K-Theory of Azumaya Algebras over Schemes

Abstract

Abstract. Let X be a connected, noetherian scheme and A be a sheaf of Azumaya algebras on X which is a locally free OX -module of rank a. We show that the kernel and cokernel of Ki (X) → Ki (A) are torsion groups with exponent a m for some m and any i ≥ 0, when X is regular or X is of dimension d with an ample sheaf (in this case m ≤ d + 1). As a consequence, Ki(X, Z/m) ∼ = Ki(A, Z/m), for any m relatively prime to a.An Azumaya algebra over a scheme is a sheaf of algebra which is one (étale) extension away from being a full matrix algebra. This should indicate that the functors arising from linear algebra would be similar over the Azumaya algebra and its base algebra. In [CW] it was shown that this is the case for Hochschild homology (over affine schemes). The aim of this note is to show that we have a similar result for K-functors. Indeed, we will show that, when X is of dimension d with an ample sheaf, e.g. affine or quasi-projective, or X is regular, and A is a sheaf of Azumaya algebras on X that is a locally free O X -module of rank a, then K i (X) is isomorphic to K i (A) up to bounded a-torsion. In order to prove these results, we first show that the kernel and cokernel of K i (X) → K i (A) are torsion groups with exponent a m for some m. When A is free over O X , this is a straightforward argument using Morita theory. When A is locally free, and X is of finite dimension with an ample sheaf, we will use an extension of Bass' result on K-theory of rings to do this. An alternative argument is given when X is regular, where we use a Mayer-Vietoris sequence to piece together local results into a global one. One reason that we focus on kernel and cokernel of K i (X) → K i (A) is that even for i = 1 and A being a division algebra the cokernel gives a very interesting group with applications to the group structure of the algebra (see Remark 16).Here K i (A) is the Quillen K-theory of the exact category of sheaves of O X − locally free, left A-modules of finite type. If A is non-commutative, we define a locally free, left A-module to be a left A-module M such that M is locally free as an O X -module, and we use the category A-mod and the corresponding exact subcategory of locally free left A-modules to calculate G i (A) and K i (A) respectively. We follow the notation in [S] throughout this note. X and Y will always denote schemes, R and S commutative O X -algebras, and A and B possibly non-commutative O X -algebras that are locally free and of finite type as O X -modules. Tensor products are over O X unless otherwise indicated. If X is a scheme and A is a not necessarily commutative sheaf of O X -algebras, we follow standard notation and denote the category of left

Noncommutative motives of Azumaya algebras

“…When 1/r ∈ k, one has moreover the following isomorphisms: The isomorphism H H * (X ) H H * (A) is well known and holds without the assumption that 1/r ∈ k. In what concerns cyclic homology, the isomorphism H C * (X ) H C * (A) was established by Cortiñas and Weibel [13] in the affine case. The algebraic K -theory isomorphism K * (X ) 1/r K * (A) 1/r was obtained recently by Hazrat and Hoobler [18] under the assumption that X is either regular Noetherian or Noetherian of finite Krull dimension with an ample sheaf. Besides these particular cases, all the remaining isomorphisms provided by Corollary 3.1 are, to the best of the authors' knowledge, new in the literature.…”

Noncommutative motives of Azumaya algebras

“…When applied to the above examples of additive invariants, Corollary 3.1 gives rise to the following (concrete) isomorphisms The isomorphism HH * (X) ≃ HH * (A) is well-known and holds without the assumption 1/r ∈ k. In what concerns cyclic homology, the isomorphism HC * (X) ≃ HC * (A) was established by Cortiñas-Weibel [12] in the affine case. The algebraic K-theory isomorphism K * (X) 1/r ≃ K * (A) 1/r was obtained recently by Hazrat-Hoobler [16] under the assumption that X is either regular noetherian or noetherian of finite Krull dimension with an ample sheaf. Besides these particular cases, all the remaining isomorphisms provided by Corollary 3.1 are, to the best of the authors knowledge, new in the literature.…”

Multiplicative groups of division rings

“…Clearly when A is a division algebra and i = 1, we get CK 1 (D) defined in (6.11). The fact that CK i (A) and ZK i (A) are torsion of bounded exponent was studied in several papers (see [8,29]). A consequence of this, is that the K-theory of A coincides with the K-theory of its base ring up to torsions.…”