Smooth and proper algebras via stable $(\infty,2)$-categories

Abstract: Since Grothendieck, the notion of an abelian 1-category has provided a natural setting to do algebra which encompasses both categories of modules and categories of sheaves. Since Lurie, the notion of a stable $(\infty,1)$-category has provided a similar setting to do derived algebra, encompassing derived categories of modules and sheaves, and improving upon the notion of a triangulated category due to Verdier.

In this talk, we discuss a few possible notions of stable $(\infty,2)$-category, motivated by enriched category theory. Examples include the $(\infty,2)$-category of dg categories, the $(\infty,2)$-category of stable $(\infty,1)$-categories, and various $(\infty,2)$-categories of stacks of stable $(\infty,1)$-categories. The intention is to provide a natural home for the study of such $(\infty,2)$-categories, which are of interest in areas such as the Geometric Langlands program, secondary algebraic K-theory, and derived algebraic geometry.

We discuss work in progress on showing that our notions of stable $(\infty,2)$-category are equivalent. As an application, we show for example that every smooth and proper algebra over a regular commutative Noetherian ring k may be constructed from $k$ by iterating two simple operations: glueing along a perfect bimodule, and 2-idempotent splitting.

algebraic geometryrepresentation theory

Audience: researchers in the topic

Algebra and Geometry Seminar @ HKUST

Series comments: Algebra and Geometry seminar at the Hong Kong University of Science and Technology (HKUST).

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