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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Is it true that $\operatorname{inj.dim}_R R= \operatorname{inj.dim}_R \widehat{R}?$ Is there a one-sided inequality?

$(R,m)$ is a local ring. Is it true that $\operatorname{inj.dim}_R R= \operatorname{inj.dim}_R \widehat{R}?$
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124 views

Question about the definition of homology

$\quad$ The functor $H_n$ measures the number of “$n$-dimensional holes” in the space (or simplicial complex), in the sense that the $n$-sphere $S^n$ has exactly one $n$-dimensional hole and no $m$-...
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Topological dimension and derham cohomological dimension

If G is a compact complex manifold then does the topological dimension bound the deRham cohomological dimension below? By derham cohomological dimension, I mean the largest extended natrual number $n$...
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22 views

Examples of d-extensions in realisation of $\operatorname{Ext}^d$

If $R$ is a commutative unital associative ring and $A$ is an $R$-algebra of dimension $d$, which is local as a ring, then from dimension theory we know that the global dimension of $A$ must be at ...
2
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1answer
108 views

Local Cohomology - Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings

This question arises in the context of Theorem 3.5.8 in Bruns and Herzog, Cohen-Macaulay Rings. Let $(R,m)$ be a local complete Cohen-Macaulay ring of dimension $d$. Denote by $H_m^d(-),\omega_R$ ...
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Surprising applications of cohomology

The concept of cohomology is one of the most subtle and powerful in modern mathematics. While its application to topology and integrability is immediate (it was probably how cohomology was born in the ...
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38 views

Question about relative singular homology groups

I know that the sphere $S^{\infty}$ is contractible, but why if $H$ is a Hilbert space then we have $$H_q(H,S^{\infty})=0, q\in \mathbb{N}?$$ Please help me Thank you
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114 views

Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
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57 views

Is it possible to compute homology groups of a space given the Pontryagin ring?

Or similarly, given the cohomology ring of a space, is it possible to compute its cohomology groups? I'm mainly interested in integer and mod 2 homology and cohomology.
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55 views

Bounds dimension, scheme and projective dimension

Is the dimension of a (commutative unital associative) algebra always bounded above by its protective (injective) dimension? If not is it always bounded above by its global dimension?
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322 views

Mayer-Vietoris sequence for local cohomology

Update 7:35pm UTC 3/23/14: I've reposted this quesion on MathOverflow here. As an assignment in my commutative algebra class, I need to prove the Mayer-Vietoris sequence for local cohomology: Let ...
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69 views

Exercise from Assem-Simson-Skowronski

I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory. The exercise in question is from chapter IV. So, let $...
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126 views

Is every module a direct limit of cyclic modules?

I want to show that $M$ is $A$-flat is equivalent to $\mathrm{Tor}_1^A(M,A/I)=0$ for every finitely generated ideal $I$. I want to show $\mathrm{Tor}^A_1(M,N)=0$ for any $A$-module $N$. Is ...
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1answer
41 views

Homotopy between two homomorphisms and homology

If I have two chain complexes $C$ and $D$ and I suppose that there is a homotopy between $\phi, \psi:C \rightarrow D$ (i.e there is a sequence of homomorphisms $(K_n: C_n\rightarrow D_{n+1})$ such ...
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2answers
160 views

If a morphism of pushouts of complexes (with one arrow monic) is composed of quasi-isos, then the induced arrow is one also

EDIT: The original title was: If a morphism of diagrams of complexes is composed of quasi-isomorphisms, is the induced arrow a quasi-isomorphism? Let $J$ be a small category and $C$ be the category ...
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48 views

existence of a cofiber sequence

can anyone help me with this problem. thanx. Show that there are cofiber sequence $S^{n+3} \to S^{n+2} \to \sum^{n}\mathbb{C}P^2$ for each $n \in \mathbb{Z}^+$. Conclude that a space of the form $S^{...
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187 views

Does $\operatorname{id} M =\dim R$ hold for finite modules of finite injective dimension?

When $\operatorname{id}R<∞$ then $\operatorname{id}R = \dim R$. The same holds for a finite free, projective or flat module instead of $R$, that is, $\operatorname{id}M = \dim R$. Does it hold for ...
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2answers
165 views

Having trouble understanding the Tor functor

I am having trouble understanding the Tor functor as presented in Dummit and Foote. Given $\dotsb\to P_n\to P_{n-1}\to\dotsb\to P_0\to B\to 0$ as a projective resolution with homomorphisms $d_i:P_n\...
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1answer
81 views

Given a torsion $R$-module $A$ where $R$ is an integral domain, $\mathrm{Tor}_n^R(A,B)$ is also torsion.

