# 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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### Topology on an Ext group

One can show that the group $\text{Ext}^1(\mathbf Q, \mathbf Z)$ (calculated in $Ab$) identifies naturally with $\mathbf A_f/\mathbf Q$, where $\mathbf A_f$ is the additive group of finite adèles. ...
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### Eilenberg-Moore Spectral Sequence for Homology with Coefficients in the Integers

I am trying to learn about the Eilenberg-Moore spectral sequence to compute homology and cohomology. I have been using Hatcher's book on spectral sequences and also McCleary's "A User's Guide to ...
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### Best approximation to an adjoint functor

I have the following question. Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best ...
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### What are some important examples of differential objects that aren't naturally graded?

[By a "differential object" I mean an object $A$ in some abelian category $\mathcal{A}$ together with a morphism $d : A \to A$ such that $d \circ d = 0$. By a "differential module" I mean a ...
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### Hypercohomology - now replaced by derived functors?

On the Wikipedia article for hypercohomology I find the following sentence. Hyperhomology is no longer used much: since about 1970 it has been largely replaced by the roughly equivalent concept ...
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### Is there an explicit description for injective sheaves?

I want to find a criterion for sheaves of modules to be injective. It would be great if one can such a criterion for sheaves of modules over a ringed space. But an answer for sheaves of abelian groups ...
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### Has this variation of Hochschild cohomology been studied?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Let $M$ be an abeliean group, and assume that it an $n$-$A$-module. That is: it has $n$ different $A$-module structures, and they are ...
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### Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
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### Leray spectral sequence for complexes

Let $f:X\rightarrow S$ be a morphism of schemes. Let $0\rightarrow C_1 \rightarrow C_2 \rightarrow C_3 \rightarrow 0$ be an exact sequence of Abelian sheaves on $X$. Is there a general procedure to ...
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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: ...
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### The cohomology of finite $G$-modules

This is to some extent a continuation of an earlier question of mine. Now that I'm all cleared up on what it means for a finite group to have periodic cohomology, I have another question; first I will ...
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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 ...
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### Connection between cobar construction of DG-coalgebra and cobar construction from monad

Given a monad $M:C\to C$ we can construct a cobar resolution from it directly as a functor $\Delta\to [C,C]$ Given a DG-coalgebra $(C,d)$ we can construct a cobar resolution $\Omega C$ of it as ...
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### Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
### Generalization for Leray Hirsch theorem for Principal $G$-bundle
This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...