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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164 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
104 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
70 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.
6
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1answer
221 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 ...
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2answers
131 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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2answers
98 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 ...
4
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0answers
75 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) ...
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1answer
99 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
134 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 ...
2
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2answers
94 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
100 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
153 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
165 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
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2answers
260 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 ...
2
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1answer
156 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 ...
3
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0answers
223 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
104 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
54 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
72 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
128 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 !?
9
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2answers
111 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 ...
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1answer
59 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
87 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
209 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 ...
2
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1answer
83 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 ...
3
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1answer
95 views

Physical interpretation of categorical structures related to Dirichlet Branes

In Dirichlet Branes and Mirror Symmetry by Aspinwall et al, section 5.9 discusses various questions that remain open. In particular they say: "There are many constructions from homological ...
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1answer
266 views

Is it true that Tensor product of injective modules is injective?

Is it true that if $M$, $N$ are injective modules over a commutative ring $R$ (with identity) then $M\otimes_R N$ is also injective ?
2
votes
2answers
285 views

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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0answers
76 views

Spectral sequences to involve together two ideals of a ring

I'm looking for spectral sequences to involve together two ideals of a ring. For instance, let $I,J$ be two ideals of Noetherian ring $R$ and $M$ be a finite $R$-module then we have the following ...
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2answers
139 views

kernel of a monic morphism

Problem Suppose $\mathscr{C}$ is an arbitrary category with zero object. $A$ and $B$ are two objects of $\mathscr{C}$. Let $f\in Mor_\mathscr{C}(A,B)$. It's given that $f$ is monic. I need to show ...
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1answer
163 views

Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
9
votes
3answers
635 views

Exactness of the Tensor Functor

This might turn out to be a very stupid question, so I apologize in advance, but it is confusing me a little bit. I know in general that if $$M'\rightarrow M\rightarrow M''\rightarrow 0$$ is an exact ...
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1answer
54 views

Quick question about chain homotopies.

In the definition of a chain homotopy (say $h$) between two chain maps (say $f$ and $g$), are the maps $h_i$ comprising the chain homotopy required to commute with all other maps involved (the $f_i$s, ...
0
votes
1answer
121 views

When do we have $m\otimes n = 0$ [duplicate]

Let $M$ and $N$ be $R$-modules ($R$ a commutative ring with identity). Let $m \in M$ and $n \in N$. Is there any necessary and sufficient condition to have $m\otimes n = 0$ (as an equation in ...
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2answers
85 views

is the pullback of the cohomology of a group to the cohomology of a subgroup surjective?

If $H$ is a subgroup of $G$, is $i^*(H^*(G)$) surjective onto the cohomology of $H$? $i$ is the inclusion of $H$ in $G$.
6
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1answer
269 views

Existence proof of the tensor product using the Adjoint functor theorem.

Can one prove the existence of the tensor product by the adjoint functor theorem? (of, say, modules over a commutative ring) If yes, how would one check the SSC (solution set condition) for the hom ...
6
votes
1answer
147 views

Ext of an $\mathfrak{m}$-primary ideal

Let $(A,\mathfrak m,k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)=0$ then ...
3
votes
0answers
76 views

Depth for intersection of prime ideals

Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring over field $K$. How can one compute $\operatorname{depth}(R/\bigcap_{i=1}^r p_j)$, where each $p_j$ is generated by some variables $x_i$ and have a ...
16
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1answer
729 views

Homological methods in algebraic geometry

This question will probably seem quite silly to those well-versed in algebraic geometry (about which I admittedly hardly know anything); in the preface of Atiyah-Macdonald's book on commutative ...
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0answers
278 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
2
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0answers
107 views

Derived functors and coboundary operator

I understand that one can define the cohomology of an object $A$ in terms of a complex (non-zero in positive degrees) in some Abelian category, together with differentials, such that the composition ...
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1answer
106 views

Dedekind ring characterization via projective modules

I am looking for a book or course notes proving the following result: Let $R$ be an integral domain. Then $R$ is a Dedekind ring if and only if every submodule of a projective $R$-module is ...
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0answers
55 views

Rank of homology group

$x_0$ is the unique a global minimum and let $c=f(x_0)$ in a Hilbert space, let $\theta$ be an other critical point of $f$ non minimum. the Morse type number of $x_0$ is ...
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1answer
125 views

free resolution of koszul complex. [closed]

Can anyone prove the following using spectral sequence? Let $f_{1},f_{2},\ldots,f_{n}$ be a regular sequence. Prove that $K(f_{1},f_{2},\ldots,f_{n})$ is a free resolution of ...
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0answers
80 views

Is the category of chain complexes complete and cocomplete in small?

Does the category of chain complexes (let's say of modules over some ring) have all small limits and colimits? What I understand is that the category of chain complexes is certainly finitely ...
3
votes
1answer
57 views

Free DG modules

Let $A$ be a DG algebra and $f : F \to M$ a morphism of DG $A$-modules such that $F$ is free and the induced map $H^{\bullet}F \to H^{\bullet}M$ vanishes. Does it follow that $f$ is nullhomotopic? My ...
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1answer
235 views

Proof of the five lemma

How to do this using the snake lemma? this is an exercise in Lang's Algebra book. It should somehow be obvious, but I don't see it
5
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1answer
71 views

Global dimension of Artin algebras over a perfect field

Let $A$ be an Artin algebra over a perfect field $k$. Suppose that the global dimension of $A$ is finite. How one can prove that $$ \operatorname{gl}(A)=\max\{i \geq 0\mid ...
3
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1answer
40 views

Question on differential modules

Let $A,B$ be differential modules with differentiation homomorphism $d$ (such that $d^2=0$). Then let say that $g$ is an epimorphism from $A$ into $B$. Then is it possible for an induced homomorphism ...
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172 views

Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...