Homology, homomorphisms and presentations of commutative algebras (Q1820207)

All rings and algebras will be commutative and with unit, and a ring homomorphism will map unit into unit. The author establishes sufficient conditions for that the subalgebra generated by a set of elements of a supplemented algebra be a polynomial algebra, e.g.: Let B be a supplemented algebra over a ring A with \(H_ 1(A,B,A)=0\). If the augmentation ideal I of B contains a family of elements \(\{x_ s| \quad s\in S\}\) such that the elements \(x_ s\) generate B as an algebra and the set \(\{x_ s+I^ 2| \quad s\in S\}\) is a basis of the A- module \(I/I^ 2\), then B is the polynomial algebra over A in the indeterminates \(\{x_ s| \quad s\in S\}.\) Let A be a PID and let B be a supplemented algebra over A given by a finite presentation \((x_ 1,...,x_ n| r_ 1,...,r_ m)\). The number n-m is called the deficiency of the presentation. The author defines def(B), the deficiency of B, to be the maximum of the deficiencies of the finite presentations of B. If M is a finitely generated A-module, by sM is denoted the minimum number of generators of M. The author proves: (1) Let B be a supplemented algebra over A such that B is finitely presented and let I be the augmentation ideal of B. Then \(def(B)\leq rank_ A(I/I^ 2)-sH_ 1(A,B,A);\) (2) Let B be a supplemented algebra over A given by a presentation with \(n+m\) generators and m relators. Suppose that \(x_ 1,x_ 2,...,x_ n\) are elements of the augmentation ideal I of B such that \(x_ 1,x_ 2,...,x_ n\) generate B as an algebra over A. Then B is the polynomial algebra over A in the indeterminates \(x_ 1,x_ 2,...,x_ n\).