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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if $F_{\bullet}$ is a complex and $r$ an integer, what is $F_{r-\bullet}$?

While reading the paper Some results and questions on the Castelnuovo-Mumford regularity, by Marc Chardin, I encountered in the proof of Theorem 5.1 the notation $F^N_{r-\bullet}$. To provide some ...
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33 views

Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ ...
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1answer
50 views

Degree theory and Invariance of domain

We'll use the Proposition (F) to show that: (Invariance of domain) Let $f: M \to N$ be a proper smooth mapping of two oriented, boundaryless, smooth manifolds of dimension $m$; furthermore, $N$ is ...
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1answer
59 views

Definition of $Hom(A,B)$

I have lots of confusion about definition of $Hom(A,B)$. I would like to ask several questions with my thoughts. Hopefully I could solve my problem. -Firstly, my book write that if $A$ and $B$ is ...
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1answer
23 views

Split Lie algebra extensions?

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras. A Lie algebra extension is a short exact sequence $$0\longrightarrow \mathfrak{h}\stackrel{\jmath}{\longrightarrow} ...
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32 views

Vanishing of generalized local cohomology modules

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $N$ be an $R$-module. Let $M$ be a projective $R$-module. Is it true that $H_{I}^i(M‎, ‎N)=0$ for all ...
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46 views

How injective $\overline{f_p}$ maps $\mathfrak{m}M_{p}/\mathfrak{m}^2M_{p}$ to $\mathfrak{m}L_{p}/\mathfrak{m}^2L_p$?

If $(R,\mathfrak{m},k)$ is a local ring, $A$ a finite $R$-module. Let $L_{\bullet}:\cdots\rightarrow L_1\xrightarrow{d_1} L_0\xrightarrow{d_0} A\rightarrow 0$ be a minimal free resolution. $\bar{d_i}$ ...
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1answer
74 views

Determine $H(\mathbb{R, Q})$ and $H(\mathbb{R, Z})$

I need to determine the relative (singular) homology groups of $\mathbb{R} \text{ mod } \mathbb{Q}$ and $\mathbb{R} \text{ mod } \mathbb{Z}$. Any hints on what I need to know for this question? ...
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1answer
15 views

Construct a free chain complex K

Let $(A_{n})_{n \in \mathbb{Z}}$ be a set of finitely presented abelian groups. Construct a chain complex $\mathbf{K}$, with each $K_{n}$ a free abelian group, such that for each $n \in \mathbb{Z}$, ...
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210 views

Computing (the ring structure of) $\mathrm{Ext}^\bullet_R(k,k)$ for $R=k[x]/(x^2)$

Let $k$ be some field (say of characteristic zero, if it matters) and define $$R=k[x]/(x^2).$$ I want to compute $$\mathrm{Ext}^\bullet_R(k,k)$$ and, in particular, the ring structure on it (though I ...
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52 views

Why $\bar{d_0}$ is injective in a minimal free resolution? [closed]

If $(R,\mathfrak{m},k)$ is a Noetherian local ring, $A$ a finite $R$-module. Let $L.:\cdots\rightarrow L_1\xrightarrow{d_1} L_0\xrightarrow{d_0} A\rightarrow 0$ be a minimal free resolution. ...
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1answer
81 views

Calculate $\operatorname{RHom}$ in a the derived category of graded $\mathbb{C}[x]$-Modules

I was trying to do the following exercise. Consider the category of graded $\mathbb{C}[x]$-Modules, it is clear that we can regard $\mathbb{C}[x]$ as a graded module setting $\operatorname{deg}(x)=1$. ...
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23 views

projective resolution for an $I$-torsion $R$-module

Let $R$ be a commutative Noetherian ring with non-zero identity, $I$ be an ideal of $R$ and $M$ be an $I$-torsion $R$-module. We know that there exists an injective resolution of $M$ in which each ...
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1answer
54 views

Is R/m a flat R-module?

