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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3
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86 views

Covariant functor, and left exact

I'm reading A first course of Homological Algebra by Northcott, and there is something that the author said it was straightforward. But for some reason, I just don't see the straightforwardness of it. ...
2
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1answer
103 views

How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?

I am trying to prove that the global dimension of the polynomial ring $F[x_1,\dots,x_n]$, where $F$ is a field , is exactly $n$. By Koszul complex, I know its global dimension is greater than or ...
4
votes
1answer
79 views

Finite Projective Dimension implies non vanishing Ext

Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$? Can't we write the free module as a direct ...
3
votes
1answer
84 views

$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$

Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true: $$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$ The ...
1
vote
1answer
222 views

What is the relation between graded modules and finitely generated modules

The reason I ask this question is I found two different statements about Hilbert's syzygy theorem from Jacobson's Basic Algebras 2nd and Wikipedia. Please have a look at the following pictures. The ...
1
vote
2answers
102 views

Questions about projective modules.

Let $P$ be a projective module and $M$ a submodule of $P$. We know that $M$ is also a projective module. Can we conclude that $P=M\oplus N$ for some module $N$? Thank you very much.
3
votes
1answer
143 views

Property of Hom-functor

How to prove $$\operatorname{Hom}_{R}(A,\operatorname{Hom}_{\mathbb{Z}}(R,B))\cong \operatorname{Hom}_{\mathbb{Z}}(A,B)$$ where $R$ is a commutative ring, $A$ an $R$-module and $B$ an abelian group? ...
1
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1answer
67 views

''Commutative'' 2-cocycles

Let ba $G$ an abelian group and $L$ is $G$-module. If $f$ is a 2-cocycle in $Z(G,L)$, is it true that $f(g,h)=f(h,g)$ for all $g,h \in G$? Or even for $\bar{f} \in H^{2}(G,L)$ is ...
2
votes
1answer
94 views

Example 1.K in A User's Guide to Spectral Sequences

I'm having trouble with Example 1.K, p.25, of John McCleary's book A User's Guide to Spectral Sequences. Specifically, I don't understand how he defines the "obvious map" in the second ...
3
votes
1answer
106 views

Spectral sequences: equivalence of exact couples and classic (?) method

By the 'classic' method I mean the construction of the spectral sequence associated to a filtration as found in Weibel's book p. 133-134. There is also the method of construction through exact couples ...
3
votes
1answer
108 views

Sufficient condition for a direct limit of abelian groups to be infinitely generated

I have the following setup. The CW-complexes $\Gamma_n$ are equipped with maps $\gamma_n\colon\Gamma_{n+1}\rightarrow\Gamma_{n}$ and it is known that the rank of their first cohomology groups is ...
8
votes
1answer
130 views

How does Local Cohomology detect UFD?

I read that Grothendieck developed Local Cohomology to answer a question of Pierre Samuel about when certain type of rings are UFDs. I know the basics of local cohomology but I have not seen a ...
3
votes
0answers
214 views

mapping cones of chain homotopic maps

Suppose that $ f $ and $ f' : C \to D $ are morphisms of chain complexes; Cone($f$) is the mapping cone of $f$; if $f$ and $f'$ are chain homotopic, what is the relation between Cone($f$) and ...
6
votes
5answers
244 views

(Elementary) applications of group (co-)homology

I am looking for an elementary example of a problem, for which one does not need many things to understand the question, but which can be solved with group homology or cohomology. My background is, ...
6
votes
2answers
300 views

Intuition behind Direct limits

Let $R$ be a commutative ring and $x\in R$ be a nonzero divisor. Then i know that the direct limit of $R\mapsto R\mapsto R\mapsto\cdots $, where each map is multiplication by $x$ is $R_x$, the ...
3
votes
1answer
80 views

Injective dimension is locally finite but not globally

Let $A$ be a commutative ring. Could someone provide me an example where $\operatorname{id}_{A_{\mathfrak p}}(M_{\mathfrak p})$ is finite for all $\mathfrak p\in \operatorname{Spec}(A)$, but ...
14
votes
2answers
198 views

Questions about Rickards proof that $D^b_\mathtt{sg}(A) \equiv \mathtt{stmod}(A)$

Setup: Let $A$ be a self-injective algebra (so projective = injective for modules) and let $D^b(A)$ and $K^b(A)$ be the bounded derived category and the full subcategory consisting of the perfect ...
9
votes
1answer
154 views

When does a cohomology theory have a ring structure?

