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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Exact Sequences in algebraic geometry [closed]

A very basic question. I am going to take my first course in Algebraic Geometry next semester and I am now repeating some commutative algebra to be prepared. I just came up to the part of Homological ...
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121 views

Mayer Vietoris for locally finite singular homology

Usually one defines the traditionnal singular homology (let's say in $\mathbb{Z}$ and on a topological space $X$) by using singular $p$-chains. A singular $p$-chain is a finite formal sum $\sum c_{\...
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Bockstein homomorphism and the universal coefficient theorem

The following statement is given in the third comment of kernel of the mod $2$ Bockstein on the first cohomology group: Statement: Let $X$ be a path-connected finite $CW$-complex. Suppose $$ H_1(X;...
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$\text{Hom}(M \otimes_A N, L) \approx \mathscr{L}(M,N; L)$ The $A$-linear homs from the tensor product into $L$ are isomorphic with bilinear maps.

Let $M,N, L$ be two $A$-modules over a commutative ring $A$. Let $\mathscr{L}(M,N;L)$ be the $A$-module of bilinear maps $M \times N \to L$. Then $\text{Hom}_A(M \otimes_A N, L) \approx \mathscr{L}(...
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Symbol for exactness in diagram

When making commutative diagrams there are many symbols one can use and even omit in favours of others. For example in the following short exact sequence $$0\to A\to B\to C \to 0$$ we can rewrite it ...
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In Kunneth Formula for Cohomology, the finitely generated condition is necessary.

Kunneth formula for cohomology: The cross product $H^*(X;\mathbb Z)\otimes H^*(Y;\mathbb Z)\to H^*(X\times Y;\mathbb Z)$ is an isomoprhism of rings if $X$ and $Y$ are CW complexes and $H^k(Y,R)$ is ...
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Generator of $\tilde{H}_n(S^n;G)$

I'm struggling with the following exercise: Let $\sigma_n: \Delta^n \to\Delta^n/ \partial \Delta^n$ be the quotient map. Show that $[\sigma_n]$ generates $\tilde{H}_n(S^n;G)$ with $G$ any abelian ...
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Are those two ways to relate Extensions to Ext equivalent?

Given an extension $\xi$ of $R$-modules $0\to B\to X\to A \to 0$, one usually associates $x\in\operatorname{Ext}^1(A,B)$ by taking the long exact sequence $$\ldots\to \operatorname{Hom}(A,X) \to \...
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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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chain map induces an isomorphism in homology, but as a cochain map, $f^*$ does not induce an isomorphism in cohomology

Let $f:(X,A)\to (Y,B)$ be a continuous map of pairs $(X,A), (Y,B)$ of topological spaces. $f$ induces a chain map on singular chain complexes $$f_*:C_*(X,A;R)\to C_*(Y,B;R),\; \sum_{\sigma}r_{\sigma}\...
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Finding an example of an extension of length 2

I am working on extensions in general but for sake of simplicity we can assume it's a module here. I am interested in an extension of the form $$0\to B \to E_2\to E_1 \to A \to 0$$ which is an ...
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137 views

Bijection between Extensions and Ext (Weibel Theorem 3.4.3)

I was wondering about one step in the proof of surjectivity of $\Theta$ constructed for Theorem 3.4.3 in Weibel's "An introduction to homological Algebra". For an extension $\xi:0\to B\to X\to A\to0$ ...
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definitition of projective resolution of $R$-modules (with homology)

Let $R$ be a commutative ring with unit $1_R$, $M$ be a $R-$module. I have a small question about different definitions of projective resolutions of $M$ (and I'm confused with the degrees of the ...
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57 views

What is a direct limit of exact sequences?

