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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A question about the universal coefficient theorem.

Or rather a couple of questions. Let $X$ be some topological space, $R$ be a (unital) PID and $G$ be an $R$-module. Am I correct in understanding that the singular cochain complexes ...
6
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2answers
102 views

Why is the definition of $\lim^1$ via a cokernel the first derived functor of $\lim$?

Let $A_*=\ldots\to A_n\to A_{n-1}\to\ldots\to A_0$ be a linear system of abelian groups. The limit of this system may be defined as the kernel of the map $$ \prod A_n\xrightarrow{g-1}\prod A_n $$ ...
2
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1answer
46 views

$K(A)\cong \mathbb Z$ for a PID $A$

In Atiyah and Macdonald, chapter 7, exercise 26, iii), it's required to show the Grothendieck group $K(A)\cong \mathbb Z$ for a PID $A$. By ii) of this problem, it's easy to show that $K(A)$ is ...
3
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1answer
38 views

Exact sequence induces exact sequences for free parts and torsion parts?

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...
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1answer
58 views

Queston on the definition of singular homology

From the Hatcher's can someone told me why $\sigma$ has singularities ? Thank you
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0answers
44 views

Faithfully flat checkable on finitely generated modules

A left $R$-module $_RM$ is said to be faithfully flat if it is flat and, for any $N_R$, $N \otimes_R M = 0$ implies $N = 0$. I would like to show that $M$ is faithfully flat if it is flat and, for any ...
7
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1answer
86 views

Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
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23 views

Relationship between homological functors and t-structures

Let $D$ be a triangulated category, $A$ an abelian category and $\pi: D \to A$ a homological functor (sending distinguished triangles to long exact sequences). Can we describe (the) obstructions to ...
2
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2answers
88 views

Represent localization as a direct limit

Let $A$ be a commutative ring with identity, $S\subset A$ a multiplicatively closed subset and $1\in S$. Does the equation $$S^{-1}A=\varinjlim_{s\in S}A_s$$ make sense? Here $A_s$ is the ...
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26 views

Exactness of derived functor [duplicate]

Is the right derived functor of a left exact functor left exact also? If now, can anything be said about its exactness in general?
2
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99 views

Duality between Tor and Ext?

Let $A$ be a $\mathbb{N}$-graded, locally finite $\Bbbk$-algebra, $\Bbbk$ being a field, $A=\oplus_{n \geq 0} \ A^n$, each $A^i$ being finitely dimensional as a $\Bbbk$ vector space. Assume also that ...
4
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1answer
91 views

Existence and homotopies of embeddings between simplicial complexes

Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$. I'm ...
0
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1answer
78 views

Kernel and direct sum

Let $R=k[x_1,\ldots,x_7]$ be a polynomial ring over field $k$ and $I=\bigcap_{i=1}^4 \mathfrak{p}_i$ where $\mathfrak{p}_1=(x_1,x_3,x_5,x_6), \mathfrak{p}_2=(x_1,x_3,x_4,x_6), ...
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36 views

Reference request: exact sequences of Lie algebras

I have a reference request: where can I read more about the following? Consider the short exact sequence $0\rightarrow \mathfrak{n}^- \rightarrow \mathfrak{gl}_n\rightarrow \mathfrak{b}\rightarrow ...
5
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61 views

Best approximation to an adjoint functor

I have the following question. Suppose I have a functor $F\colon C\to D$ between two categories. I would like it to have an adjoint (say, right), but it doesn't. Is there a way to define a "best ...
3
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0answers
81 views

Good textbooks on homological algebra

Can someone give me a recommendation on homological algebra textbooks? I would like something that are accessible to beginners and that have 1) a brief look at preadditive, additive, monoidal, ...
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1answer
115 views

Universal Coefficient Theorem - what kind of morphisms?

Let $G$ be an $R$-module, where $R$ is a P.I.D., and let $X$ be a topological space. We have the exact sequence $$0 \rightarrow H_n(X) \otimes G \rightarrow H_n(X; G) \rightarrow ...
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28 views

Show that there is a natural isomorphism $\mathrm{Ext}_{R}^n(M, M') \cong \mathrm{Ext}_{R}^n(\Omega M, \Omega M')$

Given an element $\rho \in \mathrm{Ext}_{R}^n(M, M')$, we can associate an exact sequence $$0 \rightarrow M' \rightarrow M_{n-1} \rightarrow \ldots \rightarrow M_0 \rightarrow M \rightarrow 0.$$ Thus, ...
4
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48 views

