Asymptotic depth of twisted higher direct image sheaves
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- by Renate Bär and Markus Brodmann PDF
- Proc. Amer. Math. Soc. 137 (2009), 1945-1950 Request permission
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 $-\infty$.References
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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
- MR Author ID: 41830
- Email: brodmann@math.unizh.ch
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1945-1950
- MSC (2000): Primary 13D45, 13E10, 14F05
- DOI: https://doi.org/10.1090/S0002-9939-08-09759-1
- MathSciNet review: 2480275