Derived analytic geometry through localized contextsMPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
Oberseminar Arithmetic Geometry and Representation Theory
Classically, HKR identifies Hochschild homology with differential
forms in the smooth setting. Ben-Zvi--Nadler interpret this
comparison through derived loop spaces in characteristic zero.
Outside characteristic zero, Raksit's nonconnective affine formalism
produces an HKR filtration on Hochschild homology whose associated
graded recovers the derived de Rham side of the theory.
In analytic geometry, the exactness needed for derived constructions
often clashes with completeness. One response is to develop
homological algebra in Quillen-exact categories. The condensed
approach replaces topology by condensed structure, yielding an
abelian category, and axiomatizes completeness through analytic
rings. This leads to the question: can one formulate HKR-type
results uniformly across derived analytic geometries?
I will report on work in progress with Oren Ben-Bassat and Jack
Kelly on a common framework for asking this question, at least in
the affine setting. The framework is based on localized contexts, a
generalization of derived algebraic contexts. I will introduce
analytic rings over localized contexts and discuss examples coming
from non-Archimedean analytic geometry, light condensed mathematics,
and adic completion. If time permits, I will indicate how cotangent
complexes and nonconnective analytic rings enter the intended route
toward unified and global HKR-type statements.