(Hatcher Section 3.3, page 243) First, recalling the definition of a directed system of groups: Suppose one has abelian groups $G_\alpha$ indexed by some partial ordered index set $I$ having the ...
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31 views

Splitting Lemma where $C=\mathbb{Z}.$

Given a short exact sequence $$ 0 \xrightarrow{\theta_3} A \xrightarrow{\theta_2} B \xrightarrow{\theta_1} \mathbb{Z} \xrightarrow{\theta_0} 0 $$ show that $B \cong A \oplus \mathbb{Z}.$ So far I ...
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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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90 views

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 bundles....
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65 views

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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Complex of banach spaces is exact if and only if its dual is exact

Let's consider two complexes of Banach spaces: $ X \rightarrow Y \rightarrow Z$, with the maps $S: X \rightarrow Y$, $T: Y \rightarrow Z$. The dual complex looks like $Z^{*} \rightarrow Y^{* } \...
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A map of complexes which is zero on cohomology but not zero in $D(\mathcal{A})$

Yesterday I asked a very similar question about an exercise of Gelfand's book "Methods of Homological Algebra". In the comments it was pointed out that there was an easier version of that exercise but ...
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Exercise from “Methods of Homological Algebra” Gelfand

I have to show that a map of complexes $f: A^{\bullet} \to B^{\bullet}$ in $Ab$ with $H^{n}(f)=0$ is not necessarily 0 in the derived category $D(\mathcal{A})$. To find this counterexample I'm given ...
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Filtered Colimit of associative $k$-algebras that are domains

Let $C$ be a filtered subcategory of the category of commutative algebras over a fixed field $k$ whose objects are all integral domains. Then the colimit of the obvious diagram is an integral domain. ...
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The cohomology of the Dirac operator $d+d^{*}$

Let $(M,g))$ be a Riemannian manifold with the Hodge dual operator $d^{*}$. Is there a name (and some computation in some reference) for the cohomology of the complex of Harmonic forms with ...
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58 views

Chain complex and free resolution

If $I \subset R = k[x_1,\dots,x_n]$ is an ideal. Then why: $0 \to C_i \to \dots\to C_0 \to R \to R/I \to0$ is a free resolution of $R/I $ if and only if $0 \to C_i \to \dots\to C_0 \to I \to 0$ is a ...
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Regular sequence in degree 1

$R$ is a graded algebra generated by $R_1$(the degree 1 piece) over $R_0=k$ where $k$ is a infinite field and R has no negative degree. Given irrelevant ideal has depth d, then is it possible to find ...
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Motivation for the mapping cone complexes

I was reading some topics in Homological Algebra when I came across the concepts of cone of a map of complexes and cylinder. My knowledge of Algebraic Topology is pretty basic so I only used these ...
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Eilenberg–Zilber as abstract nonsense - why is it important?

The Eilenberg–Zilber theorem in singular homology, relating the monoidal structure of the category of chain complexes with the chain complex of the cartesian product of the underlying spaces, is used ...
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How to show that $(\Lambda^2(g))^g = H^2(g)$?

Let $g$ be a semisimple Lie algebra and $\Lambda^2(g) = g \wedge g \subset g \otimes g$ the exterior square of $g$. Consider the adjoint action of g on $g \wedge g$ and let $$(\Lambda^2(g))^g = \{x \...
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Terminology question: “boundary map” in homology theory

In homology theory the same name and symbol is used to describe two different maps: 1: the boundary maps $\partial_n: C_n(X) \to C_{n-1}(X)$ appearing in a chain complex of (say) singular n-simplices....
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What does formality of a chain complex mean topologically?

I've been told that every topological abelian group is a product of Eilenberg-Mac Lane spaces, but I don't have a reference for this fact. This confuses me because via the Dold-Kan correspondence, ...
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Is homology of a chain complex a universal delta-functor?

Let $\mathcal{A}$ be an abelian category and let $Ch(\mathcal{A})$ be the category of homologicaly, non-negatively graded chain complexes in $\mathcal{A}$. The sequence of homology functors $H_n:Ch(\...
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Is the functor category $[\mathsf C,\mathsf{Ab}]$ algebraic?

