all $i \in [d]$. 
When ${\bf a} \in {\mathbb Z}^d$ is squarefree, we sometimes identify ${\bf a}$ with  
$\operatorname{supp} ({\bf a})$. For example, if $F \subset [d]$, then 
$S(-F)$ means the free module $S(-{\bf a})$, where ${\bf a} \in {\mathbb N}^d$ 
is the squarefree vector with $\operatorname{supp}({\bf a}) = F$. 

\begin{dfn}[\cite{Y}]
We say $M \in \operatorname{*gr} S$ is {\it squarefree}, 
if $M$ has a presentation of the form 
$$\bigoplus_{F \subset [d]} S(-F)^{m_F} \to 
\bigoplus_{F \subset [d]} S(-F)^{n_F} \to M \to 0$$
for some $m_F, n_F \in {\mathbb N}$. 
\end{dfn}

The above definition seems different from 
the original one given in \cite{Y}, but they coincide. 
Stanley-Reisner rings (that is, the quotient rings of $S$ by 
squarefree monomial ideals) and many modules related to them 
are squarefree. 
Here we summarize basic properties of squarefree 
modules. See \cite{Y, Y5} for further information. 
Let $\operatorname{Sq}(S)$ be the full subcategory of $\operatorname{*gr} S$ consisting of 
squarefree modules. 
Then $\operatorname{Sq}(S)$ is closed under kernels, cokernels, and 
extensions in $\operatorname{*gr} S$. Thus $\operatorname{Sq}(S)$ is an abelian category. Moreover, we have  
$D^b(\operatorname{Sq}(S)) \cong D^b_{\operatorname{Sq}(S)}(\operatorname{*Gr} S)$. 
If $M$ is squarefree, then each term in a 
${\mathbb Z}^d$-graded minimal free resolution of $M$ is of the form 
$\bigoplus_{F \subset [d]} S(-F)^{n_F}$. Hence we have $\operatorname{reg}(M) \leq  d$. 
Moreover, $\operatorname{reg}(M) = d$ if and only if $M$ has a summand 
which is isomorphic to $S(-{\mathbf 1})$. 

\begin{dfn}[R\"omer~\cite{R0}]\label{sqE}
We say $N \in \operatorname{*gr} E$ is 
{\it squarefree}, if $N = \bigoplus_{F \subset [d]} N_F$ 
(i.e., if ${\bf a} \in {\mathbb Z}^d$ is not squarefree, then $N_{\bf a} = 0$). 
\end{dfn}

A monomial ideal of $E$ is always a squarefree $E$-module. 
Let $\operatorname{Sq}(E)$ be the full subcategory of $\operatorname{*gr} E$ consisting of 
squarefree modules. Then $\operatorname{Sq}(E)$ is an abelian category with 
$D^b(\operatorname{Sq}(E)) \cong D^b_{\operatorname{Sq}(E)}(\operatorname{*gr} E)$.  
If $N$ is a squarefree $E$-module, 
then so is ${\bf D}_E(N)$. That is, ${\bf D}_E$ gives an exact duality functor 
from $\operatorname{Sq}(E)$ to itself.  We have functors 
${\mathcal S}: \operatorname{Sq}(E) \to \operatorname{Sq}(S)$ and ${\mathcal E}: \operatorname{Sq}(S) \to \operatorname{Sq}(E)$ giving an 
equivalence $\operatorname{Sq}(S) \cong \operatorname{Sq}(E)$. Here 
${\mathcal S}(N)_F = N_F$ for $N \in \operatorname{Sq}(E)$ and $F \subset [d]$, 
and the multiplication map ${\mathcal S}(N)_F \ni z \mapsto x_i z \in 
{\mathcal S}(N)_{F \cup \{ i \}}$ for $i \not \in F$ is given by 
${\mathcal S}(N)_F =N_F \ni z \mapsto (-1)^{\alpha(i, F)} y_i z 
\in N_{F \cup \{ i \}}= {\mathcal S}(N)_{F \cup \{ i \}}$, where 
$\alpha(i,F) =  \# \{ \, j \in F \mid j < i \, \}$. 
See \cite{R0, Y5} for detail. 
Since a free module $E({\bf a})$ is {\it not} squarefree 
unless ${\bf a} = 0$, the syzygies of a squarefree $E$-module are 
{\it not} squarefree. 

