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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Some question in relative homology

When we consider the pair (X,A) in relative homology, do we assume A is a sub complex of X? And why don't we just consider X/A instead of (X,A)? Is there an better advantage to consider (X,A) ...
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124 views

Sheaf cohomology via resolutions vs. derived categories

So I know that when introducing sheaf cohomology, there are two main approaches via derived categories, and a perhaps more "down to earth" method of resolving by acyclic, fine, soft, sheaves. I'm ...
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35 views

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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28 views

Composition of stable-pseudomonomorphisms

Terminology Let $\mathbf{C}$ be a finitely-complete finitely-cocomplete category with zero object (not necessarily additive!). A morphism $f\colon A\rightarrow B$ is a pseudomonomorphism iff $(0\...
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109 views

Direct proof that infinite product of copies of $\mathbb{Z}$ is not projective

It is well-known that the abelian group $$A = \prod_{n=1}^\infty \mathbb{Z}$$ is not free (see, for example this MO question), and that over a PID being free is equivalent to being projective (see ...
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93 views

When is $\operatorname{Hom}(M, E)$ injective? [closed]

Let $R$ be a commutative Noetherian ring with non-zero identity, $M$ be an $R$-module, and $E$ be an injective $R$-module. When is $\operatorname{Hom}(M, E)$ an injective $R$-module?
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14 views

An algorithm for determining if a tensor is pure?

Suppose I have an category whose objects are free $R$-modules (R a polynomial ring) and whose morphism-spaces $\mathrm{Hom}(A,B)$ between objects $A$ and $B$ are spanned by a finite set of module-...
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1answer
55 views

Composition factors of injective indecomposable and projective indecomposable modules

Let $A$ be a finite-dimensional algebra over an arbitrary field $K$. Let $L_1$ and $L_2$ be simple modules such that $L_1 \not \cong L_2$. Let the $A$-module $Q_1$ be the injective hull of $L_1$, ...
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61 views

If $R$ is $I$ and $J$-adically complete, then it is $(I+J)$-adically complete.

Let $R$ be a Noetherian ring with ideals $I$ and $J$. I already proved the following: Lemma: If $I \subseteq J$ and $R$ is $J$-adically complete, then $R$ is $I$-adically complete. And now I'm ...
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31 views

Minimal graded free resolution and matrix representations

In a graded $R$-module, let $$0 \to C_p \xrightarrow{\phi_p} C_{p-1} \xrightarrow{\phi_{p-1}}C_{p-2} \to \dots \to C_1 \xrightarrow{\phi_1} C_0 \xrightarrow{\psi} M \to 0$$ be a minimal graded free ...
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45 views

finding high weight vector in Verma module

Let $\frak{g}$ be a (semi-)simple lie algebra. Let $\lambda$ be a dominant integral weight. Denote $L(\lambda)$ to be the irreducible representation of highest weight $\lambda$. From BGG resolution, ...
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59 views

Minimal graded free resolution of the ideal $I = (x^r, y^s) \subset k[x,y]$

What is the minimal free-graded resolution of the ideal $I = (x^r, y^s) \subset k[x,y]=R$ for $r,s \in \mathbb{N}$? I tried reducing this down to $r = s = 1$ and I think it is $$0 \to R(-2) \to R(-1)...
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1answer
22 views

Existence of homomorphism in diagram

Consider the following diagram where each complex is an $R-$module and the rows are exact and the maps $g,h$ are $R-$mod homomoprhisms and the right square commmutes $h(p(b)) = q(g(b)), \forall b \in ...
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1answer
32 views

Direct limit of a direct system looking like a cochain complex of objects.

I would like to ask you about a special kind of direct systems $ (A_i, f_{i}^{j} )_{ i,j \in ( I , \leq ) } $ looking like a cochaîn complex $ (A_i , f_{i}^{j} )_{ i,j \in ( \mathbb{N}^* , \leq ) } $ ...
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102 views

Kähler differential over a field

I have been working with Kähler differentials, and I have $\Omega^1_{B/k}$, where B is a commutative $k$-algebra, and $k$ is a field. I was wondering that for $d(b)=0$, does this imply that $b\in k$? ...
2
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1answer
91 views

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: $$S^{-1}\mathrm{Tor}...
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47 views

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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37 views

