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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Finite projective dimension and vanishing of ext on f.g modules
Let $A$ be a commutative noetherian ring.
Suppose $M$ is a finitely generated $A$-module.
Let $n>0$ be an integer. It is well known that if $Ext^n(M,N) = 0$ for all $A$-modules $N$, then $M$ has ...
2
votes
2answers
66 views
Show that $[l_1 \cdot l_2 \cdot l_3 ] = [l_1 + l_2 + l_3] \in H_1(X)$ The first Homology group of X
Let $l_1$ , $l_2$ and $l_3$ be three paths in X with $l_1 (0) =
l_3 (1)$, $l_1 (1) = l_2 (0)$ and $l_2 (1) = l_3 (0)$. Define the loop $l = l_1 \cdot l_2 \cdot l_3 $ (based at $l_1 (0)$).
Show that ...
4
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1answer
55 views
Another basic short exact sequence problem
In the following commutative diagram of R-modules, all of the rows and columns are exact. Prove that $K$ is isomorphic to $L$.
\begin{array}{ccccccccccc} &&&&&&&&0 ...
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0answers
26 views
Support of a direct sum of local cohomology modules
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$. Let $M$ be a finitely generated $R$ module. How can we show the following:
$$\operatorname{Supp}(\bigoplus_{j\ge ...
2
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2answers
32 views
Set of Homomorphisms as an $R-$ module
$\require{AMScd}$
I'm reading A first course of homogocial algebra by D.G. Northcott, and I don't quite get the Example 1 on page 25. Here's what it says:
Example 1
Let the module $A$ belongs to ...
3
votes
1answer
42 views
All local cohomology modules being zero
Let $R$ be a Noetherian ring with unit, $I$ be an ideal of $R$ and let $M$ be a finitely generated $R$-module. Suppose $H_{I}^j(M)=0$ for all $j$, then how can one show that $M=IM$?
The converse of ...
7
votes
2answers
203 views
Vanishing of a certain Tor
I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes
and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
1
vote
1answer
19 views
Submodules of homology modules
I have been dealing with certain subgroups of group cohomology, and the following general question comes to my mind. Suppose $C$ is a chain complex of $R$-modules and $H_n(C)$ its $n$-th homology ...
1
vote
1answer
34 views
Group extension reference request
I'm looking for a reference for the following "well known" result
Let $C$ be an abelian group and $G$ a finite group, and let $$0 \rightarrow C \rightarrow W \rightarrow G \rightarrow 0$$ be a ...
3
votes
2answers
37 views
Covariant functor, and left exact
I'm reading A first course of Homological Algebra by Northcott, and there is something that the author said it was straightforward. But for some reason, I just don't see the straightforwardness of it.
...
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0answers
32 views
How to prove the global dimension of the polynomial ring $F[x_1,…,x_n]$ is $n$?
I am trying to prove that the global dimension of the polynomial ring $F[x_1,...,x_n]$, where $F$ is a field , is exactly $n$. And by Koszul Complex, I know its global dimension is greater than or ...
3
votes
1answer
41 views
Finite Projective Dimension implies non vanishing Ext
Suppose the projective dimension of a module $M$ is $n < \infty$. Does there exist a free $R$-module $F$ such that $\operatorname{Ext}^n(M, F) \not = 0$?
Can't we write the free module as a direct ...
3
votes
1answer
59 views
$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$
Let $R$ be a Noetherian ring and $I$ an ideal of $R$. If $n$ is the cohomological dimension of $I$, then why is the following isomorphism true:
$$H_{I}^{n}(M)\cong H_{I}^{n}(R)\otimes_R M.$$
The ...
1
vote
1answer
19 views
What is the relation between graded modules and finitely generated modules
The reason I ask this question is I found two different statements about Hilbert's syzygy theorem from Jacobson's Basic Algebras 2nd and Wikipedia. Please have a look at the following pictures. The ...
1
vote
2answers
53 views
Questions about projective modules.
Let $P$ be a projective module and $M$ a submodule of $P$. We know that $M$ is also a projective module. Can we conclude that $P=M\oplus N$ for some module $N$? Thank you very much.
3
votes
1answer
59 views
Property of Hom-functor
How to prove $$\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
vote
1answer
35 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
55 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
58 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
44 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 ...
7
votes
1answer
54 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 UFD's.
I know the basics of local cohomology but I have not seen a ...
2
votes
0answers
42 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 ...
7
votes
5answers
103 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, ...
5
votes
2answers
68 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
0answers
38 views
Injective dimension is locally finite but not globally
Let $R$ 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}(R)$, but ...
10
votes
2answers
94 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 ...
0
votes
0answers
44 views
Why Syz algorithm for module have dimension problem, how to divide vector and multiply vector at final step to get 1*6 Vector
refer to page 162 in an introduction to grobner basis, william W. Adams
https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!338
https://skydrive.live.com/redir?resid=E0ED7271C68BE47C!339
when ...
6
votes
0answers
78 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 ...
0
votes
0answers
45 views
How to calculate Grobner basis for Syz(i,f1,f2,g1,g2)
refer to page 174 in an introduction to grobner bases, william A. Adams
no matter choose first row or second row of i, f1, f2, g1, g2, the maple code not the same as in the book
...
2
votes
2answers
82 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'
...
1
vote
0answers
43 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
106 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 ...
5
votes
0answers
63 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 ...
1
vote
1answer
24 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 ...
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vote
1answer
40 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
25 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
62 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$. ...
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votes
0answers
27 views
Spectral sequence for computation homologies of kernel of homomorphism
Is there a spectral sequence, which can help to calculate homologies of a kernel of morphism of 2 groups with coefficient in $\mathbb{Z}$?
4
votes
0answers
33 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 ...
0
votes
1answer
24 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.
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vote
1answer
31 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
84 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 ?
2
votes
0answers
36 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
50 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
70 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
64 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
127 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 ...
3
votes
1answer
85 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
59 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 ...
1
vote
1answer
54 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 ...
