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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When does homology commutes with arbitrary direct sums

Is it necessary to have the criteria that the direct sum of a collection of monics is a monic, to show that homology commutes with arbitrary direct sums? Because when I tried to prove the result, I ...
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Injective dimension and Krull dimension of a module

Let $R$ be a regular local ring and $M$ an $R$-module (not necessarily finite), then the injective dimension $\operatorname{id}_R(M)$ of $M$ is finite. When $M$ is finitely generated, we have ...
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80 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 ...
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Finite injective dimension

Let $A$ be a commutative noetherian ring. Is it true that if $A$ is regular then any module over it has a finite injective dimension? What if $A$ is Gorenstein? Any reference who discuss this?
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Another description for the map $\text{Ext}^1_\mathbb{Z}(A,G)\to H^2(G,A)$

Group extensions of $G$ by $A$ $0\to A\to E\to G\to 0$ up to equivalence (where $G$ and $E$ may be nonabelian) are in bijection with the second group cohomology ...
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27 views

homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's Intro to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that cokernels commute with ...
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On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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118 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...
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30 views

Basic computation of exact sequence

Given a long exact sequence of vector spaces: $$...\longrightarrow V_1 \overset{f}{\longrightarrow}V_2\overset{g}{\longrightarrow}V_3\longrightarrow...$$ Given another vector space $W$, is the ...
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66 views

Minimal injective resolution of a module

Let $R$ be a commutative Noetherian ring and $M$ an $R$-module. Let $0\rightarrow M \rightarrow E^{\bullet}$ be a minimal injective resolution of $M$ and $0\rightarrow M\rightarrow I^{\bullet}$ be an ...
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Extension to rational and real chains

In the paper on stable commutator length, D. Calegari says that generalized $\operatorname{scl}$ function can be extended by linearity to rational group $1$-chains and by continuity to real chains ...
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Additive, covariant functor commutes direct limits, then it commutes with direct sums?

Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ ...
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Question about a theorem from Chang's book: Methods in Nonlinear analysis

I have this this theorem from Chang's book: Methods in Nonlinear analysis, with it's proof, but i don't understand it, for example what it means $K(f_{\sigma_i})$ ? Please help me thank you
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258 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 ...
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84 views

Question about Homology from the Chang's book: Methods in Nonlinear analysis

In the K.C.Chang's book in page $336$ of the book this corollary without prove there is a theorem before it but I don't know if it is a corollary of this theorem, how I can prove this ...
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35 views

Question about Chains complexes

I have $\mathcal{U}=\lbrace X-U, A\rbrace$ such that $\overline{U}\subset \overset{º}{A}$ and $X=\overset{º}{(X-U)}\cup \overset{º}{A}$ where $X$ is a topological space, $A$ is a subset of $X$ and ...
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An example of short exact sequence which is not exact triangle.

Let $0 \to \mathbb Z/2\mathbb Z \to \mathbb Z/4\mathbb Z\to \mathbb Z/2\mathbb Z \to 0$ be a short exact sequence of complexes concentrated in degree $0$. How can I prove that it cannot be made into ...
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Derived Functors and nice Resolutions

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following: "Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor ...
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Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
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85 views

Composition of bicartesian squares

A commutative square is called bicartesian when it is both pull-back and push-out. In an abelian category, consider two pull-back squares $(X)$ and $(Y)$: $$ \begin{array}{ccccc} A & ...
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56 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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Example of Tor-Rigid Module

Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ ...
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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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Action of $G/H$ on $H_n(H;M)$

I'm currently studying group cohomology and have trouble with the Hochschild-Serre spectral sequence. My problem is this: Given a short exact sequence of groups $$ 0 \to H \to G \to G/H \to 0$$ how ...
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Question about the definition of homology

$\quad$ The functor $H_n$ measures the number of “$n$-dimensional holes” in the space (or simplicial complex), in the sense that the $n$-sphere $S^n$ has exactly one $n$-dimensional hole and no ...
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Generalisation of cochain complexes and “curvature”

Someone has mentioned to me that generalizations of co-chain complexes and their cohomology have been studied, where instead of $d^2 = 0$ we have something like $d^2 \alpha = q \alpha $, which is ...
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37 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that ...
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Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $

Let $\mathbb{P}^1 $ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \ $ we can define some quasi-coherent sheaves on $\mathbb{P}^1 $. We can ...
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221 views

Injective Maximal Cohen-Macaulay modules

Let $R$ be a Gorenstein (not necessarily commutative) ring and let $I$ be an injective finitely generated module over $R$. Is it true that if $\operatorname{Ext}_R^i(I, R)=0$ for $i > 0$, then $I$ ...
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231 views

The relationship of free, divisible, projective, injective, and flat modules.

In general, we have that: free $\Rightarrow$ projective $\Rightarrow$ flat injective $\Rightarrow$ divisible ( ($\Rightarrow$) be ($\Leftrightarrow$) in PIDs) Simple Counter-examples: projective ...
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Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
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The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
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49 views

Short exact sequences and finite injective dimension

Say that $0 \to M \to N \to L \to 0$ is a short exact sequence of modules in a Noetherian local ring and that inj dim$(M)$, inj dim$(N) < \infty$. Does this imply that $L$ also have finite ...
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Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
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Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
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51 views

Question about the Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
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The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) ...
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Universal property of tensor product of dg algebras

Does the tensor product of dg algebras have a universal property? I have not seen anything about this in the literature.
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402 views

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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1answer
36 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
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1answer
40 views

Weibel “Introduction to homological algebra” Main Theorem 4.4.16

I can't understand the proof of Main Theorem 4.4.16 from Weibel's book "An Introduction to homological algebra". The Theorem states Let $R$ be a local noetherian commutative ring, then $R$ is ...
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40 views

Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
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31 views

Calculating the intersection product in CH(X)

Let CH$(X)$ be the Chow-Ring of a projective,smooth variety with cycles modulo rational equivalence. Lets assume Kunneth-Formula holds. There is an intersection product CH$^a(X) \otimes $ CH$^b(X) ...
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Ext functor commutes with connecting homomorphisms?

Suppose we have an exact sequence $0 \to L \to M \to N \to 0$ and a morphism $f \colon A \to B$ of $R$-modules. If $\delta \colon \text{Ext}^{i}_{R}(B,N) \to \text{Ext}^{i+1}_{R}(B,L)$ and $\delta' ...
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Hom and $\otimes$ functors on chain complexes.

I can't solve the exercise $2.7.3$ from Weibel's book "An Introduction to homological algebra": Let $P,Q$ be right and left $R$-module chain complexes, $I$ be a cochain complex of abelian groups. ...
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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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$i^{-1} F$ 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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412 views

What are $E_\infty$-rings?

I've been working with DG-algebras for the last year, and was able to obtain using them some nice commutative homological algebra results. However, I keep hearing about a (more general???) concept of ...
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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 ...
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132 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. }\; ...