Let $\mathsf{A}$ be an abelian category, such as $R \mathsf{Mod}$ for concreteness. We can think of the category of $\mathbb{Z}$-graded objects in $\mathsf{A}$ as the functor category $\mathsf{A}^\mathbb{Z}$ where $\mathbb{Z}$ is viewed as a discrete category. This works for $\mathbb{N}$-gradings as well. The category of filtered objects in $\mathsf{A}$ in the classical sense may (potentially) […]

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 […]

The standard categorical definition of image is that it is the cokernel of the kernel. Under what nice conditions does this definition coincide with kernel of the cokernel? It coincides for abelian groups, modules,… and I’d dare to say that over any category of sets with “some additional structure” as it is usually vaguely defined […]

Let $C$ be an abelian category. Assuming that $\hom_C(A,B)$ has an abelian group structure, prove that the zero map $0_{AB}:A\to B$ is the neutral element of this group. I know that the group operation can be defined as $f+g:=\nabla_B\circ(f\oplus g)\circ \Delta_A$ for every $f, g\in\hom_C(A, B)$, where $\oplus$ is the biproduct, $\Delta_X:X\to X\oplus X$ is […]

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence associated to a short exact sequence for $\operatorname{Hom}_{\mathcal{A}}$ where $\mathcal{A}$ is an abelian category. To apply this construction I have proven the Chinese remainder […]

Let $E^\bullet$ and $F^\bullet$ be complexes on an abelian category; what does it mean to say that $E^\bullet$ and $F^\bullet$ are quasi-isomorphic? Does it only mean that there is a map of complexes $f:E^\bullet \to F^\bullet$ that induces isomoprhisms between the cohomology objects? Or does it also guarantee the existence of a map of complexes […]

I have some trouble in proving Exercise A3.51 of Eisenbud’s book “Commutative Algebra with a view toward Algebraic Geometry”, pag. 688. The solution is sketched at pag. 754 at the end of the book. The only step that is not clear to me is how to prove that the maps labelled $h$ e $k$ (of […]

It’s truly bizarre that there exists no full modern exposition of this theorem, as noted elsewhere. Anyway, I thought I’d poke through and see if I could get the gist of how it works as somebody who has familiarity with categorical techniques, if not abelian techniques. Here’s how it goes, following Swan and wikipedia. We […]

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, Ch. II, §4] asserts that the following data define a model structure on $\mathcal{M}$: The weak equivalences are the quasi-isomorphisms (= homology isomorphisms). The cofibrations are the monomorphisms […]

This question is the categorical version of this question about splitting up long exact sequences of modules into short exact sequence of modules. I want to understand the general mechanism for abelian categories. Here’s the decomposition I managed to get: But it doesn’t involve any cokernels like the module version does. The closest I get […]

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