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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$i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
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179 views

Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
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313 views

Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
2
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1answer
121 views

The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
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185 views

Definition of Hochschild homology in terms of Tor functor (bar resolutions)

I had 2 kind of dumb questions about the definition of Hochschild homology in terms of the Tor functor: 1 - Let $R$ be a $k$-algebra and $M$ an $R$-bimodule, let $H_*(R,M)$ be the Hochschild homology ...
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62 views

Global dimension of translation algebra

What is the Hochschild cohomological dimension of the "translation algebra": $\mathbb{C}\langle x,y\rangle/(xy-yx-x)$? I expect it to be $2$, but I haven;t found a serious argument as to why this ...
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54 views

Exact sequence from Serre spectral sequence

let me say first that I don't know homological algebra very well, so I apologize in advance if my question is stupid.. It regards the Serre spectral sequence associated to a fibration $0\rightarrow F ...
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218 views

Kunneth formula for group homology

I'm trying to prove Kunneth formula for group homology. $$ 0 \to \bigoplus_p H_p(G,M)\otimes H_{n-p}(G',M') \to H_n(G\times G',M \times M') \to \bigoplus_p Tor_1^{\mathbb Z}(H_p(G,M),H_{n-p-1}(G',M')) ...
3
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1answer
49 views

Computing the order of the first cohomology group $|H^1(S_n, \mathbb F_p^n)|$

Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action. ...
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2answers
68 views

Why does tensoring a projective resolution with a flat module give another projective resolution?

This question came up in this thread: Proving that tensoring a projective module with a flat module gives a projective module? Assume $\left\{P_i\right\}$ is a projective resolution of an $R$-module ...
5
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1answer
155 views

Projective modules over $kG$ equivalent to injective.

Let $k$ be a field and $G$ is finite group. I want to prove that a $kG$ module $P$ is projective iff it's injective. I proved that if module is projective then it's injective. 1) $kG$ is injective ...
5
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1answer
87 views

$L\otimes_{\Delta}\text{Hom}_{\Delta}(M,\Delta)\cong \text{Hom}_{\Delta}(M,L)$

This is exercise 5 in maximal orders by I.Reiner. This is not homework though. Let $\Delta$ be a ring $L_{\Delta}$ be any module, and let $M_{\Delta}$ be a finitely generated and projective. ...
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2answers
191 views

What is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$?

In the category of $\mathbb{Z}$-modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$? I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is ...
0
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1answer
82 views

Using Tor to find the Torsion submodule

Say $R$ is an integral domain with field of fractions $F$. I need to show that, for any $R$-module $B$, $Tor_1^R(F/R, B)\cong t(B)$, where $t(B)$ is the torsion submodule of $B$. So say $$\cdots\to ...
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1answer
248 views

Finite pushforward commute with taking cohomology

Let $f: X \to Y$ be a finite morphism of schemes. How one can show that $f_*H^i(G) \cong H^i(f_* G)$ for any $G \in D(X)$ and any $i \in \mathbb{Z}$? In english, $G$ is a complex of quasi-coherent ...
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1answer
56 views

Empty set in a simplicial complex

Should the empty set be considered a simplex in a simplicial complex? Which justifications exist for the answer? I guess it is somewhat comparable to $1$ not being a prime number.
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1answer
72 views

Possible examples where the Zero Divisor Conjecture does not hold

Given a ring $R$ with a nonzero zero divisor $x$, it is easy to show that if $M$ is a nonzero $R$-module, then there exists $y\in R-\{0\}$ such that $ym=0$ for some $m\in M-\{0\}$. I was ...
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39 views

Graduations and filtrations for localizations

I'm trying to answer the following questions: Let $A$ be a (not necessarily commutative) $\mathbb{Z}$-graded ring and $S$ a multiplicative subset of $A$ such that $AS^{-1}$ exists. Is $AS^{-1}$ a ...
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1answer
104 views

Characterization of the kernel and cokernel of the natural homomorphism between a module and its double dual. [closed]

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Suppose $$ G \overset{\varphi}{\rightarrow} F \to M \to 0$$ is exact where $F,G$ are finite free modules. Suppose ...
6
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1answer
291 views

Singular $\simeq$ Cellular homology?

Given an arbitrary CW-complex, are the singular chain complex $S_\ast(X)$ and cellular chain complex $C_\ast(X)$ homotopy equivalent or just quasi-isomorphic (some chain map induces isomorphisms on ...
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65 views

Hochschild dimension

I'm curious; if $A$ ia a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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1answer
296 views

When does a functor commute with colimits?

Is it true that an additive functor between abelian categories commutes with colimits if it's right-exact and commutes with (arbitrary) direct sums? If yes, does someone know a good source of a ...
4
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1answer
336 views

Homology of mapping telescope

It is stated here http://math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf that if $X$ is an increasing union of the type $X=\bigcup_{i \in I}X_i$ (where $X_i \subset X_{i+1}$), then we have an ...
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50 views

Free resolution by Groebner basis

I am studying approaches of Groebner basis in Homological and commutative algebra. I am so confused how can I find the minimal resolution for the below ideal $$I=\langle ...
4
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2answers
114 views

Hochschild cohomology of a formal quantization of an associative algebra

Let $A$ be a commutative associative $k$-algebra and let $A[[\hbar]]$ be the formal deformation of $A$. I would like to know if there is a relation between the Hochschild co-homologies ...
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21 views

Hochschild (co)-homology of a formal quantization of an associative algebra [duplicate]

Let $A$ be a commutative associative $k$-algebra and let $A[[\hbar]]$ be the formal deformation of $A$. I would like to know if there is a relation between the Hochschild co-homologies ...
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1answer
139 views

What can we say about groups $G$ with $H_3(G)=0$?

