פעילויות השבוע

In the last twenty years algebraic geometry has evolved rapidly, from the geometry of schemes and stacks, to the derived algebraic geometry (DAG) of today. The flavor of contemporary DAG is very homotopical, in the sense that is largely based on simplicial sets and Quillen model structures.

This talk is on another approach to DAG, of a very algebraic flavor, which avoids simplicial methods and model structures altogether. Instead, the fundamental concept is that of DG rings, traditionally called unital associative cochain DG algebras. DG rings are of two distinct kinds: noncommutative and commutative. These two kinds of DG rings interact, primarily through central DG ring homomorphisms; and this interaction is quite fruitful. The main tool for studying DG rings, DG modules over them, and the associated derived categories, is the construction and manipulation of resolutions. Hence ”constructive approach“.

I will define the notions mentioned above, and state several results, among them: (1) The squaring operation and Van den Bergh‘s rigid dualizing complexes in the commutative arithmetic setting; (2) Theorems on derived Morita theory; (3) Duality and tilting for commutative DG rings. I will try to demonstrate that this constructive approach is very amenable to calculation. I will also mention work of Shaul, within this framework, on derived completion of DG rings and on the derived CM property.

The talk will conclude with a couple of conjectural ideas: (a) A structural description of the derived category of DG categories; (b) A construction of the cotangent DG module within the framework of commutative DG rings, without any arithmetic restrictions.

Some of this work is joint with J. Zhang, L. Shaul, M. Ornaghi and S. Singh.

Slides for the talk are available here:

https://sites.google.com/view/amyekut-math-bgu/home/lectures/constr-der-algebra

(updated 15 March 2022)

For an action of a countable amenable group $G$ on a compact metric space $X$, the mean dimension $mdim (G, X)$ was introduced by Lindenstrauss and Weiss, for reasons unrelated to $C^*$-algebras. The radius of comparison $rc (A)$ of a $C^*$-algebra $A$ was introduced by Toms, for use on $C^*$-algebras having nothing to do with dynamics.

A construction called the crossed product $C^* (G, X)$ associates a $C^*$-algebra to a dynamical system. There is significant evidence for the conjecture that $rc ( C^* (G, X) ) = (1/2) mdim (G, X)$ when the action is free and minimal. We give the first general partial results towards the direction $rc ( C^* (G, X) ) \geq (1/2) mdim (G, X)$. We don‘t get the exact conjectured bound, but we get nontrivial results for many of the known examples of free minimal systems with $mdim (G, X) > 0$. The proof depends, among other things, on Cech cohomology, and uses something we call the mean cohomological independence dimension. Unlike the currently known results in the other direction, it works for all choices of $G$.

The talk will include something about the crossed product construction; no previous knowledge of it will be assumed.

This is joint work with Ilan Hirshberg.