Given an integral domain $R$, and a left torsion $R$-module $A$ (i.e. $\forall{a}\in A,\exists{r}\in R$ such that $ra=0$) how would you show that $\mathrm{Tor}_n^R(A,B)$ is also a torsion $R$-module?
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91 views

Can group cohomology be used to study fiber bundles?

Is (non-abelian) cohomology used to study vector and principal bundles? Can you give me a text or an article? For example: Consider a vector bundle $E$ with fiber $V$ and base manifold $M$. Consider ...
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167 views

An easy infinite free resolution

I'm doing exercise 1.23 on Eisenbud's Commutative algebra, and I have the following situation: let $k$ be a field and $R = k[x]/(x^n)$. They ask for a free resolution of $R/(x^m)$, for some $m \leq n$....
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1answer
58 views

Acyclic resolution but not projective

Suppose $\mathfrak{C}$ is an abelian category which does not have enough projectives and we're interested in computing the right derived functors of some covariant functor $F$. If however, every ...
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52 views

Does additivity of (equivariant) cohomology hold at the algebra level?

The additivity property of many (co)homology theories is that if $X = \bigsqcup_{i \in I} X_i$ then $H^*(X) = \bigoplus_{i\in I} H^*(X_i)$. This is usually either an axiom of the theory, can be proven ...
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147 views

Existence of module of finite injective dimension

At p. 107 of the book Cohen-Macaulay Rings by Bruns and Herzog, the authors write "any module of finite projective dimension (over a Gorenstein ring $R$) has finite injective dimension as well, ...
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239 views

Isomorphism in cohomology is an isomorphism in homology

Let $f:X \to Y$ be a continuous map between topological spaces and $R$ some coefficients. From the universal coefficient theorem for homology we immediatly get, that if $H_*(f,\mathbb{Z})$ is an ...
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1answer
310 views

Homology out of Smith normal form: simultaneous or independent diagonalization?

Let $R$ be a PID (such as $\mathbb{Z}$) and $R^m\overset{A}{\longrightarrow} R^n\overset{B}{\longrightarrow} R^o$ matrices with $BA=0$ and Smith normal forms $A=P\underbrace{\mathrm{diag}(a_1,\ldots,...
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1answer
326 views

How to calculate $Ext(M,N)$?

I am confused about the calculation of $\text{Ext}(M,N)$. If $N$ is a fixed module and if we consider the projective resolution $$\cdots \to C_1 \to C_0 \to M \to 0,$$ then $\text{Ext}_n(M,N)$ is the $...
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1answer
65 views

Map induced by localization on categories

I have been doing some reading in Hartshorne's Algebraic Geometry on derived functors and subsequent results in cohomology. Given $A$ an abelian category of groups, I have seen that the map $\mathrm{...
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50 views

Under what conditions are the resolutions of two modules subcomplexes of the resolution of the tensor product?

I have that $S=k[x_1, \dots, x_n]$, $I$ is a lattice ideal, and $J$ is a monomial ideal. I am interested in the resolution of $S/(I+J)\cong S/I\otimes S/J$. In particular, I am interested in knowing ...
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146 views

Showing $M\cong M'\oplus M''$ given an exact sequence

I am struggling with the following question: $R$ is a ring. $$M'\overset{f}{\longrightarrow} M\overset{g}{\longrightarrow} M''$$ are homomorphisms of $R$-modules such that for any $R$-module $N$, the ...
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1answer
44 views

When are maps between Hom sets induced?

