We have constructed the bijection ψ: e(C, A) →Ext^1(C, A) using a projective resolution of C. De

step 1 In this question it is told that if a function is defined from set A to B and it is a bijective

We have constructed the bijection ψ: e(C, A) →Ext^1(C, A)...

We have constructed the bijection $\psi: e(C, A) \rightarrow \operatorname{Ext}^{1}(C, A)$ using a projective resolution of $C$. Define a function $\psi^{\prime}: e(C, A) \rightarrow$ $\operatorname{Ext}^{1}(C, A)$ using an injective resolution of $A$, and prove that $\psi^{\prime}$ is a bijection.

Key Concepts

Injective Resolution

An injective resolution is an exact sequence where a given module is embedded into an injective module, followed by subsequent modules that continue the sequence. This construction is essential for computing right derived functors such as Ext, and it serves as a dual to the projective resolution commonly used for left derived functors.

Projective Resolution

A projective resolution is an exact sequence that begins with the module in question and continues with projective modules mapping onto that module. This resolution is the standard method for computing derived functors like Ext by facilitating the lifting of maps through these well-behaved projective modules.

Ext Groups

Ext groups are algebraic constructs in homological algebra that measure the obstructions to extending modules. Specifically, Ext^1 groups classify equivalence classes of short exact sequences (extensions) of modules, providing a precise formulation of how non-split extensions can be characterized by their corresponding Ext elements.

Bijection Between Extensions and Ext^1

The bijection between the set of equivalence classes of extensions (denoted by a set like e(C, A)) and the group Ext^1 exemplifies the deep connection between exact sequences and derived functors. It establishes that every extension corresponds uniquely to a class in Ext^1, and this correspondence can be constructed using either projective or injective resolutions. The proof hinges on demonstrating that the map constructed via an injective resolution (?') is both well-defined and invertible, mirroring the classical Yoneda interpretation of Ext.

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