Derived categories in algebra, geometry, and topology

The Department of Mathematics “Federigo Enriques” of Università degli Studi di Milano is organizing a one day mini workshop on Derived categories in algebra, geometry, and topology. It will be held on June 19th, 2025 at via Saldini, 50 20133 Milano (MI), Italy from 10am to 4pm CET.

Please see below for title and abstracts of our speakers:

Speaker: Alicia Lamarche (BIMSA, Beijing)
Details: 10am to 11am CET in Aula 8
Title: Arithmetic and Derived Categories

Abstract

Are derived categories a suitable tool for solving arithmetic problems? We will discuss this question by highlighting the works of many authors who have found success in extracting arithmetic data from derived categories.

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Speaker: Paul Balmer (UCLA, USA)
Details: 11:30am CET in Aula 8
Title: Completion in tensor-triangular geometry

Abstract

In recent joint work with Beren Sanders, we consider the completion of a tensor-triangulated category with respect to a Thomason subset of its spectrum. (This should not be confused with completion with respect to a metric.) We prove that this construction recovers usual I-adic complextion of commutative rings in reasonable cases. Among the properties of the spectrum of the completed category, we prove what we call the Tate Intermediate Value Theorem.

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Speaker: Jan Šťovíček (Charles University, Prague)
Details: 2:30pm CET in Aula 9
Title: Dualizable torsion objects in tensor triangular geometry

Abstract

Suppose that T is a nice enough tensor-triangulated category with an action of a (graded) commutative noetherian ring R (such as the derived category of a commutative noetherian ring, noetherian scheme or the homotopy category of injective representations of a finite group over a field). If V is a closed subset of Spec(R), the category of dualizable objects in the full subcategory of V-torsion objects was recently studied by Benson, Iyengar, Krause and Pevtsova for group algebras and local rings and by Balmer and Sanders for general commutative rings. The interest stems from the fact that the Balmer spectrum in both cases corresponds to the spectrum of the V-adic completion of R. We contribute by proving that, if T possesses a strong compact generator g, then the V-torsion part of g strongly generates all the dualizable V-torsion objects. This is a joint work with Jun Maillard.

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Inquiries for any details can be directed to

• Pat Lank, email: plankmathematics ‘at’ gmail ‘dot’ com
• Luigi Lombardi, email: luigi ‘dot’ lombardi ‘at’ unimi ‘dot’ it