On a theorem of Hazrat and Hoobler

We prove the following theorem.

Theorem 1.1. Let X be a d-dimensional scheme of finite type over a field k, and let A be an Azumaya algebra on X of constant degree n. Let B A : G i (X) → G i (X; A) and B A : K i (X) → K i (X; A) be the homomorphisms induced by the functor F → A ⊗ OX F . Then, 1. the kernel and cokernel of B A : G i (X) → G i (X; A) are torsion groups of exponents dividing n 2d+2 ;

2. the kernel and cokernel of B A : K i (X) → K i (X; A) are torsion groups of exponents dividing n 2d+2 if X is regular.

A is an Azumaya algebra of constant degree n over a scheme X of finite type over a field k, then the base extension homomorphism

The theorem above should be compared to the following two theorems, which motivated us in the first place.

Theorem 1.3 ). If A is an Azumaya algebra of constant degree n which is free over a noetherian affine scheme X, then

has torsion kernel and cokernel of exponents at most n 4 . Theorem 1.4 ). Let X be a d-dimensional noetherian scheme, and let A be an Azumaya algebra on X of constant degree n. Then,

and the cokernel is torsion of exponent dividing n 4d+2 ;

if X is regular, and the cokernel is torsion of exponent dividing n 4d+2 in this case;

3. the kernel and cokernel of B A : K i (X) → K i (X; A) are torsion groups of exponent dividing n 2d+2 if X has an ample line bundle.

Since a degree n Azumaya algebra is locally split by degree n extensions, it is expected that the base extension map

should be an isomorphism.

Here is a partial history of results and techniques in this direction. Wedderburn’s theorem [10] easily implies that K 0 (k) → K 0 (A) is injective with cokernel isomorphic to Z/m, where A ∼ = M m (D) for a central k-division algebra D.

Green-Handelman-Roberts [5] proved that the map B A is an isomorphism when A is a central simple algebra of degree n over a field. They used the Skolem-Noether theorem. That case has also been proven by Hazrat [7] using the fact that A is étale locally a matrix algebra.

The theorem of Hazrat-Millar quoted above uses the opposite algebra. The theorem of Hazrat-Hoobler uses Bass-style stable range arguments and Zariksi descent for Gtheory.

Our result uses twisted versions of Bloch’s cycle complexes. These twisted cycle complexes and the twisted motivic spectral sequence that relates them to G-theory are due to Kahn and Levine [11]. It is possible that our result could be extended to essentially smooth schemes over Dedekind rings by a combination of the work of Kahn and Levine [11] and Geisser [4].

The following is an interesting corollary of our approach: there are natural filtrations of length d on G i (X) and G i (X; A) coming from [11]. The map B A : G i (X) → G i (X; A) respects the filtrations. We show that the induced maps on each of the d + 1 slices have kernel and cokernel groups of exponent at most n 2 .

It is worth mentioning two other functors on Azumaya algebras with values in abelian groups where the base extension maps are isomorphisms. Dwyer and Friedlander [3, 2.4, 3.1] showed that

is an isomorphism in some cases (all of which are Azumaya algebras over a noetherian ring), where K ét denotes étale K-theory, as, for instance, in Thomason [12]. In this direction, it is possible to show (for instance, in the setting of Antieau [1]) that K ét (X; A) is an invertible object (in the sense of the Picard group) over K ét (X) in the category of étale sheaves of K ét -module spectra on a scheme X.

Finally, Cortiñas and Weibel [2] proved the base extension maps induce isomorphisms in Hochschild homology over a field k.

Let X in Sch/k be an integral k-scheme of finite type, and let A be a sheaf of Azumaya algebras on X of rank n 2 . The degree of A is defined to be the integer n. Let E be a left A-module which is locally free and finite rank na as an O X -module. For generalities on Azumaya algebras, which as O X -modules are always locally free and of finite rank, see [6].

As in Kahn-Levine [11], define the cycle complex of X with coefficients in A as follows. Let S X (s) (t) denote the set of closed subsets W ⊂ X × k ∆ t such that

for all faces F of ∆ n . Taking inverse images, S X (s) ( * ) becomes a simplicial set. Let X s (t) denote the subset of irreducible W in S X (s

See [11, Definition 5.6.1]. Kahn and Levine show that this actually becomes a complex, z s (X, * ; A), and they define the higher Chow groups with coefficients in A as

There are maps relating the complex z r (X, * ; A) to z r (X, * ), the untwisted complex that computes Bloch’s higher Chow groups. These are induced by the base-change map B E and the forgetful map F on K-theory.

The map B E takes a k(W )-vector space and tensors with E k(W ) to produce a left A k(W ) -module. The norm map F simply forgets the A⊗ k(W ) -module structure

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