I've looked around and I can't quite seem to find an answer to this question. When does a cohomology theory admit a non trivial product structure? I was trying to compute a cohomology ring from a CW ...
3
votes
2answers
314 views

Characterization of short exact sequences

The following is the first part of Proposition 2.9 in "Introduction to Commutative Algebra" by Atiyah & Macdonald. Let $A$ be a commutative ring with $1$. Let $$M' ...
0
votes
1answer
106 views

For $R$-modules $M,N$, what are sufficient conditions for $\operatorname{Supp}(M\otimes_R N)\subseteq \operatorname{Supp}(\operatorname{Hom}_R(M,N))$?

Let $R$ be a commutative ring, $M$ and $N$ be finitely generated $R$-modules. What additional conditions will ensure $\operatorname{Supp}(M\otimes_R N)\subseteq ...
4
votes
1answer
190 views

Groups acting on polytopes

I am currently reading the paper "Polytopal Resolutions for Finite Groups" [1] by Graham Ellis, James Harris and Emil Skoeldberg and have a question regarding an early remark of theirs. Their basic ...
6
votes
0answers
186 views

Description of $\mathrm{Ext}^1(R/I,R/J)$

Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$? What do I mean by a nice description? For example ...
3
votes
1answer
128 views

Proving Two Complexes are Not Quasi-Isomorphic

In Richard Thomas' paper "Derived Categories for the Working Mathematician" he mentions (page 6) that the two complexes $$ \begin{align*} C^\bullet&= \mathbb{C}[x,y]^{\oplus ...
1
vote
1answer
83 views

Which Short Exact Sequences Can I Extract From A Doubly Infinite Exact Sequence?

I know how if we have a short exact sequence of $R$ modules, $0 \rightarrow A_1 \rightarrow A_2 \rightarrow A_3 \rightarrow 0$ , we can deduce properties about the known modules from the unknown ...
1
vote
1answer
82 views

Change of base rings for exterior algebra

This may be not a good question. But I really get tough. I am studying basic knowledge about homological algebras and I am dealing with Koszul's Complex and Hilbert's Syzygy Theorem. At the very ...
1
vote
1answer
101 views

Homotopical equivalence of complexes

Let $f : C_{\bullet} \to D_{\bullet} $ be a chain map (in the category of $R$-Mod, for example) and suppose that $f_ {*}: H_ { n}(C_{\bullet}) \simeq H_ {n}(D_{\bullet})$ is invertible for all $n$. ...
5
votes
0answers
64 views

Flatness and tensor product of rings

Let $R_1$ and $R_2$ be two subrings of a ring $R$ (not necessarily commutative) which commute in $R$ so that we have a ring homomorphism $R_1\otimes_\mathbb{Z} R_2\rightarrow R$ and $R$ is a module ...
1
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1answer
59 views

Simple modules preserved, if exact sequences preserved by functor

I have the following question: If a functor between two categories sends exact sequences to exact sequences, how does it follow that it preserves simple modules as well? Thanks for the help.
1
vote
1answer
146 views

Tensor algebra of Dg-algebra

Suppose that $k$ is commutative ring and $A=(A,d)$ is Dg-algebra over k. How can one define Dg-algebra structure on $T(A)$ where $T(-)$ is tensor algebra? Secondly how is defined tensor product in ...
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votes
2answers
188 views

$\operatorname{Ext}$ and injectives, respectively projectives

If $\operatorname{Ext}^{ 1}_{R}(A,B) = 0 $ for all $R$-Mod $A$ then $B$ is injective ? If $\operatorname{Ext}^{ 1}_{R}(A,B) = 0 $ for all $R$-Mod $B$ then $A$ is projective ?
4
votes
1answer
149 views

Modules with maximal submodules and projective dimension

If $R$ is a left noetherian ring, then every finitely generated left $R$-module $M$ is noetherian, and hence every proper submodule of $M$ is contained in some maximal submodule of $M$. Is it ...
3
votes
1answer
97 views

Universal coefficient theorem for homology

When Hatcher discusses the universal coefficient theorem for homology (section 3.A, pg. 261), he first takes the exact sequence of chain complexes $$0 \rightarrow Z_n \xrightarrow{i_n} C_n ...
2
votes
2answers
163 views

Equivalence of categories and derived functors.

Don't know if this kind of a dumb question but let $A$ and $B$ be abelian categories and suppose they're equivalent: there are two functors $P: A \rightarrow B$ and $Q: B \rightarrow A$ satisfying the ...
1
vote
1answer
126 views

Artinian ring with zero finitistic dimension

Let $R$ be a left artinian ring with identity. Suppose $R$ contains copies of all its simple right $R$-modules. Is it true that every left $R$-module of finite projective dimension is projective (so ...
5
votes
2answers
224 views

Applications of Mitchell's embedding theorem

I don't understand what is the advantage of viewing a particular category as a category of modules over some ring. Can anybody tell me some application of Mitchell's embedding theorem so that I can ...
4
votes
1answer
224 views

Algebraic Topology Double Complexes

I am going through Bott and Tu and trying to do Exercise 9.13 which says When a homomorphism $f: K \rightarrow K'$ of double complexes induces $H_d$-isomorphism, it also induces $H_D$-isomorphism. ...
6
votes
1answer
88 views

Is $\operatorname{Tor}_i(M,N)$ of finite length?