Let $(R,\frak m)$ be a commutative Noetherian local ring. Is $R/\frak m$ a flat $R$-module? Thanks.
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1answer
69 views

Compute Ext with Macaulay2

I want to compute Ext with Macaulay2. I see in the website they write how to do but I can not do. Can anyone help me with an example? For example, let $S=k[x,y,z,t]$. How compute ...
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3answers
182 views

Why are we interested in cohomology?

I've been studying algebraic topology for over half a year now and came across alot of different topics of it (fundamental groups, Van Kampen, singular homology, homology theory, Mayer Vietoris, ...
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44 views

Homomorphisms of Chain Complexes

Let $(K, d^{K})$ and $(L, d^{L})$ be chain complexes. For $n \in \mathbb{Z}$ define $$ \mathrm{Hom}(K, L)_{n} := \prod_{j \in \mathbb{Z}} \mathrm{Hom}(K_{j}, L_{j+n})$$ and $$ d_{n}^{K,L} \ \colon ...
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Is this functor representable?

Fix a group $G_0$ and $R$ a subset of $G_0$. Consider the functor $F$ from $\textbf{Grps}$ to $\textbf{Sets}$, sending every object $G$ in $\textbf{Grps}$ to $F(G)$, the subset of $\varphi \in ...
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21 views

A short exact sequence that cannot be made into an exact triangle. (Weibel 10.1.2)

The following exercise is in Weibel Chapter 10. Regard the groups $\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{Z}/4\mathbb{Z}$ as cochain complexes in degree 0. Show that the short exact sequence $$ 0 ...
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1answer
35 views

Projective Dimension and Schanuel's Lemma

Let $R$ be a ring and $M$ a (say, left) $R$-module of projective dimension $n$. According to Noncommutative Noetherian Rings, any projective resolution of $M$ can be terminated at length $n$, and this ...
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47 views

Are product / coproduct projections / inclusions 'semistrict'?

Let $\mathbf{C}$ be a category with zero object, kernels, and cokernels. Then, a morphism $f\colon A\rightarrow B$ in $\mathbf{C}$ is semistrict iff the canonical map $\operatorname{Coker}(\ker ...
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40 views

Proof of Birger Iversen “Cohomology of Sheaves” Theorem 6.8

I am having troubles completing the proof of theorem 6.8 (page 44) from Birger Iversen, Cohomology of Sheaves. (pdf here) Previously we had constructed a functor $\rho$ from $K^+(A)$ (the homotopy ...
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1answer
74 views

Does trivial cohomology imply trivial homology? Does $\operatorname{Hom}(A,\mathbb Z) = \operatorname{Ext}^1(A, \mathbb Z) = 0$ imply $A = 0$?

Is there a topological space $X$ such that $H^i(X; \mathbb{Z}) = 0$ for all $i > 0$, but $H_n(X; \mathbb{Z}) \neq 0$ for some $n > 0$? In his answer to the question Is homology determined ...
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31 views

Any characterization for commutative rings over which “projective modules” equal “free modules”?

As far as I know, over any PID, an polynomial rings over a field, or an local ring, projective modules are always free. This kind of results make me curious about if there are any overall ...
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40 views

Augmented graded algebras - properties

Let $A$ be an augmented graded unital algebra over field $k$. Define $A_+=\bigoplus\limits_{i\ge 1}A^{(i)}$. I'm trying to show that $\sum\limits_{i+j>k}A_+^i\otimes ...
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2answers
51 views

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) ...
4
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1answer
110 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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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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1answer
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 ...
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1answer
102 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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1answer
86 views

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

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 ...
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1answer
50 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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36 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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1answer
30 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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1answer
41 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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54 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 ...
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1answer
21 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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96 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 $b\in k$? ...
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1answer
84 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: ...
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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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36 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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1answer
44 views

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 ...
2
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1answer
67 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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2answers
52 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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66 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 ...
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1answer
42 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 ...
2
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0answers
30 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 ...