I'm trying to better understand $R$-module homomorphisms, and I know that say, an $\, f:M\to N$ induces $\, f_*:Hom_R(V,M)\to Hom_R(V,N)$ or $\, f^*:Hom_R(N,V)\to Hom_R(M,V)$. What I'm wondering is, ...
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69 views

Tor dimension in polynomial rings over Artin rings

I found this tricky problem in trying to understand some properties of local rings at non-smooth points of embedded curves. But this would be a very long story. So I make it short and I try to go ...
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1answer
88 views

tensor, symmetric, exterior power of a module over a PID

Let $R$ be a PID and $M\cong R^r\!\oplus\bigoplus_{i=1}^s\!R/Ra_i$. Denote the tensor, symmetric, exterior power of $M$ by $T^nM=\bigotimes_{k=1}^nM$ and $S^nM= T^nM/\langle x_{\sigma1}\!\otimes\cdots\...
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159 views

Existence of finite projective resolution

The situation I'm considering is quite involved. All rings are noetherian commutative with $1$. All modules are finitely generated. First of all we fix a non reduced local ring $A$ where all zero ...
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61 views

The differentials of a spectral sequence

Suppose we are on the $E_r$ page and the lattice either consists of 0 or $\mathbb{Q}[x,y]$ in each entry. Suppose in particular that the points $(p,q)$ and $(r, s)$ (and "their codomains") are equal ...
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Reference request: where can I find illustrative, concrete examples of the use of the Eilenberg–Moore spectral sequence?

Pursuant to advice at When does cohomology take pullbacks to pushouts?, I tried to use the Eilenberg–Moore spectral sequence in the simplest conceivable example, for the Hopf bundle $S^3 \to S^2$...
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Question on the fill-in morphism in a triangulated category

Let $$ \begin{array}{rcl} A&\to& B\\ \downarrow & &\downarrow\\ A'&\to& B' \end{array} $$ be a commutative diagram in a triangulated category. By the axioms of a triangulated ...
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38 views

The Poincare series for a bigraded vector space

I don't understand this computation (this is from McCleary's book on spectral sequences, p.15): The Poincare series of a (locally finite) bigraded vector space $E^{\ast,\ast}$ is defined as $P(E^{\...
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2answers
48 views

Extending monics in a commutative diagram

Given a commutative diagram in a Grothendieck category $\mathscr{A}$ \begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & \overset{p}{\...
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1answer
216 views

Determinant of long exact sequence

Let the following be a long exact sequence of free $A$-modules of finite rank: $$0\to F_1\to F_2\to F_3\to...\to F_n\to0$$ I want to show that $\otimes_{i=1}^n (\det F_i)^{-1^{i}} \cong A$, where ...
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1answer
54 views

Extensions of G-modules

Let $G$ be a finite group of order $n$ and $\Lambda={\mathbb{Z}}[G]$ the group ring of $G$. Let $A$ be a finitely generated free abelian group on which $G$ acts. Let $B$ be a finitely generated $\...
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What does the $\Omega$ represent in $\Omega S^{n}$?

To put my question in context, I'm reading Hatcher's book on Spectral sequences is which is say " The suspension homomorphism $E$ is the map on $pi_{i}$ induced by the natural inclusion map $S^{n}\...
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short exact sequence

Let $0 \rightarrow L \stackrel{\alpha}\rightarrow M\stackrel{\beta}\rightarrow N \rightarrow 0$ be an exact sequence, and $M_1$, $M_2$ be two submodules of $M$; then whether the follwing implications ...
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237 views

Relative homology of ball and sphere

What is the result of $H_k(B^n,S^{n-1}; \mathbb{A })$ and in any book can i found the proof ? And what about $H_n(S^{n};\mathbb{A})$ (sigular homology of the sphere )?? Please help me. Thank you
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Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and ...
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1answer
38 views

Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that $\mathrm{...
14
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609 views

Long exact sequence in cohomology associated to a short exact sequence of *functors*

In homological algebra, when you have a left exact functor $F$ From an abelian category $\mathcal{A}$ to an abelian category $\mathcal{B}$ and you have enough injectives in $\mathcal{A}$, then you ...
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Mapping cones and resolutions

Let me preface my question by acknowledging the vagueness of it. I am hoping to find some information in the form of references as opposed to a hard and fast solution. Suppose that $S=k[x_1, \dots, ...
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63 views

Showing that an epimorphism to a free module of finite rank splits

Let $M$ be an $R$-module and let $F$ be a free $R$-module of finite rank. Let $\phi : M \to F$ be an epimorphism. Then show that $M$ has a submodule $F' \cong F $ such that $M=F' \oplus \ker\phi$. I ...
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How to compute Ext over an exterior algebra

I found this question in several places (even on mathoverflow and mathstackexchange), but I never found a satisfying answer. Let $k$ be a field and $V$ a finite dimensional $k$-vectorspace. I ...