Purely inseparable ring extensions and Azumaya algebras (Q2711336)

Main result: Theorem 5. Let \(B\) be an Azumaya \(Z\)-algebra, \(D\) a derivation of \(B\), and \(\delta=D|Z\). Assume that \(Z/Z^\delta\) is a purely inseparable extension of exponent one with \(\delta\), and \(\delta\) satisfies the minimal polynomial \(t^{p^e}+t^{p^{e-1}}\alpha_e+\cdots+t^p\alpha_2+t\alpha_1\) (\(\alpha_i\in Z^\delta\)). If \(D^{p^e}+\alpha_eD^{p^{e-1}}+\cdots+\alpha_2D^p+\alpha_1D=0\), then there hold the following:NEWLINENEWLINENEWLINE(1) \(B=B^DZ=B^D\otimes_{Z^\delta}Z\), \(_{B^D}B\) is a finitely generated projective module.NEWLINENEWLINENEWLINE(2) \(B^D\) is an Azumaya \(Z^\delta\)-algebra, and \(V_B(B^D)=Z\).NEWLINENEWLINENEWLINE(3) \(\Hom(_{B^D}B_{B^D},{_{B^D}B_{B^D}})=Z[D]=Z\oplus ZD\oplus ZD^2\oplus\cdots\oplus ZD^{p^{e-1}}\).NEWLINENEWLINENEWLINE(4) \(\text{Der}_{B^D}(B)=ZD\oplus ZD^p\oplus\cdots\oplus ZD^{p^{e-1}}\). In particular, \(\text{Der}_{Z^\delta}(Z)=Z\delta\oplus Z\delta^p\oplus\cdots\oplus Z\delta^{p^{e-1}}\).NEWLINENEWLINENEWLINE(5) \(B[X;D]\) and \(Z[X;\delta]\) are Azumaya \(Z^\delta[f]\)-algebras and \(B[X;D]=Z[X;\delta]\oplus_{Z^\delta}B^D=Z[X;\delta]\oplus_{Z^\delta[f]}B^D[f]\), where \(f=X^{p^e}+X^{p^{e-1}}\alpha_e+\cdots+X^p\alpha_2+X\alpha_1\).