Normality of algebras over commutative rings and the Teichmüller class. II: Crossed pairs and the relative case
The paper under review contains the second part of a research presented in three papers (whose sections share a common numeration), which is devoted to the study of the generalized Teichmüller cocycle map associated with normal algebras over commutative unitary rings (for the other two parts, see [the author, J. Homotopy Relat. Struct. 13, No. 1, 1--70 (2018; Zbl 1439.13025); ibid. 13, No. 1, 127--142 (2018; Zbl 1428.13011)]). Let \(S\) be a commutative ring with a unit, \(U(S)\) the group of invertible elements of \(S\), \(Q\) a group acting on \(S\) by ring automorphisms via a homomorphism \(k_ {Q} : Q \to\mathrm{Aut}(S)\), and let \(R\) denote the subring of those elements of \(S\) which are fixed under \(Q\). Recall that a normal \(Q\)-algebra means a pair \((A, \sigma )\), where \(A\) is a central \(S\)-algebra and \(\sigma \) is a homomorphism of \(Q\) into the group Out\((A)\) of outer automorphisms of \(A\) that lifts on \(A\) the action of \(Q\) on \(S\) so that the composite of \(\sigma \) with the obvious map Out\((A) \to\mathrm{Aut}(S)\) coincides with \(k_ Q\). The action of \(Q\) on \(S\) induces on \(U(S)\) a structure of a\(Q\)-module, which allows one to associate with \((A, \sigma )\) its Teichmüller complex, a certain crossed \(2\)-fold extension \(e(A, \sigma )\) starting at \(U(S)\) and ending at \(Q\). This crossed \(2\)-fold extension represents a the Teichmüller class of \((A, \sigma )\) in the third cohomology group \(H ^ 3(Q, U(S))\). In the paper under review, the author introduces the concept of a \(Q\)-normal Galois extension of commutative rings, and associates to such an extension \(T \vert S\) a structure extension \(e_ {(T \vert S)}: N \rightarrowtail G \twoheadrightarrow Q\) of \(Q\) by the Galois group \(N =\mathrm{Aut}(T \vert S)\) and an action of \(G\) on \(T\) by ring automorphisms. Using this, he develops the relative theory of the generalized Teichmüller cocycle map. One of the basic constructions of the theory is the \(Q\)-normal crossed product algebra \((A_ e, \sigma_ \psi )\), which is associated to a crossed pair \((e, \psi )\) with respect to \(e_ {(T \vert S)}\) and \(U(T)\), the latter being endowed with the structure of a \(G\)-module coming from the \(G\)-action on \(T\); the crossed product algebra \((A_ e, \sigma_ \psi )\) represents a member of the kernel of the ``scalar extension'' homomorphism of the crossed Brauer group \(XB(S, Q)\) into the crossed Brauer group \(XB(T, G)\). This result is interpreted in terms of the Deuring embedding problem of a central \(T\)-algebra into a central \(S\)-algebra. The reviewed paper also considers the behaviour of the crossed Brauer group under \(Q\)-normal Galois extensions as well as the equivariant Brauer group from the perspective of the relative theory. Finally, the author constructs an eight term exact sequence involving the relative Teichmüller cocycle map and suitable relative versions of generalized Brauer groups, and compares the theory with the eight term exact sequence in the cohomology of a group extension. He also considers some more sophisticated versions of the ordinary, equivariant and crossed relative Brauer groups and shows that the resulting exact sequences behave better compared to the theory with group cohomology than do the usual notions of the generalized relative Brauer groups.
scientific article; zbMATH DE number 6860557
scientific article