\begin{prop}[{Herzog-R\"omer, \cite[Corollary~3.3.5]{R02}}]
\label{Tim} If $N$ is a squarefree $E$-module 
(e.g., $N = E/J$ for a monomial ideal $J$), 
then we have $\operatorname{lpd}(N) \leq d-1$. 
\end{prop}

This result easily follows from  
Theorem~\ref{wKos=regH} and the fact that 
$H^i({\mathcal F}^*(N'))(-{\mathbf 1})$ is a squarefree $S$-module for all $i$  
and $H^i({\mathcal F}^*(N')) = 0$ unless $0 \leq i \leq d$. 
(Recall the remark on the regularity of squarefree modules 
given before Definition~\ref{sqE}, and note that 
$M:=H^d({\mathcal F}^*(N'))(-{\mathbf 1})$ is generated by $M_0$). 

We also remark that \cite[Corollary~3.3.5]{R02} just states 
that $\operatorname{lpd} (N) \leq d$. 
But their argument actually proves that $\operatorname{lpd} (N) \leq d-1$.   
In fact, they showed that $$\operatorname{lpd}(N) \leq \operatorname{proj.dim}_S {\mathcal S}(N).$$ 
But, if $\operatorname{proj.dim}_S {\mathcal S}(N) = d$ then ${\mathcal S}(N)$ has a summand   
which is isomorphic to $K = S/(x_1, \ldots, x_d)$ and hence 
$N$ has a summand which is isomorphic to 
$K = E/(y_1, \ldots, y_d)$. But $K \in \operatorname{Sq}(E)$ has 
a linear resolution and irrelevant to $\operatorname{lpd}(N)$.  
\medskip

To refine Proposition~\ref{Tim}, we need further properties of 
squarefree modules. 


If $M^\bullet \in D^b(\operatorname{Sq}(S))$, then 
$\underline{\Ext}^i_S(M^\bullet, {\mathcal D}^\bullet_S)$ is squarefree for all $i$. Hence 
${\mathcal D}^\bullet_S$ gives a duality functor on $D^b(\operatorname{Sq}(S))$. On the other hand, 
${\mathbf {A}} := {\mathcal S} \circ {\bf D}_E \circ {\mathcal E}$ is an exact 
duality functor on $\operatorname{Sq}(S)$ and it induces a duality functor on $D^b(\operatorname{Sq}(S))$.  
Miller \cite[Corollary~4.21]{Mil} and 
R\"omer \cite[Corollary~3.7]{R1} proved that 
$\operatorname{reg}({\mathbf {A}}(M)) = \operatorname{proj.dim}_S M$ for all $M \in \operatorname{Sq}(S)$. 
I generalized this equation to a complex $M^\bullet \in D^b(\operatorname{Sq}(S))$ in 
\cite[Corollary~2.10]{Y7}.
 
\begin{lem}\label{sqf formula}
Let $N \in \operatorname{Sq}(E)$ and set $N' := {\bf D}_E(N)$. Then we have 
\begin{equation}\label{1st}
\operatorname{reg}(H^i({\mathcal F}^*(N'))) = - \operatorname{depth}_S 
( \, \operatorname{Ext}_S^{d-i}( {\mathcal S}(N'), \, S ) \, )  
\end{equation} and
\begin{equation}\label{2nd}
\operatorname{lpd}(N) = \max \{ \, i - \operatorname{depth}_S 
( \, \operatorname{Ext}_S^{d-i}( {\mathcal S}(N'), \, S ) \, )  \mid  
0 \leq i \leq d  \, \}.
\end{equation}
Here we set the depth of the 0 module to be $+\infty$.
\end{lem}


If $M :=  \operatorname{Ext}_S^{d-i}( {\mathcal S}(N'), \, S ) \ne 0$,  
then $\operatorname{depth}_S M \leq \dim_S M \leq i$. 
Therefore all members in the set of the right side of 
\eqref{2nd} are non-negative or $- \infty$.  


