Reduced \(K\)-theory of Azumaya algebras.
Consider an Azumaya algebra \(A\) over a commutative ring \(R\), and let \(K_i\) denote the Quillen \(K\)-functor. For \(i\geq 0\), one defines \(ZK_i(A)\) and \(CK_i(A)\) as the kernel and co-kernel of the map \(K_i(R)\to K_i(A)\), induced by the embedding \(R\to A\). The author defines a \(D\)-functor from a category of algebras to Abelian groups, as a functor \(G\) satisfying the following properties: (1) the composition \(G(A)\to G(M_m(A))\to G(A)\) (for suitable maps) is multiplication by \(m\); (2) the kernel of the above map \(G(M_m(A))\to G(A)\) is \(m\)-torsion; (3) if \(S\) is a finite free module over the center \(R\) of \(A\), then the composition \(G(A)\to G(A\otimes S)\to G(A)\) (for suitable maps) is multiplication by \([S:R]\); and (4) \(G\) is trivial on commutative rings. It is then noted that \(CK_i\) and \(ZK_i\), as well as \(SK_1(A)=A^1/A'\) (\(A^1\) being the group of elements of norm \(1\)) are \(D\)-functors. Some useful properties are proven for a \(D\)-functor \(G\) defined on Azumaya algebras over local rings. For example, \(G(A\otimes B)=G(A)\times G(B)\) when \(A\) and \(B\) have co-prime indices. Those techniques are used to prove the following: if \(A\) is a tame Azumaya algebra over a Henselian ring \(R\), then \(SK_1(A)\) is equal to \(SK_1\) of any quotient ring of \(A\). Also, if \(R\) is local and the rank of \(A\) is prime to \(\text{char\,}\overline R\), then \(CK_1(A)\cong CK_1(\overline A)\), where \(CK_1(A)=A^*/R^*A'\). The final section provides some interesting computational observations on \(CK_i\) and \(ZK_i\) for \(i=1,2\), for algebras over Pythagorean fields.
scientific article; zbMATH DE number 5078299
scientific article