Baer multiplication

A binary operation on the set of classes of extensions of modules, proposed by R. Baer [1]. Let $A$ and $B$ be arbitrary modules. An extension of $A$ with kernel $B$ is an exact sequence:

$$\tag{1 } 0 \rightarrow B \rightarrow X \rightarrow A \rightarrow 0.$$

The extension (1) is called equivalent to the extension

$$0 \rightarrow B \rightarrow X _ {1} \rightarrow A \rightarrow 0$$

if there exists a homomorphism $\alpha : X \rightarrow X _ {1}$ forming part of the commutative diagram

$$\begin{array}{ccccc} {} &{} & X &{} &{} \\ {} &\nearrow &{} &\searrow &{} \\ B &{} &\downarrow &{} & A \\ {} &\searrow &{} &\nearrow &{} \\ {} &{} &X _ {1} &{} &{} \\ \end{array}$$

The set of equivalence classes of extensions is denoted by $\mathop{\rm Ext} (A, B)$. The Baer multiplication on $\mathop{\rm Ext} (A, B)$ is induced by the operation of products of extensions defined as follows. Let

$$\tag{2 } 0 \rightarrow B \mathop \rightarrow \limits ^ \beta X \mathop \rightarrow \limits ^ \alpha A \rightarrow 0,$$

$$\tag{3 } 0 \rightarrow B \rightarrow ^ { {\beta _ 1} } Y \rightarrow ^ { {\alpha _ 1} } A \rightarrow 0$$

be two extensions. In the direct sum $X \oplus Y$ the submodules

$$C = \{ {(x, y) } : { \alpha (x) = \alpha _ {1} (y) } \}$$

and

$$D = \{ {(-x, y) } : { x = \beta (b),\ y = \beta _ {1} (b) } \}$$

are selected. Clearly, $D \subset C$, so that one can define the quotient module $Z = C/D$. The Baer product of the extensions (2) and (3) is the extension

$$0 \rightarrow B \rightarrow ^ { {\beta _ 2} } Z \rightarrow ^ { {\alpha _ 2} } A \rightarrow 0,$$

where

$$\beta _ {2} (b) = \ [ \beta (b), 0] = \ [0, \beta ^ \prime (b)],$$

and

$$\alpha _ {2} [x, y] = \ \alpha (x) = \ \alpha _ {1} (y).$$

References

 [1] R. Baer, "Erweiterung von Gruppen und ihren Isomorphismen" Math. Z. , 38 (1934) pp. 374–416 [2] H. Cartan, S. Eilenberg, "Homological algebra" , Princeton Univ. Press (1956)
How to Cite This Entry:
Baer multiplication. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Baer_multiplication&oldid=45581
This article was adapted from an original article by V.E. Govorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article