The functor $\underline{\mathbf{R}}^if_*$

Let $f: X \to Y$ be a proper morhpism of varieties, and $\mathcal{F}$ be a sheaf on $X$. Then we have $f_* \mathcal{F}$ as a sheaf on Y and we also have a higher derived functor $\mathbf{R}^i ...
0
votes
1answer
23 views

Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...
2
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73 views

Cartan and Eilenberg Homological Algebra

OK, I am looking at Cartan and Eilenberg Homological Algebra book (1956, 1973 printing). Chapter V.9, p97 they define functors T(-,-) of type L$\Sigma$ and R$\prod$. T is of type L$\Sigma$, if T(A,C) ...
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1answer
37 views

can the projective dimension be read from any projective resolution?

Let $P_{\bullet}, P'_{\bullet}$ be two projective resolutions of an $R$-module $M$. Denote their differentials by $d,d'$ respectively. Define $M_i = \operatorname{ker} d_{i-1}, M'_i = ...
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1answer
31 views

Small question on relative holology

if $Y\subset X$ , what is $\ker \delta$ such that $\delta: H_k(X,Y)\rightarrow H_k(Y)$ ? is it $\ker \delta = H_k(X,Y)$ ? $\delta$ is the usual connecting homomorphism from the long exact sequence ...
2
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1answer
63 views

homotopy equivalence of projective resolutions

Let $P_{\bullet}$ and $P'_{\bullet}$ be projective resolutions of a module $M$ over a commutative ring $R$. Then $P_{\bullet}$ and $P'_{\bullet}$ are homotopy equivalent (see e.g. Matsumura, CRT, ...
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2answers
55 views

Right homotopic maps iff chain homotopic

Assume the model structure on $Ch(R)$ (chain complexes of left modules over the ring $R$) in which fibrations are dimensionwise epimorphisms (i.e. surjections) and weak equivalences are homology ...
0
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1answer
66 views

Homology and Exact Sequence

I have this exact sequence: $$0\stackrel{f}{\rightarrow} H_k(X,C)\stackrel{g}{\rightarrow} H_k(X,A)\stackrel{h}{\rightarrow} 0$$ Can I say that $H_k(X,A)=H_k(X,C)$ and why? Please; Thank you.
5
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1answer
116 views

On Gorenstein ring of dimension zero

Let $R$ be an Artinian local ring. Then $R$ is a Gorenstein ring (i.e., $R$ is an injective $R$-module) iff for any ideal $I$ of $R$, Ann$($Ann$(I))=I$. Why? (We call $R$ Gorenstein if injective ...
0
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2answers
57 views

Is complex exact if its Euler characteristic is zero?

For a bounded complex $M$ of finite-dimensional $k$-vector spaces we define its Euler characteristic as $$ \chi=\sum_{n\in \mathbb{Z}} (-1)^n\dim(M_n) $$ In particular, if complex is exact then its ...
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59 views

$\text{Ext}^1(-,B)$, independence of projective resolution.

If we want to compute the group $\text{Ext}^1(A,B)$ we take a projective resolution of $A$ $$\cdots\to P_2 \to P_1 \to P_0 \to A \to 0,$$ apply the contravariant functor $\text{Hom}(\cdot,B)$ to it ...
3
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49 views

Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
2
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1answer
80 views

Question concerning Eisenbud's theorem on matrix factorisations

I have the following question: Let $S$ be a commutative regular local ring and $\mathfrak{n}$ be its maximal ideal. Let $f\in\mathfrak{n}$ be a non zero-divisor in $S$ and let $m\geq 1$ ne a natural ...
2
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1answer
70 views

Homology of Chain Complexes from Free Resolution

Suppose I have an $R$-module $M$ and a free resolution $$ \ldots \to F_2 \to F_1 \to M \to 0. $$ I apply an additive functor $f$ in $R$-$\mathbf{Mod}$ to the free resolution to get $$ \ldots \to ...
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2answers
57 views

Inductive definition of group cohomology?

At the start of Atiyah and Wall's section on group cohomology (in the Cassels-Frhlich collection of Algebraic Number Theory notes) they, of course, define group cohomology (actually, a 'cohomological ...
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1answer
51 views

Behaviour of Betti tables with exact sequences

Let $0 \to M' \to M \to M'' \to 0$ be an exact sequence of finitely generated graded $S$-modules, where $S=k[x_1, \ldots, x_n]$ is a polynomial ring in $n$ variables. Let $\beta_{i,j}(M)$ denote the ...
3
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1answer
69 views

Is the derived category of a commutative ring monoidal?

