# 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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### Variation on localization of Tor

It is known that Let $R$ be a commutative ring with unit and $S \subset R$ a multiplicative sistem. If $M$ and $N$ are $R$-modules there is a isomorphism of $S^{-1}R$-modules: ...
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### Long exact sequence to short exact sequence [duplicate]

$$\dots\xrightarrow{p_*}\pi_{n+1}(B)\xrightarrow{\partial}\pi_n(F)\xrightarrow{\text{inc}_*}\pi_n(E)\xrightarrow{p_*}\pi_n(B)\xrightarrow{\partial}\pi_{n-1}(F)\xrightarrow{\text{inc}_*}\dots$$ From ...
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### $\mathcal{T}^i(X/Y,\mathcal{F})$ forms a sheaf

In Hartshorne's Deformation Theory, given an $A$-algebra $B$ and a $B$-module $M$, he defines these functors $T^i$ for $i=0,1,2$ that outputs $B$-modules $T^i(B/A,M)$. In Exercise 3.5, he asks the ...
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### An identity map which is not null-homotopic

I have some problems in understanding how the definition of a null-homotopic cochain map actually works. Maybe I lack concrete examples. Let $f^{.}:X^{.}\longrightarrow Y^{.}$ a cochain map of ...
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### Analogue of the trivial extension for higher Ext.

I've been doing some homological algebra and some work on showing some extensions are equivalent, and a thought just came to me, which is that I didn't know how to write down what the analogue of the ...
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### Hochschild homology of dgas with nontrivial differential

In this question, we see how to compute the Hochschild homology of a dga with zero differential: it's just the same as computing its Hochschild homology as a graded algebra. I want to know about ...
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### Where can I found an explanation of group cohomology from the point of view of invariants?

I heard once that we can view group cohomology as the right derived functor quantifying precisely (i.e. by the usual long exact sequence) how much the functor of "taking the invariants" is not right ...
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### Estimating regularity of sheaves with rank of certain modules and zeroth cohomology

I'm studying Eisenbud's book "Geometry of syzygies", in particular the Gruson-Lazarsfeld-Peskine theorem for Castelnuovo-Mumford regularity. I'm concerned about an intermediate step in the proof. Let ...
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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 ...
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### A tensor identity - $\text{Hom}_{R}(A,B \otimes_S C) \cong \text{Hom}_{R}(A,B)\otimes _SC$

Let $R,S$ be associative algebras over $\mathbb{C}$. Let $A$, $B$ and $C$ be, a left $R$-module, a $(R,S)$-bimodule, a left $S$-module, respectively. Assume that $B\otimes_S C$ is finite-dimensional. ...
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### $C_\infty$ analog of the correspondence between $A_\infty$-alg. structures on $A$ and dg coalg. strucures on $(\bar T(sA),\Delta)$

There is a 1-1-correspondence between $A_\infty$-algebra structures on a graded vector space $A$ and dg. coalgebra structures on the bar construction $(\bar T(sA),\Delta)$. My question: Is there any ...
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### Relation between ranks of free sheaves and cohomology

Suppose that $\mathbb{P}^r=\mathbb{P}^r_K$ is the projective space over a field $K$. Let $\mathcal{O}_{\mathbb{P}^r}(-1)^n\longrightarrow \mathcal{O}_{\mathbb{P}^r}^m$ be a morphism of vector ...
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### Show that a sequence is a free resolution

Let $I \subset R = k[x_1,\dots,x_n]$ be an ideal and $f \in R$ such that $I = \left < f \right >$ ($k$ is a field, so R is commutative ring). How do I show that (1) $I$ has a free resolution ...
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