Lisa Reiner: Introduction to Homological Algebra, up to the Definition Ext_R^n (M, N)

Abstract.

The contravariant functor \(\operatorname{Hom}(\square, D)\) is additive, and moreover left exact, but is not exact. In this thesis we define a sequence of functors \(\operatorname{Ext}^n_R (\square , D)\) which in some sense measure the failure of the Hom functor to be exact. The functors \(\operatorname{Ext}^n_R (\square , D)\) are called the derived functors of Hom. They are defined as the cohomology groups of the cochain complex \(\operatorname{Hom}(P_\bullet , D)\), where \(P_\bullet\) is a choice of projective resolution of the source variable. The definition is independent of the choice of \(P_\bullet\), because \(P_\bullet\) is unique up to chain homotopy, and cohomology groups are homotopy-invariant.

When the functors \(\operatorname{Ext}^n_R (\square , D)\) are applied to a short exact sequence of modules they generate a long exact sequence of cohomology groups. The functors \(\operatorname{Ext}^n_R (\square , D)\) are characterized by this long exact sequence of cohomology groups together with the natural isomorphisms \(\operatorname{Ext}^0_R (\square, D) \cong \operatorname{Hom}(\square , D)\) and \(\operatorname{Ext}^n_R (Q , D)=0\) when Q is projective and \(n > 0\).

The functor \(\operatorname{Hom}(\square, D)\) is exact if and only if D is injective. It follows that an R-module D is injective if and only if \(\operatorname{Ext}^n_R (B,D)=0\) for all modules B and \(n > 0\). One of the first applications of Ext groups stems from the fact that there is a bijection between equivalence classes of extensions of A by C and the group \(\operatorname{Ext}^1_R(C, A)\). We can define the bijection by using the existence of a chain-map \(a_n\) between the projective resolution \(P_C\) of C and the extension A by C. Since a chain-map implies a commuting diagram of the complexes involved, \(a_1 d_2 \colon P_2 \to P_1 \to A\) is equal to \(0 \colon P_2 \to 0 \to A\) and \(a_1\) can be viewed as an element of the kernel of the induced map \(d_2^* : \operatorname{Hom}(P_1 , A) → \operatorname{Hom}(P_2 , A)\).

Since \(a_1\) belongs to the kernel of \(d_2^*\), \(a_1\) is a representative of a coset in \(\operatorname{Ext}^1_R(C, A)\). The map is defined by mapping the extension class represented by the extension A by C to the coset of \(\operatorname{Ext}^1_R(C, A)\) represented by \(a_1\). The inverse is defined by choosing a representative of a coset of \(\operatorname{Ext}^1_R(C, A)\) and a projective resolution of C. Using these two objects an extension A by C is constructed as a second row in a commutative diagram where all second rows are equivalent extensions. The inverse then maps the coset represented by \(a_1\) to the extension class A by C.