On a theorem of Hazrat and Hoobler

We use cycle complexes with coefficients in an Azumaya algebra, as developed by Kahn and Levine, to compare the G-theory of an Azumaya algebra to the G-theory of the base scheme. We obtain a sharper version of a theorem of Hazrat and Hoobler in certain cases.


Introduction
Let K * (X; A) be the K-theory of left A-modules which are locally free and finite rank coherent O X -modules; let G * (X; A) be the K-theory of left A-modules which are coherent O X -modules.
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.

Corollary 1.2. If
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 ;

the kernel of B
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 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 isétale 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 .

Twisted higher Chow groups and twisted G-theory
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) (t) such that dim k W = s + t. Define, for t ≥ 0, z s (X, t; A) = W ∈Xs(t) 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 CH s (X, t; A) = H t (z s (X, * ; A)).
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.

4
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 on a vector space.
In particular, the kernels of these maps are zero, and the cokernels of the maps are over k(W ). Proof. This follows immediately from Equation (2).

Corollary 2.2.
The cokernel of F : z s (X, t; A) → z s (X, t) is a torsion group of exponent bounded above by n 2 , and B E : z s (X, t) → z s (X, t; A) is a torsion group of exponent bounded above by na.
Proof. In the first case, one always has ind(A k(W ) )|n, so the statement follows from Equation (2). Similarly, na ind(A k(W ) ) 2 |na, so the second statement follows. Proof. This follows immediately from Lemma 2.1.
Here is our main theorem.
Theorem 2.4. Let X be a d-dimensional scheme of finite type over a field, and let A be an Azumaya algebra on X. Then, the kernels and cokernels of and of F : G r (X; A) → G r (X) are groups of exponent bounded above by (na) d+1 for all r ≥ 0.
Proof. Kahn and Levine [11] show that there is a convergent spectral sequence There is also the motivic spectral sequence The functors B E : G(X) → G(X; A) and F : G(X; A) → G(X) are compatible with these spectral sequences and the functors B E and F on higher Chow groups. Note that E p,q 2 = E p,q 2 (A) = 0 whenever q < 0, −p < 0, or q > d. We will prove the theorem for the kernel of the functor B E . The other cases are entirely similar. On the E ∞ -page, the composition functor F •B E is still multiplication by na, so the kernels and cokernels of B E on E ∞ are still of exponent at most na. The spectral sequences abut to filtrations F s G r (X; A) and F s G r (X) where The filtration looks like The filtration F s G r (X) is of length d by the vanishing statements. Let z ∈ G r (X) be in the kernel of F , and let z be the image of z in E −r−d,d ∞ . Then, by hypothesis, z is in the kernel of F , so that na · z = 0. Thus, na · z is contained in F −d+1 G r (X). Continuing in this way, we see that (na) d+1 · z is contained in F 0 G r (X) = 0. So, (na) d+1 · z = 0.
Corollary 2.5. The same result holds for K-theory when X is regular. Corollary 2.6. The maps B E : F (s/s+1) G r (X; A) → F (s/s+1) G r (X) F : F (s/s+1) G r (X) → F (s/s+1) G r (X; A) have torsion kernels and cokernels of exponent at most na.
Proof. This follows from the proof of the theorem. Corollary 2.7. For any integer j prime to na, the maps B E : z s (X, * ; Z/j) → z s (X, * ; A; Z/j) B E : G r (X; Z/j) → G r (X; A; Z/j) F : z s (X, * ; A; Z/j) → z s (X, * ; Z/j) F : G r (X; A; Z/j) → G r (X; Z/j) are isomorphisms.
It is interesting that this method proves the isomorphisms by means of an isomorphism of cycle complexes, not just a quasi-isomorphism.