an:03977123
Zbl 0605.13010
Wiegmann, Klaus Werner
A theory of relative extensions for subalgebras and submodules
EN
Quaest. Math. 9, 471-493 (1986).
00152380
1986
j
13D03 13B02 18G15
Relative extensions; complex analytic deformation; relative derivation
Relative extensions of algebras and submodules are studied. These extensions appear naturally in complex analytic deformation theory. Let \((A,B)\) and \((G,f)\) be two pairs, where \(A\subset B\) is a subalgebra over the commutative ring \(K\), \(F\) is a \(B\)-module, \(G\subset F\) is a sub-\(A\)-module. Consider all relative extensions
\[
\begin{matrix} 0 & \to & F \to \tilde B & \to & B &\to 0 \\ && \cup && \cup && \cup \\ 0 & \to & G & \to & \tilde A & \to & A & \to & 0 \end{matrix}
\]
such that both rows are (\(K\)-split) extensions. Let \(\text{Ex}(A,B;G,F)\) denote the class of all relative extensions. The equivalence relation may be established on \(\text{Ex}(A,B;G,F)\). In this way we obtain a set \(\text{Ex}(A,B;G,F)\) of equivalence classes, which endowed with two operations: \(K\times \text{Ex}(\cdot)\to \text{Ex}(\cdot),\) \(\text{Ex}(\cdot)\times \text{Ex}(\cdot)\to \text{Ex}(\cdot)\) becomes a \(K\)-module.
Proposition 1. \(\text{Ex}(A,B;G,F)\) is isomorphic to the \(K\)-module
\[
[C^ 1(A,F/G) \times_{Z^ 2(A,F/G)} Z^ 2(B,F)]/C^ 1(B,F).
\]
The notations according to \textit{S. MacLane} [Homology. Berlin etc.: Springer-Verlag (1963; Zbl 0133.26502); 3rd edition (1975; Zbl 0328.18009)] are used here. If \(K\) is a field, the vector space \(\text{Ex}(A,B;G,F)\) is isomorphic to \(Z^ 2(A,B;G,F)/C^ 1(A,B;G,F)\). Using this isomorphism, the author establishes some long exact sequences of relative derivation and extension modules which interlock in an interesting way.
V. Sharko
Zbl 0328.18009; Zbl 0133.26502