\begin{proof}
By Theorem~\ref{wKos=regH}, \eqref{2nd} follows from \eqref{1st}. 
So it suffices to show \eqref{1st}. 
By \cite[Proposition~4.3]{Y5}, we have ${\mathcal F}^*(N') \cong 
({\mathbf {A}} \circ {\bf D}_S \circ {\mathcal S} (N'))({\mathbf 1})$. (The degree shifting 
``$({\mathbf 1})$" does not occur in \cite[Proposition~4.3]{Y5}. 
But $E$ is a negatively graded ring there, 
and we need the degree shifting in the present convention.) 
Since ${\mathbf {A}}$ is exact, we have 
\begin{eqnarray*}
H^i({\mathcal F}^*(N'))) \cong H^i( \, {\mathbf {A}} \circ {\bf D}_S \circ {\mathcal S} (N') \, ) ({\mathbf 1}) 
&\cong& {\mathbf {A}} ( \, H^{-i}({\bf D}_S \circ {\mathcal S} (N')) \, ) ({\mathbf 1})\\ 
&=& {\mathbf {A}} ( \, \underline{\Ext}_S^{-i} ({\mathcal S}(N'), \, {\mathcal D}^\bullet_S) \, ) ({\mathbf 1}).
\end{eqnarray*}
Recall that $\operatorname{reg}({\mathbf {A}}(M)) = \operatorname{proj.dim}_S M$ for $M \in \operatorname{Sq}(S)$. 
On the other hand, since $M$ is finitely generated, 
the underlying module of $\underline{\Ext}_S^{-i}(M,{\mathcal D}^\bullet_S)$ is 
isomorphic to $\operatorname{Ext}_S^{d-i}(M, S)$. 
So \eqref{1st} follows from these facts and 
the Auslander-Buchsbaum formula. 
\end{proof}


\begin{cor}\label{comp lin}
For $N \in \operatorname{Sq}(E)$, $N$ is weakly Koszul (over $E$) 
if and only if ${\mathcal S}(N)$ is weakly Koszul (over $S$). 
\end{cor}

In \cite[Corollary~1.3]{R1}, it was proved that $N$ has a linear 
resolution if and only if so does ${\mathcal S}(N)$. 
Corollary~\ref{comp lin} also follows from this fact and 
(the squarefree module version of) \cite[Proposition~1.5]{HH}. 

\begin{proof}
We say $M \in \operatorname{gr} S$ is {\it sequentially Cohen-Macaulay}, 
if for each $i$ $\operatorname{Ext}_S^i(M,S)$ is either the zero module or 
a Cohen-Macaulay module of dimension $d-i$ 
(c.f. \cite[III. Theorem~2.11]{St}). 
By Lemma~\ref{sqf formula}, $N$ is weakly Koszul 
if and only if ${\mathcal S}(N')$ ($\cong {\mathbf {A}} \circ {\mathcal S} (N)$) 
is sequentially Cohen-Macaulay.  
By \cite[Theorem~4.5]{R1}, the latter condition holds  
if and only if ${\mathcal S}(N)$ is weakly Koszul. 
\end{proof}

Many examples of squarefree monomial ideals of $S$ 
which are weakly Koszul (dually, Stanley-Reisner rings which are 
sequentially Cohen-Macaulay) are known. 
So we can obtain many weakly Koszul monomial 
ideals of $E$ using Corollary~\ref{comp lin}. 

\begin{prop}
For an integer $i$ with $1 \leq i \leq d-1$,  
there is a squarefree $E$-module $N$ such that 
$\operatorname{lpd} N = \operatorname{proj.dim}_S {\mathcal S}(N) = i$. 
In particular, the inequality of 
Proposition~\ref{Tim} is optimal. 
\end{prop}

\begin{proof}
Let $M$ be the ${\mathbb Z}^d$-graded $i^{\rm th}$  syzygy of 
$K = S/{\mathfrak m}$. Note that $M$ is squarefree. We can easily 
check that $N:={\bf D}_E \circ {\mathcal E}(M) \in \operatorname{Sq}(E)$ satisfies the expected 
condition.   In fact, $\operatorname{proj.dim}_S {\mathcal S}(N) 
= \operatorname{proj.dim}_S {\mathbf {A}} (M) = \operatorname{reg} M = i$.  On the other hand, 
since  $\operatorname{Ext}^{d-i}_S({\mathcal S}(N'), S) = \operatorname{Ext}^{d-i}_S(M, S) = K$,  
$\operatorname{Ext}^j_S({\mathcal S}(N'), S)=0$ for all $j \ne d-i, 0$, and 
$\operatorname{depth}_S (\operatorname{Hom}_S({\mathcal S}(N'), S)) = d-i+1$, we have $\operatorname{lpd} N = i$. 
\end{proof}

The above result also says that the inequality 
$\operatorname{lpd}(N) \leq \operatorname{proj.dim}_S {\mathcal S}(N)$ of \cite[Corollary~3.3.5]{R02}
is also optimal. 
But for a monomial ideal $J \subset E$, the situation is different.  

\begin{prop}\label{d-2}
If $d \geq 3$, then we have 
$\operatorname{lpd} (E/J) \leq d-2$ for a monomial ideal $J$ of $E$. 
\end{prop}

\begin{proof}
If $d=3$, then easy computation shows that 
any squarefree monomial ideal $I \subset S$ is weakly Koszul. 
Hence $J$ is weakly Koszul by Corollary~\ref{comp lin}. 
So we may assume that $d \geq 4$. 

