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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

Galois structure of homogeneous coordinate rings


Authors: Frauke M. Bleher and Ted Chinburg
Journal: Trans. Amer. Math. Soc. 360 (2008), 6269-6301
MSC (2000): Primary 14L30; Secondary 14C40, 13A50, 20C05
DOI: https://doi.org/10.1090/S0002-9947-08-04436-X
Published electronically: July 21, 2008
MathSciNet review: 2434287
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Abstract: Suppose $G$ is a finite group acting on a projective scheme $X$ over a commutative Noetherian ring $R$. We study the $RG$-modules $\mathrm {H}^0(X,\mathcal {F} \otimes \mathcal {L}^n)$ when $n \ge 0$, and $\mathcal {F}$ and $\mathcal {L}$ are coherent $G$-sheaves on $X$ such that $\mathcal {L}$ is an ample line bundle. We show that the classes of these modules in the Grothendieck group $G_0(RG)$ of all finitely generated $RG$-modules lie in a finitely generated subgroup. Under various hypotheses, we show that there is a finite set of indecomposable $RG$-modules such that each $\mathrm {H}^0(X,\mathcal {F} \otimes \mathcal {L}^n)$ is a direct sum of these indecomposables, with multiplicities given by generalized Hilbert polynomials for $n >> 0$.


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Additional Information

Frauke M. Bleher
Affiliation: Department of Mathematics, University of Iowa, Iowa City, Iowa 52242-1419
Email: fbleher@math.uiowa.edu

Ted Chinburg
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email: ted@math.upenn.edu

Keywords: Group actions on schemes, Euler characteristics, homogeneous coordinate rings, Riemann-Roch Theorems, Grothendieck groups
Received by editor(s): May 11, 2006
Received by editor(s) in revised form: October 26, 2006
Published electronically: July 21, 2008
Additional Notes: The first author was supported in part by NSF Grants DMS01-39737 and DMS06-51332 and NSA Grant H98230-06-1-0021. The second author was supported in part by NSF Grants DMS00-70433 and DMS05-00106.
Article copyright: © Copyright 2008 Frauke M. Bleher and Ted Chinburg