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# PICARD GROUPS IN TRIANGULAR GEOMETRY AND APPLICATIONS TO MODULAR REPRESENTATION THEORY

PAUL BALMER
Transactions of the American Mathematical Society
Vol. 362, No. 7 (JULY 2010), pp. 3677-3690
Stable URL: http://www.jstor.org/stable/25677843
Page Count: 14
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## Abstract

For a tensor triangulated ${\Bbb Z}/p\text{-category}\ \scr{K}$ , with spectrum ${\rm Spc}(\scr{K})$ , we construct an injective group homomorphism $\check{{\rm H}}^{1}({\rm Spc}(\scr{K}),{\Bbb G}_{{\rm m}})\otimes {\Bbb Z}[1/p]\hookrightarrow {\rm Pic}(\scr{K})\otimes {\Bbb Z}[1/p]$ , where ${\rm Pic}(\scr{K})$ is the group of $\otimes \text{-invertible}$ objects of $\scr{K}$ . 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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