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Workshop on birational geometry, March 2019
25 марта 2019 г. 15:30–16:30, г. Москва, Room 427
 


The Hochschild-Kostant-Rosenberg theorem fails in characteristic $p$ (after Akhil Mathew)

Vadim Vologodsky

HSE

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Аннотация: Let $X$ be a smooth algebraic variety over a field $K$, and let $\Delta: X \to X \times X$ be the diagonal embedding. Then the cohomology sheaves of the complex $\mathrm L\Delta^*\Delta_*\mathcal O_X$ are canonically identified with the sheaves of differential forms on $X$. In particular, there is a spectral sequence from the Hodge cohomology of $X$ to the hypercohomology of the complex $\mathrm L\Delta^*\Delta_*\mathcal O_X$ . If the characteristic of the base field $K$ is $0$ or larger then $\dim X$, the complex $\mathrm L\Delta^*\Delta_*\mathcal O_X$ is formal, i.e. quasi-isomorphic to the direct sum of its cohomology sheaves. It follows that in this case the above spectral sequence degenerates at the first page. It has been a longstanding question whether this degeneration holds in any characteristic. I will explain a recent result of Akhil Mathew showing that the analogous spectral sequence fails to degenerate for the classifying stack of the finite group scheme $\mu_p$ over $\mathbb F_p$. This easily yields an example of a smooth projective projective variety $X$ such that the spectral sequence does not degenerate.

Язык доклада: английский

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