$\mathcal{T}^i(X/Y,\mathcal{F})$ 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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I'm searching for $X$ and $V$ such that the evaluation map $f:H^n(X,A;V)\to \hom_\mathbb{Z}(H_n(X,A;\mathbb{Z}),V)$ is not injective

Let $V$ be a $\mathbb{Z}$-module $(X,A)$ a pair of topological spaces. We define the pairing-map in singular co- homology$$H^n(X,A;V)\times H_n(X,A;\mathbb{Z})\to V$$ $$([\xi ],[\alpha])\mapsto \xi (\...
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1answer
39 views

Contractibility of an exact chain complex

How can one prove that an exact (acyclic) chain complex of projective modules that is trivial in negative degrees is contractible? I would appreciate some nudges in the right direction more than ...
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1answer
85 views

A chain complex is split if and only if it splits as a direct sum.

This is the first part of Exercise 1.4.2 in An Introduction to Homological Algebra by Weibel. The first part is showing that a chain complex, $C$, with boundaries $B_n$ and cycles $Z_n$ in $C_n$ is ...
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64 views

Counterexample to, if $f$ is acyclic then $kerf$ and $cokerf$ acyclic.

This is the second half of exercise 1.3.5 in An Introduction to Homological Algebra by Weibel, it simply asks if this statement is true of false and I believe it is false but cannot construct a ...
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69 views

Why is Hom$_A(M,A)$ a right $\Gamma$ comodule?

I'm reading through appendix I (Hopf algebroids) of Ravenel's green book, and I came across a line I can't understand in a proof. The part of the lemma I'm interested in states: $\mathbf{Lemma A1....
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1answer
48 views

Baer Sum notation requires clearence.

I am working on Baer sum and I have my book by Rotman, Introduction to Homology, and also MacLanes book Homology and they use notation I am puzzled on. I have understood baer sum of extensions ...
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32 views

Compute directly that the mapping cone of a homotopy equivalence is contractible

Let's consider the category $Ch_R$ of cochain complexes of modules over a commutative ring $R$. I'm trying to prove that if the chain map $\phi:M\rightarrow N$ is a homotopy equivalence then its ...
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25 views

Automorphism of semisimple Lie algebra corresponding to a simple reflection

Let $\mathfrak{g}$ be a complex, finite-dimensional Lie algebra. Let $\mathfrak{h}\subset \mathfrak{g}$, $W$ and $\Pi$ be a Cartan subalgebra, its Weyl group and the set of all simple roots, ...
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69 views

Baer sum of $\mathbb{Z}_9$ and $\mathbb{Z}_9$

I am working on trying to figure out the third extension of $\mathbb{Z}_3$ by $\mathbb{Z}_3$, I know one is $\mathbb{Z}_9$ and the neutral element (with respect to baer sum) $\mathbb{Z}_3\oplus \...
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45 views

Getting a double complex that computes Ext

Suppose $C$ is an abelian category and I am trying to compute $Ext^i(M,N)$ for some objects $M,N$. Suppose there is an exact sequence $0 \rightarrow A_1 \rightarrow A_2 ... \rightarrow A_n \...
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$T^i$ functors in Hartshorne's Deformation Theory

In chapter 3 of Hartshorne's Deformation Theory, he defines functors $T^i$ for $i=0,1,2$ that take as input a ring homomorphism $A\rightarrow B$ and a $B$-module $M$ and outputs $T^i(B/A,M)$, a $B$-...
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An exact sequence of inverse systems of $R$-modules

Let $$0\longrightarrow \big\{A_n,f_{mn}\big\}_{m \leq n} \overset{\Phi}\longrightarrow \big\{B_n,g_{mn}\big\}_{m \leq n} \overset{\Psi}\longrightarrow \big\{C_n,h_{mn}\big\}_{m \leq n} \...
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1answer
44 views

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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33 views

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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53 views

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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52 views

Is $M_R\otimes _R {_R}N\cong M_{\mathbb Z}\otimes_{\mathbb Z} {_{\mathbb Z}}N$?

Suppose $M$ is a right $R-$module and $N$ is a left $R-$ module. Also $M$ and $N$ are naturally $Z-$ module, both in left and right side. So we will denote $M_R$, $M_{\mathbb Z}$, and $_RN, _{\mathbb ...
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1answer
76 views

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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1answer
124 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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75 views

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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57 views

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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1answer
46 views

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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1answer
44 views

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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40 views

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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3answers
144 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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1answer
31 views

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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1answer
61 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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1answer
41 views

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 ...