Note that ${\mathbf {A}} \circ {\mathcal S}(E/J)$ is isomorphic to a squarefree monomial 
ideal of $S$. We denote it by $I$. 
By Lemma~\ref{sqf formula}, 
it suffices to show that $\operatorname{depth}_S (\operatorname{Hom}_S(I,S)) \geq 2$ 
and $\operatorname{depth}_S (\operatorname{Ext}_S^1(I,S)) \geq 1$. 
Recall that $\operatorname{Hom}_S(I,S)$ satisfies  
Serre's condition $(S_2)$, hence its depth is 
at least 2. Since $\operatorname{Ext}_S^1(I,S) \cong 
\operatorname{Ext}_S^2(S/I,S)$, it suffices to prove 
that $\operatorname{depth}_S (\operatorname{Ext}_S^2(S/I, S)) \geq 1$. 

If $\operatorname{ht} (I) > 2$, then we have $\operatorname{Ext}_S^2(S/I,S) = 0$.   
If $\operatorname{ht}(I) = 2$, then $\operatorname{Ext}_S^2(S/I,S)$ satisfies $(S_2)$ as an 
$S/I$-module and $\operatorname{depth}_S \operatorname{Ext}_S^2(S/I,S) \geq \min \{ \, 2, 
\dim (S/I) \, \} \geq 2$. So we may assume that $\operatorname{ht}(I)=1$. 
If the heights of all associated primes of $I$ are 1, 
then $I$ is a principal ideal and $\operatorname{Ext}^i_S(S/I, S) = 0$ 
for all $i \ne 1$. So we may assume that $I$ has an prime of larger 
height. 
Then we have ideals $I_1$ and $I_2$ of $S$ such that $I = I_1 \cap I_2$ 
and the heights of any associated prime of $I_1$ 
(resp. $I_2$) is 1 (at least 2).
Since $I$ is a radical ideal, we have $\operatorname{ht} (I_1 + I_2) \geq 3$. 
Hence $\operatorname{Ext}^2_S(S/(I_1 + I_2), S) = 0$ and 
$\operatorname{Ext}^3_S(S/(I_1 + I_2), S)$ is either the zero module or 
satisfies $(S_2)$ as an $S/(I_1+I_2)$-module.  
In particular, if $\operatorname{Ext}_S^3(S/(I_1 + I_2), S) \ne 0$ 
(equivalently, if $\dim (S/(I_1 + I_2)) = d-3$) then  
$\operatorname{depth}_S (\operatorname{Ext}^3_S(S/(I_1 + I_2), S)) 
\geq \min\{ 2, d-3 \} \geq 1$. Note that  
$\operatorname{depth}_S ( \operatorname{Ext}_S^2(S/I_2, S) ) \geq 2$. 
From the short exact sequence 
$$0 \to S/I \to S/I_1 \oplus S/I_2 \to S/(I_1 + I_2) 
\to 0$$ and the above argument, we have the exact sequence  
\begin{equation}\label{last}
0 \to \operatorname{Ext}^2_S(S/I_2, S) \to \operatorname{Ext}^2_S(S/I, S) \to 
\operatorname{Ext}^3_S(S/(I_1 + I_2), S).
\end{equation}  
We have $\operatorname{depth}_S (\operatorname{Ext}^2_S(S/I,S)) \geq 1$ by \eqref{last}, 
since the modules beside this module have positive depth. 
\end{proof}

\section*{Acknowledgments}
This research  started during my stay at Instituto de Matem\'aticas 
de la UNAM (Morelia, Mexico) in the summer of 2004. 
I am grateful to JSPS and CONACYT for financial support, 
and the institute for warm hospitality.  
Special thank are due to Professor Robert Martinez-Villa for 
stimulating discussion and encouragement. 
I also thanks Professor Izuru Mori for valuable information on AS regular 
algebras. 

