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Castelnuovo-Mumford regularity of deficiency modules amd boundedness of cohomology | Markus Brodmann
; Maryam Jahangiri
; Cao Huy Linh
; | Date: |
6 Jan 2009 | Abstract: | Let $dinN$ and let $D^d$ denote the class of all pairs $(R,M)$ in which
$R$ is a Noetherian homogeneous ring with Artinian base ring and $M$ is a
finitely generated graded $R$-module of dimension $leq d$. For $(R,M)inD^d$
let $(M)$, $g(M)$ and $
^k(M)$ respectively denote the beginning, the
generating degree and the Castelnuovo-Mumford regularity at and above level $k
in N_0$ of $M$ and set $
(M):=
^0(M)$. If $iinN_0$ and $nin$, let
$d^i_M(n):=length_{R_0}(D^i_{R_+}(M)_n)$, where $D^i_{R_+}(-)$ denotes the
$i$-th right-derived functor of the $R_+$-transform functor $D_{R_+}(-)$. If
the base ring $R_0$ of $R$ is in addition local, we write $K^i(M)$ for the
$i$-th deficiency module of $M$. Our first main result says, that $
(K^i(M))$
is bounded in terms of $(M)$ and the "diagonal values" $d^j_M(-j)$ with $j =
0,..., d-1$. As an application of this we get that $
(K^i(M))$ is bounded in
terms of $(M)$, $
^2(M)$ and the Hilbert polynomial $p_M$ of $M$. From this
we prove a number of further bounding results for the invariants $
(K^i(M))$.
The cohomology table of a pair $(R,M)inD^d$ is defined as the family
$d_M:=(d^i_M(n))_{(i,n)in{0,..., d-1} imes}$. We say that a subclass $C$
of $D^d$ is of finite cohomology if the set ${d_M mid(R,M)inD^d}$ is
finite. A set $S sb{0,..., d-1} imes$ is said to bound cohomology, if for
each family $(h^s)_{sin S}$ of non-negative integers, the class ${(R,M)in
D^dd^i|_M(n)leq h^{(i,n)} for all(i,n)inS}$ is of finite cohomolgy. We
prove, that this is the case if and only if $S$ contains a quasi diagonal,
that is a set of the form ${(i,n_i)|i=0,...,d-1}$ with integers $n_0> n_1 >
>... > n_{d-1}$.We then deduce a number of further finiteness results for the
cohomology of certain subclasses of $D^d$. | Source: | arXiv, 0901.0690 | Services: | Forum | Review | PDF | Favorites |
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