Articles of homological algebra

A particular isomorphism between Hom and first Ext.

Let $R$ commutative ring and $I$ an ideal of $R$. How do I prove that $\operatorname{Ext}^1_R(R/I,R/I)$ isomorphic to $\operatorname{Hom}_R(I/I^2,R/I)$ ? This question is an exercise of the course but has a chance of being false.

Exactness of short exact sequences

Let $A$ be a ring and let $P$ be a projective $A$-module. Then, the exactness of the sequence: $$0\longrightarrow M_1 \overset{f}{\longrightarrow}M_2\overset{g}{\longrightarrow}M_3\longrightarrow 0 \tag{1}$$ implies the exactness of the induced $\mathbb{Z}$-module sequence: $$0\longrightarrow \text{Hom}_A(P,\,M_1) \overset{f_\bullet}{\longrightarrow} \text{Hom}_A(P,\,M_2)\overset{g_\bullet}{\longrightarrow} \text{Hom}_A(P,\,M_3)\longrightarrow 0 \tag{2}$$ Does the exactness of (2) imply that $g$ is an epimorphism? Let $x\in M_3$. Suppose that there […]

Monic and epic implies isomorphism in an abelian category?

This question already has an answer here: Some questions on abelian category 2 answers

Finite projective dimension and vanishing of ext on f.g modules

Let $A$ be a commutative noetherian ring. Suppose $M$ is a finitely generated $A$-module. Let $n>0$ be an integer. It is well known that if $Ext^n(M,N) = 0$ for all $A$-modules $N$, then $M$ has a finite projective dimension. What happens if we only know this for finitely generated $N$? That is, suppose that for […]

when to use projective vs. injective resolution

I am a bit confused about when I should use projective versus injective resolutions to calculate derived functors. Am I correct in thinking that for right exact functors, the left derived functor is defined using projective resolutions and for left exact functors, the right derived functor is defined using injective resolutions? Is this true regardless […]

When is the kernel pair of a finite presentation of algebraic structures finitely generated?

Fix an algebraic theory; denote its free models by $T^k$. There are two possible definitions of what it means for a coequalizer $T^m\twoheadrightarrow M$ to be a finite presentation of $M$. $f\colon T^m\twoheadrightarrow M$ is the coequalizer of some pair of morphisms $T^k\rightrightarrows T^m\twoheadrightarrow M$. This is the standard definition in category theory textbooks. If […]

Rank of a cohomology group, Betti numbers.

How is the rank of a cohomology group computed and what does it convey? I am trying to understand the concept behind betti numbers in a simplicial homology. Edited with details: Given a set of nodes/vertices/points X, let $C_{l}(X)$ denote the subsets of $X$ with cardinality $|C_{l}(X)|=l+1$. $\partial_{l}$ and $\delta_{l}$ are bounded linear maps with […]

The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) \stackrel{\text{def}}{=} \sup(\{ i \in \mathbb{N} \mid {H_{I}^{i}}(M) \neq 0 \}). $$ […]

Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of $R$-modules $\{N_i\}_{i\in I}$ the canonical map of abelian groups $$\bigoplus\limits_{i\in I}\text{Hom}_R\left(M,N_i\right)\longrightarrow\text{Hom}_R\left(M,\bigoplus\limits_{i\in I} N_i\right)$$ is an isomorphism. Example: Any finitely generated $R$-module is […]

If two chain maps over a PID induce the same homomorphism, then they are homotopic

If two chain maps $f,g:\mathcal{X} \rightarrow \mathcal{Y}$, where $\mathcal{X},\mathcal{Y}$ are chain complexes with free modules $X_p$ and $Y_p$ over a PID, $R$, induce the same homomorphism in the homology, then how to prove that they are homotopic?