Verify the basic properties of E(A, B). In particular, show that the Baer sum respects the equiv

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Verify the basic properties of E(A, B). In particular, show...

Verify the basic properties of $\mathrm{E}(A, B)$. In particular, show that the Baer sum respects the equivalence relation and that addition is associative and commutative. Furthermore, show that the split extensions form a single class, which corresponds to the zero element of $\mathrm{E}(A, B)$.

Key Concepts

Split Extensions and the Zero Element

A split extension is an extension that is isomorphic to the direct sum of its subobjects, thereby trivializing the connecting maps. In the group E(A, B), all split extensions form a single equivalence class, and this class acts as the zero element. This means that when any extension is added to a split extension (via the Baer sum), the result is the original extension, reflecting the identity property in an abelian group.

Associativity and Commutativity of Addition

The group operation on extension classes, defined via the Baer sum, must satisfy the conditions of being associative and commutative. Associativity means that the order in which the operations are performed on three extension classes does not affect the outcome, while commutativity ensures that the sum of two classes does not depend on the order. These properties are verified by carefully analyzing the pushout and pullback constructions underlying the Baer sum.

Baer Sum

The Baer sum is an operation that combines two extensions (or equivalence classes of extensions) of a given pair of objects into a new extension. It provides the addition law in the group structure of extension classes. This sum is constructed via pushouts and pullbacks that merge the exact sequences, and it ensures that the group operation is well defined, meaning that it respects the equivalence relation between extensions.

Equivalence of Extensions

In the study of extensions, two short exact sequences are considered equivalent if there exists a morphism between them making the obvious diagrams commute. This equivalence relation is essential in forming the group E(A, B) since the Baer sum is defined on these equivalence classes rather than on individual sequences. It ensures that the algebraic structure on extensions is independent of the choice of representatives within each class.

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