Derived analytic geometry through localized contextsMPIM

by Jeroen Hekking (Stockholms University)

Tuesday Jul 7, 2026, 3:05 PM → 5:00 PM Europe/Berlin

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.