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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Galois structure of homogeneous coordinate rings
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by Frauke M. Bleher and Ted Chinburg PDF
Trans. Amer. Math. Soc. 360 (2008), 6269-6301

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
  • 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.
  • © Copyright 2008 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
  • MathSciNet review: 2434287