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

Bachelor thesis

Time: Thu 2023-10-26 09.00 - 10.00

Location: Albano, Cramer Room

Respondent: Lisa Reiner

Supervisor: Gregory Arone

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.