On a theorem of Hazrat and Hoobler¹¹1This material is based upon work supported by the NSF under Grant RTG DMS 0838697.

Let $\mathbf{K}_{*}(X;\mathcal{A})$ be the $K$-theory of left $\mathcal{A}$-modules which are locally free and finite rank coherent $\mathcal{O}_{X}$-modules; let $\mathbf{G}_{*}(X;\mathcal{A})$ be the $K$-theory of left $\mathcal{A}$-modules which are coherent $\mathcal{O}_{X}$-modules.

We prove the following theorem.

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

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 $\mathbf{K}_{0}(k)\rightarrow\mathbf{K}_{0}(A)$ is injective with cokernel isomorphic to $\mathds{Z}/m$, where $A\cong M_{m}(D)$ for a central $k$-division algebra $D$.

Green-Handelman-Roberts [5] proved that the map $B_{\mathcal{A}}$ is an isomorphism when $\mathcal{A}$ 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 $G$-theory.

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 $\mathbf{G}_{i}(X)$ and $\mathbf{G}_{i}(X;\mathcal{A})$ coming from [11]. The map $B_{\mathcal{A}}:\mathbf{G}_{i}(X)\rightarrow\mathbf{G}_{i}(X;\mathcal{A})$ 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 $\mathbf{K}^{\mathrm{\acute{e}t}}$ 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 $\mathbf{K}^{\mathrm{\acute{e}t}}(X;\mathcal{A})$ is an invertible object (in the sense of the Picard group) over $\mathbf{K}^{\mathrm{\acute{e}t}}(X)$ in the category of étale sheaves of $\mathbf{K}^{\mathrm{\acute{e}t}}$-module spectra on a scheme $X$.

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

Let $X$ in $\mathbf{Sch}/k$ be an integral $k$-scheme of finite type, and let $\mathcal{A}$ be a sheaf of Azumaya algebras on $X$ of rank $n^{2}$. The degree of $\mathcal{A}$ is defined to be the integer $n$. Let $\mathcal{E}$ be a left $\mathcal{A}$-module which is locally free and finite rank $na$ as an $\mathcal{O}_{X}$-module. For generalities on Azumaya algebras, which as $\mathcal{O}_{X}$-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 $\mathcal{A}$ as follows. Let $S^{X}_{(s)}(t)$ denote the set of closed subsets $W\subset X\times_{k}\Delta^{t}$ such that

for all faces $F$ of $\Delta^{n}$. 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 $t\geq 0$,

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

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

The map $B_{\mathcal{E}}$ takes a $k(W)$-vector space and tensors with $\mathcal{E}_{k(W)}$ to produce a left $\mathcal{A}_{k(W)}$-module. The norm map $F$ simply forgets the $\mathcal{A}\otimes_{k(W)}$-module structure on a vector space.

In particular, the kernels of these maps are zero, and the cokernels of the maps are

over $k(W)$.

Here is our main theorem.

It is interesting that this method proves the isomorphisms by means of an isomorphism of cycle complexes, not just a quasi-isomorphism.