# Tagged Questions

Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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### Why are spherical objects so named?

Let $S$ be an object in an abelian category. Then we say S is spherical if $Ext^p(S,S)$ is 0 unless $p = 3$. I know that the cohomology of the three sphere bears some formal resemblence, but it doesn'...
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### Minimal injective resolutions isomorphism [closed]

How can I prove that given an $A$-module $M$ two injective resolutions of $M$ are isomorphic as complexes? Thank you, have a nice day Asdrubale
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### Are “noncommutative resolutions” a thing?

I am interested in looking at "resolutions" of modules in noncommutative rings, making some obvious necessary modifications to the definition of a resolution. However, whenever I try to search the ...
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### Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
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### Ideals and Tensor Products

I'm reading Osbourne's Basic Homological Algebra, and on page 18 he has this situation where we've got a ring $R$ and a right-ideal $I$, and some left $R$-module $B$. He says $I\otimes B$ is not a ...
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### Generalization of Universal Coefficient Theorem

Suppose we are in an abelian category $\mathscr{A}$. Given a fixed monomorphism $A \overset{i}{\hookrightarrow} B$, and an object $C$, I would like to express concisely the notion of the group of maps ...
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### Why does this homological lemma hold?

Let $A$ be a noetherian ring; let $C^{\boldsymbol\cdot}$ be a bounded above complex of flat $A$-modules in positive degrees, let $L^{\boldsymbol\cdot}$ be a bounded above complex of free $A$-modules ...
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### Homotopy Colimit of Truncations

Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ...
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### Homological dimension of categories of modules

Let $A$ be a Noetherian ring. We have two categories: (a) category of $A$-modules (b) category of finite type $A$-modules. Do their homological dimensions agree? The homological dimension of an ...
Given continuous maps $f_i : X_i \to Y_i$ ($i=1, 2$) we may consider the singular chain cocomplexes $$C^n(Y_i) \oplus C^{n-1}(X_i)$$ with boundary operator: $$(u^n, v^{n-1}) \mapsto (-\delta u^n, ... 1answer 59 views ### Ext and Tor over noncommutative rings This might be a stupid question, but I could not find a good reference that thoroughly explains the matter. I will start with some lengthy introduction. If \mathcal{A} is any abelian category, then ... 1answer 33 views ### Homology group versus group homology If we have a simplical complex K, then we are able to define C_i(K) as the free abelian group over \mathbb Z_2 with the basis of all i-dimensional simplices. By using the boundary map we are ... 1answer 95 views ### 0\to C'\to C\to C''\to0 splits if C\cong C'\oplus C'' as a chain complex? Question Given a unitary ring A and an exact sequence$$0\to C'\xrightarrow iC\xrightarrow pC''\to0$$in the Abelian category of chain complexes over A, where C,C',C'' are chain complexes of ... 1answer 2k views ### Abstract nonsense proof of snake lemma During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ... 3answers 342 views ### How to do diagram chasing effectively? I am trying to teach myself some homological algebra, and the book I am using is Aluffi's wonderful Algebra: Chapter 0, which introduces homology at the end of chapter 3. I have spent a lot of time ... 1answer 78 views ### Is a kernel in a full additive subcategory also a kernel in the ambient abelian category? Setting: Let \mathscr{C}\subset \mathscr{A} be a full additive subcategory of an abelian category. Let C,C' be objects of \mathscr{C} and let f\in \operatorname{Hom}_\mathscr{C}(C,C')=\... 0answers 24 views ### Definition of contractible chain complex A relatively simple question. A book I'm reading states "a complex homotopic to the zero complex is called contractible"... but I don't understand the statement. I know what it means for chain maps ... 1answer 43 views ### Faithfully flat descent of projectivity and freeness I am reading this paper. It is proven there that if f:A\rightarrow B is a faithfully flat morphism of rings and M an A-module such that the B-module M\otimes_A B is projective, then M ... 0answers 30 views ### Interaction of a functor with internal hom An additive functor between abelian categories F: \mathscr{C} \to \mathscr{D} induces a functor on categories of chain complexes F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet. The internal hom ... 1answer 38 views ### On selfinjectivity of Hopf algebras Any group algebra kG of a finite group is selfinjective. More generally Gentile proves that for a group ring RG with R commutative and torsion free as a \Bbb Z-module, RG is selfinjective ... 1answer 609 views ### Projective resolution of tensor product Let M,N are R modules and P^\bullet, Q^\bullet are their projective resolutions. Can we obtain projective resolution M\otimes N using P^\bullet, Q^\bullet. If i understand correctly homology ... 2answers 60 views ### Cohomology of free group acting trivially I read here the formula for cohomology of a free group with trivial action:$$H^q(G, M) = \begin{cases} M &\text{for } q = 0\\ M^n &\text{for } q = 1\\ 0 &\text{for } q \geq 2\\ \end{cases}...
Consider the internal hom of chain complexes $A^\bullet$ and $B^\bullet$ $$\operatorname{Hom}^\bullet(A^\bullet, B^\bullet):=\{\text{degree n maps}\} \qquad df:= d^B\circ f - (-1)^{|f|}f \circ d^A$$...