Subintegrality and invertible modules in graded rings
An extension \(A\subseteq B\) of rings is said to be subintegral if for each \(b\in B\) there exists a finite sequence \(A=C_0\subseteq C_1\subseteq \cdots \subseteq C_r\subseteq B\) of extensions such that \(b\in C_r\) and \(C_{i+1} =C_i[a]\) for some \(a\) with \(a^2\), \(a^3\in C_i\) and for all \(i\). For an extension \(A\subseteq B\), \({\mathcal F}(A,B)\) denotes the multiplicative group of all invertible \(A\)-submodules of \(B\). For a positively graded ring \(A\), \(\widehat A\) is the completion of \(A\) with respect to the ideal \(\bigoplus_{n\geq 1}A_n\). By \(A_*\) is denoted the obvious subring of \(\widehat A\) where the multiplication \(*\) on homogeneous elements is given by \(f*g={m+n\choose m}fg\) where \(m=\deg(f)\), \(n=\deg(g)\) and extended by bilinearity and completion. The main result of the paper is the construction of a functorial isomorphism \(\zeta_{B/A}: B/A\to {\mathcal F} (A_*,B_*)\) for any subintegral extension \(A\subseteq B\) of positively graded rings with \(A_0=B_0\). Moreover the author shows that \({\mathcal F}(A_*,B_*)\) can be exhibited as the associated graded of \({\mathcal F}(A,B)\) with respect to some natural filtrations and he investigates special properties of this isomorphism for \(A\) containing a field.
scientific article; zbMATH DE number 1340604