Back to Algebraic Geometry Seminar: Fall 2022
Algebraic Geometry Seminar
Date: Tuesday, Sep 27, 2022
Time: 3:30PM - 4:30PM
Location: LCB 323
Pat Lank
University of South Carolina
Title |
Generation of derived categories in prime characteristic |
Abstract |
Within this talk, which is joint work with Matthew Ballard, we will study the bounded derived category of coherent sheaves $D^b (\operatorname{coh}X)$ on an $F$-finite Noetherian scheme of prime characteristic $p$. It will be shown that $F_\ast^e (\operatorname{perf}X)$ strongly generates $D^b (\operatorname{coh}X)$ for $e\gg 0$ where $F_\ast^e$ denotes the $e$-th iterate of the Frobenius pushforward functor on $X$. Immediate applications to this is a new proof of Kunz’s theorem, introduces new invariants for singularity types in prime characteristic coming from derived categories, categorical resolution of singularities, and produces interesting structural insight into this category. This motivates a notion of $F$-thickness; namely, when $F_\ast^e \mathcal{O}X$ strongly generates $D^b (\operatorname{coh}X)$. Lastly, in the case of smooth Fano projective complete intersections $X$, it will be shown that $F\ast^e \mathcal{O}_X$ decomposes into direct summands indexed by components in Kuznetsov/Orlov style semiorthogonal decompositions. Particularly, when $p$ is sufficiently large, $X$ is globally $F$-split. |
Algebraic Geometry