Noncommutative motives of Azumaya algebras

TL;DR

This paper proves that noncommutative motives of Azumaya algebras over a scheme are isomorphic under certain conditions, leading to the equivalence of their R-linear additive invariants and enabling computations for various algebraic and geometric objects.

Contribution

It establishes isomorphisms of noncommutative motives for Azumaya algebras and applies these results to compute invariants of complex algebraic structures.

Findings

Noncommutative motives of Azumaya algebras are isomorphic when 1/r is in R.

All R-linear additive invariants of X and A coincide under the isomorphism.

Explicit calculations of invariants for differential operators, cubic fourfolds, Severi-Brauer varieties, and more.

Abstract

Let k be a base commutative ring, R a commutative ring of coefficients, X a quasi-compact quasi-separated k-scheme, A a sheaf of Azumaya algebras over X of rank r, and Hmo(R) the category of noncommutative motives with R-coefficients. Assume that 1/r belongs to R. Under this assumption, we prove that the noncommutative motives with R-coefficients of X and A are isomorphic. As an application, we show that all the R-linear additive invariants of X and A are exactly the same. Examples include (nonconnective) algebraic K-theory, cyclic homology (and all its variants), topological Hochschild homology, etc. Making use of these isomorphisms, we then computer the R-linear additive invariants of differential operators in positive characteristic, of cubic fourfolds containing a plane, of Severi-Brauer varieties, of Clifford algebras, of quadrics, and of finite dimensional k-algebras of finite global dimension. Along the way we establish two results of independent interest. The first one asserts that every element of the Grothendieck group of X which has rank r becomes invertible in the R-linearized Grothendieck group, and the second one that every additive invariant of finite dimensional algebras of finite global dimension is unaffected under nilpotent extensions.