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A theory of relative extensions for subalgebras and submodules. (English) Zbl 0605.13010
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 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.
Reviewer: V. Sharko
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13B02 Extension theory of commutative rings
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
Full Text: DOI
[1] DOI: 10.1080/16073606.1979.9631569 · Zbl 0399.18008 · doi:10.1080/16073606.1979.9631569
[2] Mac Lane S., Homology. Berlin (1975)
[3] DOI: 10.1007/BF01432934 · Zbl 0268.32009 · doi:10.1007/BF01432934
[4] Wiegmann, K. W. 1972.Einbettung des relativen Wolffhardt- Raumes. Sb. Bayer. Akad. Wiss.81–86. Math.Nat. K1 · Zbl 0281.32014
[5] DOI: 10.1007/BF01367772 · Zbl 0233.32019 · doi:10.1007/BF01367772
[6] DOI: 10.2307/2373544 · Zbl 0159.37901 · doi:10.2307/2373544
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