Let $A$ be a commutative ring, and consider the derived category $D(A)$. Is this a symmetric monoidal category? We have an obvious product, that is $-\otimes^L_A - $, and it is clear that we have an ...
3
votes
1answer
83 views

vanishing of Tor and regular sequences

Let $R$ be a Noetherian ring and $M$ a finite $R$-module. Let $x=x_1,\dots,x_n$ be an $R$-sequence such that it is also an $M$-sequence and let $I=(x_1,\dots,x_n)$. Question: Is it true that ...
3
votes
2answers
75 views

Definition of exact split complex

I am reading "an introduction to homological algebra by Charles A.Weibel" and the author deifnes split exact complex to be exact complex $\{C_n,d_n:C_n\to C_{n-1}\}$ such that there exists a sequence ...
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0answers
21 views

The Existence of Pure Resolutions, Given a Degree Sequence?

I have been trying to understand the proof of the following theorem for the last month, I read some basics of sheaves theory and their cohomology, but still can't get the idea of this important ...
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1answer
38 views

Coeffaceable implies universial $\delta$-functor

My question is essentially about Grothendieck's Tohoku paper Proposition 2.2.1 but in the context of coeffaceable instead of effaceable. Grothendieck's paper does not give much suggestions to my ...
2
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0answers
90 views

Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
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2answers
62 views

The singular homology and cohomology of a topological space with coefficients in a zero characteristic field.

I have a field with zero characteristic, like $K=\mathbb{C},\mathbb{R}$ and I want to show that the homology groups and cohomology groups with coefficients in these fields satisfy: $$H_n(X,K) \approx ...
3
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1answer
40 views

Question from Cartan-Eilenberg, Chapter 6, exercise 5

The exercise problem is this; consider a unital ring $A$. For each right $A$-module $M$ and left ideal $I$ of $A$, TFAE. (a) For each relation $\:\sum _{i} a_iu_i=0 \:(a_i\in M, u_i\in I)$ there ...
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0answers
46 views

Hochschild homology with trivial coefficients: how to make $K$ an $M_n(K)$-module

Let $R$ be a ring, $A$ an associative $R$-algebra, and $M$ an $A$-$A$-bimodule. Then the Hochschild homology of $A$ with coefficients in $M$, denoted $HH_\ast(A)$, is the homology of the chain complex ...
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1answer
44 views

On maximal submodules of projective modules

I know that any non-zero projective module has a maximal submodule. But is it true that any proper submodule is contained in a maximal submodule !?
8
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1answer
80 views

An explicit imbedding of $(R\mathbf{-Mod})^{op}$ into $S\mathbf{-Mod}$

Given a ring $R$ consider $(R\mathbf{-Mod})^{op}$, the opposite category of the category of left $R$-modules. Since it is the dual to an abelian category and the axioms of abelian categories are ...
0
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1answer
39 views

degree zero term of minimal free resolution

Let $R=k[x_{1},\ldots,x_{n}]$ where $k$ is a field, and let $I$ be a homogenous ideal. Suppose that $\cdots\to R_{1}\to R_{0}\to R/I\to 0$ is a (the) graded minimal free resolution of $R/I$. Is it ...
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1answer
66 views

On a particular $K[x,y]$-module

This is a follow up from HERE. Suppose $K$ is a field and consider $K$ as a $K[x,y]-$module where the scalar product is defined by $f(x,y)\cdot k = f(0,0)\cdot k$. Is $K$ injective or flat as ...
3
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1answer
80 views

Projective object in the category of chain complexes

I have the following sequence of projective $\mathbb{Z}$-modules: $\cdots \rightarrow 0 \rightarrow \mathbb{Z} \overset{\times 2}\rightarrow \mathbb{Z} \rightarrow 0 \rightarrow \cdots $ This is ...
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74 views

Exact sequences and spectral sequences

We have the well-known theorem for cohomological spectral sequences as follows: Theorem: Let $(E_r , d_r )$ be a third quadrant spectral sequence and let $E^{p,q}_2‎\Rightarrow‎ H^n(Tot(M)$. a) If ...
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1answer
61 views

Cohomology of finite p-groups

Given a finite abelian $p$-group $A$ acted on by a finite $p$-group $G$. Under the assumption $\operatorname{H}^1(G,A_1)=0$, where $A_1$ is the set of elements of $A$ having order at most $p$, what ...