Let $A$ be a regular local ring, and let $M$ and $N$ be two finitely generated $A$-modules such that $M\otimes N$ is of finite length, and let $i$ be the largest integer such that ...
2
votes
1answer
392 views

does every complex have a quasi-isomorphic projective complex?

Let $C^{\textbf{.}}$ be a complex in some abelian category (edit: assuming it has enough projectives). I would like to know if there exist a complex $X^{\textbf{.}}$ consisting of projective objects ...
7
votes
1answer
817 views

Equivalences and isomorphisms of short exact sequences

In case it's necessary, I'm working in the category $\mathbf{Ab}$ of abelian groups. My question concerns what I find to be a strange way of viewing the elements of the Ext group $\mbox{Ext}(A,B)$ of ...
2
votes
1answer
142 views

Spectral Sequence involving “Triple Tor”

Can someone help me with the first 4 lines of Page 111 of Local Algebra by Serre? I would like to know which spectral sequence is being used. Initially I thought it is the Grothendieck ...
3
votes
0answers
97 views

cellular chain complex of sphere

The cellular chain complex $C_{\ast}(X)$ of an $n$-sphere $X=S^{n}$ (with any CW-complex structure), gives rise to an exact sequence $$ 0 \rightarrow \mathbb{Z} \rightarrow C_{n}(X) \rightarrow ...
2
votes
1answer
67 views

Is a functor category of an $\mathbf{Ab}$-category an $\mathbf{Ab}$-category itself?

In Weibel's An introduction to homological algebra, exercise 2.6.4 reads Show that $\operatorname{colim}$ is left adjoint to $\Delta$. Conclude that $\operatorname{colim}$ is a right exact ...
6
votes
1answer
222 views

Formulation of Künneth theorems (definition of $\mathrm{Hom}$ and $\otimes$ of complexes)

In Rotman's An Introduction to Homological Algebra, there is written: Questions: Let $\mathbf{A}$ and $\mathbf{A'}$ be chain complexes with differentials $\partial$ and $\partial'$ respectively. ...
4
votes
2answers
126 views

Does $A\!\leq\!M$ and $B\!\leq\!N$ imply $A\!\otimes_R\!B\hookrightarrow M\!\otimes_R\!N$? (tensor product of modules)

Let $R$ be a commutative unital ring. What would be an example of a $R$-modules $M,N$ with submodules $A,B$, such that there does not exist an embedding of $R$-modules $$A\!\otimes_R\!B\hookrightarrow ...
3
votes
1answer
99 views

Tor dimension of a field

Let $k$ be a field. How can I explicitly compute $$ Tor_{R}^*(k,k) $$ over the ring $R=k[x_1,\dots,x_n]$. After playing around with it for small $n$ and an unreasonable long time, I think the answer ...
6
votes
0answers
108 views

Sequences of maps between modules such that $\ker(d_n) \subseteq \text{im}(d_{n+1})$

Consider a sequence of maps between $R$ modules (where $R$ is a ring with unity) $$\cdots \rightarrow M_{n+1} \xrightarrow{d_{n+1}} M_{n} \xrightarrow{d_{n}} M_{n-1} \rightarrow \cdots$$ such that ...
2
votes
1answer
101 views

when to use projective vs. injective resolution

I am a bit confused about when I should use projective versus injective resolutions to calculate derived functors. Am I correct in thinking that for right exact functors, the left derived functor is ...
5
votes
1answer
206 views

Resolutions of bimodules as $R^e$-modules.

Let $k$ be a commutative ring, let $R$ be a $k$-algebra, a $R$-Bimodule $M$ over $R$ is a $k$-module with two actions of $R$ on $M$, on the left and on the right, the classical example of this being ...
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1answer
102 views

Homomorphic image is projective

I am reading about projective modules. I am kind of wondering if there exist an abelian group which is not projective but its homomorphic image is projective. Thanks. Edit. It seems that the ...
3
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0answers
115 views

A question of extension of vector bundles.

Fix $p \in \mathbb{P}^1$. Let $X=\mathbb{P}^1\times \mathbb{P}^1$, $C_1=\mathbb{P}^1\times \{p\}$ and $C_2=\{p\}\times \mathbb{P}^1$. Since $\mathrm{Ext}^1(\mathcal{O}_{C_2},\mathcal{O}_{C_1})\cong ...