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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic depth of twisted higher direct image sheaves


Authors: Renate Bär and Markus Brodmann
Journal: Proc. Amer. Math. Soc. 137 (2009), 1945-1950
MSC (2000): Primary 13D45, 13E10, 14F05
Published electronically: December 17, 2008
MathSciNet review: 2480275
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Abstract: Let $ \pi:X \rightarrow X_{0}$ be a projective morphism of schemes, such that $ X_{0}$ is Noetherian and essentially of finite type over a field $ K$. Let $ i \in \mathbb{N}_{0}$, let $ {\mathcal{F}}$ be a coherent sheaf of $ {\mathcal{O}}_{X}$-modules and let $ {\mathcal{L}}$ be an ample invertible sheaf over $ X$. Let $ Z_{0} \subseteq X_{0}$ be a closed set. We show that the depth of the higher direct image sheaf $ {\mathcal{R}}^{i}\pi_{*}({\mathcal{L}}^{n} \otimes_{{\mathcal{O}}_{X}} {\mathcal{F}})$ along $ Z_{0}$ ultimately becomes constant as $ n$ tends to $ -\infty$, provided $ X_{0}$ has dimension $ \leq 2$. There are various examples which show that the mentioned asymptotic stability may fail if $ \dim(X_{0}) \geq 3$. To prove our stability result, we show that for a finitely generated graded module $ M$ over a homogeneous Noetherian ring $ R=\bigoplus_{n \geq 0}R_{n}$ for which $ R_{0}$ is essentially of finite type over a field and an ideal $ \mathfrak{a}_{0} \subseteq R_{0}$, the $ \mathfrak{a}_{0}$-depth of the $ n$-th graded component $ H^{i}_{R_{+}}(M)_{n}$ of the $ i$-th local cohomology module of $ M$ with respect to $ R_{+}:=\bigoplus_{k>0}R_{k}$ ultimately becomes constant in codimension $ \leq 2$ as $ n$ tends to $ -\infty$.


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Additional Information

Renate Bär
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Address at time of publication: Kapellenweg 5, CH-8572 Berg, Switzerland
Email: renatebaer@gmx.ch

Markus Brodmann
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
Email: brodmann@math.unizh.ch

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09759-1
PII: S 0002-9939(08)09759-1
Keywords: Local cohomology, graded modules, depth, projective schemes, ample invertible sheaves, higher direct images.
Received by editor(s): April 23, 2008
Received by editor(s) in revised form: August 26, 2008
Published electronically: December 17, 2008
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.