In this MO question, the functor category $[\mathsf C,\mathsf{Ab}]$ for $\mathsf C$ a small abelian category is examined. Reading about the acyclic models theorem, I'm wondering about the same ...
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Splitting of Short Exact sequence

Suppose $d_1d_2=n$ and let $0 \to d_1\mathbb Z_n \overset {i} \to \mathbb Z_n \stackrel {d_2\cdot} \to d_2\mathbb Z_n\to 0$ be a short exact sequence. Show that sequence splits iff $\gcd(d_1,d_2)=1$. ...
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$E_{p, 0}^2$ and $E_{0, 1}^2$ terms in sequence, in terms of homology of $K(G, 1)$, homology of $K(R, 1)$, and action of $G$ on $R$ by conjugation?

This is a followup to my previous question, reproduced here. Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R,...
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Need Counterexample to show Koszul complex is not minimal free resolution?

Recall that Koszul Complex $K.(f,g)$ of polynomials $f,g \in k[x_1,\ldots,x_n]=:R$ is defined as:$$0 \to R \overset{\phi_1} \to R^2 \overset{\phi_2} \to R \to 0$$ where $\phi_1$ and $\phi_2$ are ...
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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 ...
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Short exact sequence of groups, is it possible to construct an associated fibration of spaces?

Let$$0 \to R \to F \to G \to 0$$be a short exact sequence of groups. Is it possible to construct an associated fibration of spaces$$K(R, 1) \to K(F, 1) \to K(G, 1)?$$
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Universal Property of free objects

I am working on free objects, I am restricting myself primarely to groups, rings and modules (with maybe algebras) so in a sense in the concrete category (if I am not mistaken. This is a thesis work I ...
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Show that the $i$th local cohomology functor is zero for $i > 0$

Let $I$ be an ideal of a Noetherian ring $R$, and let $M$ be a module over $R$. Let $\Gamma_I(M)$ be the set of all elements $m$ of $M$ for which $I^n m = 0$ for some $n \geq 1$. Then $\Gamma_I(-)$ ...
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1answer
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Enough Projectives in Category of Groups

Working on homology and completion a question has arisen in my head. I know that $R$-mod as a category has enough projectives in it, and as such the category of abelian groups has it as they are in $\...
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1answer
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A lemma from Hilton & Stammbach's book A Course in Homological Algebra

In orde to prove the set of equivalence classes of extensions of $A$ by $B$ is a contravariant functor of the first component and covariant functor of the second. The authors give us three lemma. I'm ...
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Why is $\ker(id\otimes \cdot b:R/(a)\otimes_R R \to R/(a)\otimes_R R)=R/(d)$?

Let $R$ be a commutative ring with unit $1_R$, $M$ a $R$-module. Let $a,b\in R\setminus \{0\}$ and $\gcd(a,b)=d$. I want to prove: $$\operatorname{Tor}_1^R(R/(a),R/(b))=R/(d).$$By definition, it is $\...
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Errata in Prof. Rotman AIHA book about projectives in the chain complex category (section 10.5)

EDIT After thinking carefully with the help of the clear answer of ZhenLin, I think I will reformulate my question the following way. The text of my original question is kept below. The claim of Prof....
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Global dimension of matrix algebra

I want to calculate the global dimension of this algebra. $$ \quad A = \begin{pmatrix} k & 0 & 0& 0 \\ k & k & 0&0\\ k&0&k&0\\ k&k&k&k \end{pmatrix} \...
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Lifting the projective property through the affine centre

Let $\mathbb{k}$ be an algebraically closed field. There are many interesting examples of $\mathbb{k}$-algebras $R$ which admit a large central subalgebra $Z_0$ such that $R$ is a free $Z_0$-module ...
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Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has Tor-...
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1answer
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Connecting homomorphism in Galois homology using the standard resolution

Let $G$ be a finite group, although this may not be necessary for almost everything that follows. One of the ways of defining Galois homology groups is using the standard resolution for the $\mathbb{Z}...