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Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

A chain complex is a sequence of abelian groups, vector spaces, or modules, with linear maps connecting them which compose to zero.

Homological algebra is the study of chain complexes and their homology groups.

5114 questions
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Is the t-truncation a functor?

One of the axioms of a t-structure in a triangulated category is that any object $X$ can be embedded inm a distingueshed triangle $$ X_0\to X\to X_1\to^+ $$ The original work by Beilinson-Bernstein-Deligne seems to suggest that this decomposition…
fosco
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Does every complex have a quasi-isomorphic projective complex?

Let $C^{\textbf{.}}$ be a complex in some abelian category (edit: assuming it has enough projectives). I would like to know if there exist a complex $X^{\textbf{.}}$ consisting of projective objects and a quasi-isomorphism $f:C^{\textbf{.}}\to…
edo arad
  • 985
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Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$

Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$ for all $n$…
user55407
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Exact sequence of double complexes induces exact sequence on total complexes

This is a homework question, so I'd appreciate hints (or perhaps explanations of concepts I've not properly digested) Anyhow: This is exercise 1.3.6 in Weibel's book on homological algebra. Let $0 \to A \to B \to C \to 0$ be an exact sequence of…
Fredrik Meyer
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Horse shoe lemma exercise from Weibel's Homological Algebra

This is regarding the exercise 2.2.4 in Weibel's Introduction to homological Algebra which is about Horse shoe lemma for projective resolutions. That is if $0 \to A' \to A \to A'' \to 0$ is an exact sequence of R modules and if ${P'}_{.}$ and…
budi
  • 1,780
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The hyper-derived functors $\mathbb L_\bullet F$ are just derived functors of $H_0F$?

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.4,2) Let $\mathbf{Ch}_{\ge0}(\mathcal A)$ be the subcategory of complexes $A$ with $A_p=0$ for $p<0$. Then the hyper-derived functors $\mathbb L_iF$ retricted to…
Yai0Phah
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Non-trivial conditions for $\mathrm{Ext}^2(A,B)=0$?

Edit: Since I had some trouble making my previous question precise without diving into details about the origin of the homological objects I'm interested in, let me ask a more open-ended question: Are there any known examples of the following…
Rasmus
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Confused about Weibel proof

In Weibel (Introduction to Homological Algebra)'s proof that left derived functors form a homological $\delta$-functor (Thm. 4.2.6), he does a lot of work that seems unnecessary to me. The relevant pages can be seen on Google…
Yuri Sulyma
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Prove that every module $M$ is epic image of a projective module.

My Proof: We knew that every module $M$ is epic image of a free module and a free module is a projective module, hence every module $M$ is epic image of a projective module. Is that true?! If false, please give me a hint.
Rachel
  • 529
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Help to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?

Consider $\mathbb{Z}_{2}$ as a $\mathbb{Z}_{4}$-module. How to compute $\mathrm{Tor}_{n}^{\mathbb{Z}_{4}}(\mathbb{Z}_{2},\mathbb{Z}_{2})$?
user6495
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Projective objects in compounded Abelian category

Suppose we have an Abelian category $\mathfrak A$ and a ring $R$. From this data we can form a new Abelian category $\mathfrak A[R]$ whose objects are objects $A\in\mathfrak A$ together with a ring homomorphism $R\to\mathfrak A(A,A)$ and whose…
Rasmus
  • 18,404
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Derived Functors

Why do we say "This functor is left exact, but not right exact" instead of "This functor preserves limits, but not colimits". It seems more natural to base the theory of derived functors on the second statement instead of the first. I've never…
six
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Ext and Tor over noncommutative rings

This might be a stupid question, but I could not find a good reference that thoroughly explains the matter. I will start with some lengthy introduction. If $\mathcal{A}$ is any abelian category, then we have functors $\mathrm{Hom}_\mathcal{A} (M,-)$…
J. Doe
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Why does the antisymmetrization map factor through $n$-forms?

Consider a $k$-algebra $A$ and a bimodule $M$. One can construct two complexes, the Hochschild complex $C_n(A,M)$ and the Chevalley-Eilenberg complex $C'_n(A,M)=M\otimes \Lambda^n(A)$. Given an element $m\otimes a_1\otimes \cdots\otimes a_n$ and a…
Pedro
  • 122,002
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Exercise 2.7.3 of Prof. Weibel H-Book is wrong. Suggestion for an errata.

In this exercise, we have to prove that there is an isomorphism $$\text{Hom}(\text{Tot}^{\oplus}(P\otimes Q),I)\cong \text{Hom}(P,\text{Tot}^{\prod}(\text{Hom}(Q,I))$$ of double complexes. But if I choose $I$ to be the cochain complex with only…
brunoh
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