Picard groups in triangular geometry and applications to modular representation theory
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- by Paul Balmer PDF
- Trans. Amer. Math. Soc. 362 (2010), 3677-3690
Abstract:
For a tensor triangulated $\mathbb {Z}/p$-category $\mathscr {K}$, with spectrum $\operatorname {Spc}(\mathscr {K})$, we construct an injective group homomorphism $\check {\operatorname {H}}^1 (\operatorname {Spc}(\mathscr {K}),\mathbb {G}_{\operatorname {m}} )\otimes \mathbb {Z}[1/p]\hookrightarrow \operatorname {Pic} (\mathscr {K})\otimes \mathbb {Z}[1/p]$, 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.References
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Additional Information
- Paul Balmer
- Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095-1555
- MR Author ID: 652084
- Email: balmer@math.ucla.edu
- Received by editor(s): June 23, 2008
- Published electronically: February 8, 2010
- Additional Notes: The author’s research was supported by NSF grant 0654397.
- © Copyright 2010 by 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
- MathSciNet review: 2601604