Let $G$ be a group. What can we say about groups such that $H_3(G)=0$? If a characterization is not possible, then knowing examples of such groups would be good? Any help is appreciated. Thanks
4
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1answer
166 views

Homological algebra (homotopical approach)

I have gone through a couple of courses in homological algebra, in the context of derived functors, abelian categories,... Now I would like to watch it from another perspective: my main interest is ...
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2answers
142 views

Soft sheaves adapted to $f_!$

I'm reading Gelfand-Manin, Homological Algebra. I understand that the class of soft sheaves is sufficiently large, because every injective sheaf is soft. Now to see that this class is adapted to ...
4
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1answer
175 views

From a vector bundle to a Koszul complex

Let $k = \mathbb C$. Given a commutative $k$-algebra $A$, an $A$-module $M$ and a homomorphism of $A$-modules $s:M \to A$, we can construct the Koszul dg algebra. $$K(A,M,s) = \wedge^{-\!*}_A(M)$$ ...
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94 views

$\operatorname{Ext}^0$ for free resolutions

I am studying homological algebra for an exam in algebraic topology, and I was wondering: Let $H,G$ be two abelian groups. What is $\operatorname{Ext}^0(H;G)$? Now here's what I have done: We ...
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3answers
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$\mathbb{Z}/n\mathbb{Z}$ projective as $\mathbb{Z}/n\mathbb{Z}$-module

$\mathbb{Z}/n\mathbb{Z}$ as $\mathbb{Z}$-module is not projective because isn't torsionfree, but is projective as $\mathbb{Z}/n\mathbb{Z}$ module ?
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0answers
287 views

Pushout and pullback of short exact sequence of groups

I think that there might be some textbooks which introduce the notions of pushout and pullback of a short exact sequence of groups. However, I cannot find any of them. To be precise, for a given ...
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1answer
64 views

computation of a group homology

A few days ago, I asked a question about a group homology, and it was actually easy. I am continuing computing group homologies, but I am stuck on this: $H_*^{\textrm {grp}}(T, \mathbb{Z}) = ...
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1answer
137 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
4
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1answer
118 views

Two definitions of homology

Let $f,g$ be arrows in an abelian category such that the composite $gf$ is defined and is given by the zero arrow. I shall try to find a definition for the quotient $\ker g /\operatorname{im} f$, ...
0
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1answer
65 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
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41 views

a group homology computation

I was assigned to compute the group homology of $\mathbb{Z}^k$ with $\mathbb Z$ as coefficient ring(with the trivial action): $H_*(\mathbb{Z}^k, \mathbb{Z})$. I know that $H_*(\mathbb{Z}^k, ...
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1answer
115 views

If two maps induce the same homomorphism, then they are homotopic

If two chain maps $f,g:\mathcal{X} \rightarrow \mathcal{Y}$, where $\mathcal{X},\mathcal{Y}$ are chain complexes with free modules $X_p$ and $Y_p$ over a PID, $R$, induce the same homomorphism in the ...
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1answer
81 views

Period of a particular finite group

Let $G$ be a group fitting in the following exact sequence: $0 \to \mathbb{Z}/p \to G \to \mathbb{Z}/q^r \to 0.$ Here $q$ and $p$ are primes (not necessarily distinct). It is easy to check (by the ...
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1answer
97 views

Obtaining Chain Complex from a Cochain Complex

In this question: Constructing a cochain complex out of a chain complex , palio asked how to construct a co-chain complex when given a chain complex as well as how to go in the opposite direction, ...
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230 views

chain homotopy equivalence between mapping cone complexes

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, ...
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41 views

An inverse limit of a certain inverse system

Let $∆$ be a directed set and $(N_i,f_{ji})_{i∈∆}$ be an inverse system of $R$-modules. Fix $α \in∆$ and consider $(M_i,g_{ji})_{i\in∆}$ as follows: $M_i=N_i$ for $i≥α$, $M_i=0$ for $i<α$, and ...
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1answer
260 views

Splitting short exact sequence of space groups

I want to prove the following: Assume we have two space groups $G,G^\prime \subseteq \text{Euc}(V) \subseteq \text{Aff}(V)$ which are affinely equivalent, $G \sim G^\prime, \; \text{ i.e. }\; ...
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1answer
92 views

An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
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1answer
99 views

When does a short exact sequence of representations exist?

The context for this question is that I am trying to determine the Grothendieck group of finite-dimensional complex representations of $T = (\mathbb{C}^*)^n$, where $\mathbb{C}^*$ denotes the ...
3
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1answer
88 views

Action of the functor Ext$_1(-,-)$ on extensions

Suppose we have an exact sequence of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & L & \overset{f}{\longrightarrow} & M & \overset{g}{\longrightarrow} & E & ...
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If $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$ then $D$ is divisible

This is Exercise 7.15(ii) from Rotman's book, Introduction to homological algebra that I'm doing. If $D$ is an abelian group and $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$, prove that $D$ is ...
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1answer
111 views

If R and S are artinian and finite dimensional algebras respectively, then the tensor product of them is artinian.

Let $R$ be an artinian algebra and $S$ be a finite dimensional algebra over the field $k$. How can i show that $R\otimes_kS$ is artinian? I know that $S$ is also artinian since it is finite ...
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A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...