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Picard groups in triangular geometry and applications to modular representation theory


Author: Paul Balmer
Journal: Trans. Amer. Math. Soc. 362 (2010), 3677-3690
MSC (2000): Primary 18E30, 20C20
DOI: https://doi.org/10.1090/S0002-9947-10-04949-4
Published electronically: February 8, 2010
MathSciNet review: 2601604
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Abstract: For a tensor triangulated $ \mathbb{Z}/p$-category $ \mathscr{K}$, with spectrum  $ \operatorname{Spc}(\mathscr{K})$, we construct an injective group homomorphism $ \check{\operatorname{H}}^1 (\operatorname{Spc}(\mathscr{K}),\mathbb{G}_{\opera... ...bb{Z}[1/p]\hookrightarrow\operatorname{Pic} (\mathscr{K})\otimes\mathbb{Z}[1/p]$, where $ \operatorname{Pic}(\mathscr{K})$ is the group of $ \otimes$-invertible objects of  $ \mathscr{K}$. In modular representation theory, we prove that this homomorphism induces a rational isomorphism between the Picard group of the projective support variety and the group of endotrivial representations.


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Additional Information

Paul Balmer
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555
Email: balmer@math.ucla.edu

DOI: https://doi.org/10.1090/S0002-9947-10-04949-4
Keywords: Picard group, gluing, support variety, triangulated category
Received by editor(s): June 23, 2008
Published electronically: February 8, 2010
Additional Notes: The author’s research was supported by NSF grant 0654397.
Article copyright: © Copyright 2010 by Paul Balmer

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