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.

learn more… | top users | synonyms

0
votes
1answer
34 views

Relation of $\operatorname{Ext}$ and projective dimension

I have some problem to understand the proof of proposition 8.38, page 473 from An Introduction to Homological Algebra by Rotman. Proposition: Let $x\in Z(R)$ be an element which is not a zero-divisor ...
0
votes
1answer
22 views

Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
1
vote
1answer
21 views

Tensor product of homology equivalences

Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same ...
3
votes
1answer
31 views

Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that ...
1
vote
1answer
35 views

Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
0
votes
1answer
37 views

Injective module and the homology of a complex

If $K$ is a complex of $R$-modules and $J$ is an injective $R$-module, prove that \begin{equation*} H^n(\operatorname{Hom}_R(K,J))\cong \operatorname{Hom}_R(H_n(K),J). \end{equation*} Thank you ...
6
votes
4answers
274 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
0
votes
0answers
24 views

How to show the homotopy category is not abelian [duplicate]

Suppose $K^+(M)$ is the category, whose objects are bounded below complex of abelian groups, morphisms are chain maps modulo homotopy equivalence. How to show the category is not abelian? Exercise ...
0
votes
1answer
41 views

Split exact sequences: a basic question.

I am a bit confused regarding the definition of a split exact sequence, whose definition is for example available here (http://ncatlab.org/nlab/show/split+exact+sequence). Let's work in an abelian ...
0
votes
0answers
21 views

Confusion about cohomology [duplicate]

Cohomology is a contravariant functor. It's easy to see that with singular cohomology, because if we have a map between cell complexes, we take $Hom(-,\mathbb R)$ (which is contrvariant) on the chain ...
3
votes
1answer
58 views

Correspondence between Ext group and extensions (from Weibel's book)

I am trying to understand the proof of Theorem 3.4.3 from Weibel's book Introduction to homological algebra. The statement is the following. Let $R$ be a ring. Given $R$-modules $A$ and $B$, an ...
0
votes
0answers
13 views

First cohomology of direct product (in the coefficients)

Let $k$ be a field and let $G = A \times B$ be the product of two algebraic groups over $k$ ($G$ is not necessarily finite nor abelian). Is there a nice way to express $H^1(Gal(k^s/k), G(k^s))$ in ...
1
vote
0answers
26 views

Connecting homomorphism of exact sequence of Ext groups

Let $0\to M\to L\to N\to0$ be an exact sequence of modules over a ring $A$. Having an $A$-module $K$ we obtain the exact sequence of Ext groups $$0\to Hom_A(N,K)\to Hom_A(L,K)\to ...
1
vote
1answer
38 views

Injective dimension $n$ implies $Ext^n$ does not vanish with an injective

Let $M$ be a finitely generated module and suppose that the injective dimension of $M$ is $n$. I want to show that there exists an injective module $I$ such that $Ext^n(I,M)\neq 0$ (and if the ...
0
votes
0answers
20 views

Minimal graded free resolution of $R/I\oplus R/J$ in terms of minimal graded free resolution of $R/I$ and $R/J$.

Let $R=k[x_1,...,x_n]$ be a graded ring over a field. Let $I,J$ be homogeneous ideals. Questions: What is the minimal graded free resolution of $R/I\oplus R/J$ (in terms of minimal graded free ...
4
votes
1answer
56 views

Derived functors - homotopical vs homological approach

In a first course in homological algebra, the lecturer introduced derived functors as universal $\delta$-functors, whose universal property is splicing short exact sequences into long ones. It so ...
1
vote
1answer
48 views

An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of ...
8
votes
0answers
123 views

Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
4
votes
0answers
55 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
2
votes
1answer
26 views

Short exact sequence of abelian groups implies long exact sequnce of cohomologies

I am trying to compute cohomologies $H^i(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}, \mathbb{Z})$. Actually it is not a big deal, because I have already computed $H^i(\mathbb{Z}/n\mathbb{Z}, ...
3
votes
1answer
82 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
0
votes
0answers
46 views

Question about Yoneda product

Let $A$ be a ring and $M,N,K$ are modules over $A$. Let $\xi\in\text{Ext}_A^1(N,M)$ and $\eta\in\text{Ext}_A^1(K,N)$ are given by $$\xi:\,\,\,0\to M\to X\to N\to0,$$ $$\eta:\,\,\,0\to N\to Y\to ...
1
vote
0answers
20 views

Cohomologies of Galois group of field extension

Let $k\subset K$ be a finite Galois extension with Galois group $G=\text{Aut}_k\,K$. How to prove that $H_i(G,K)=H^i(G,K)=0$ for all $i>0$?
1
vote
0answers
18 views

Naturality of connecting homomorphisms

Let $\mathcal{F}$ be a right-exact additive functor on the category of R-modules (R a fixed ring). Proposition A3.17(d.) in Eisenbud's Commutative algebra with a view towards algebraic geometry states ...
0
votes
0answers
19 views

Induced homology morphism of invertible linear transformation

I'm doing some excercises from Hatcher. I'm dealing with excercise 7 in section 2.2 (page 164 in PDF file): For an invertible linear transformation $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ show that ...
1
vote
2answers
31 views

exact sequence in directed limit

I want to show that proposition$5.33$ in introduction to homological algebra rotman :let $I$ be a directed set , and let $\{A_i,\alpha_j^i\}$, $\{B_i,\beta_j^i\}$, and $\{C_i,\gamma_j^i\}$ be directed ...
0
votes
0answers
64 views

Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation

The biggest obstacle for me to learn geometry and topology is the haziness of textbooks. I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to ...
3
votes
0answers
37 views

The relation between homotopy equivalence and contractible mapping cone?

In this MO thread, the OP claimed that it is obvious that homotopy equivalence implies the mapping cone contractible, whereas the converse proposition is wrong. I hate to admit that it's not obvious ...
4
votes
1answer
42 views

What is higher kernel explicitly?

Let $\mathcal{A}$ be an abelian category (for simplicity you can think that $\mathcal{A}$ is the category of modules over ring $R$). Let $[1]$ be the category with two objects and one arrow between ...
1
vote
1answer
25 views

Direct limit of modules: a property.

Suppose $A$ to be a ring and $M_i$ the indexed $A$-modules used to build the direct limit of modules $M \doteq \lim{M_i}$. Let $f_{ij}: M_i \to M_j$ the transition maps and $\phi : M_i \to M$ the ...
0
votes
1answer
32 views

Definition of (left) resolution

Let $\mathsf C$ be an abelian category. A (left) resolution of an object $A$ is a nonnegative chain complex $$\cdots \rightarrow P_2\rightarrow P_1\rightarrow P_0\rightarrow 0\rightarrow \cdots$$ ...
1
vote
0answers
57 views

“Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
2
votes
0answers
34 views

Question about computing cohomology of trivial action on $\mathbb{Z}_{4}$

I'm currently considering the trivial action of the group $G = \mathbb{Z}_{2}$ on the group $A = \mathbb{Z}/4\mathbb{Z}$. It is easy to show that $|C^{2}(G,A)|$ = $2^{8}$ and that $|B^{2}(G,A)| \leq ...
1
vote
0answers
24 views

Hochschild homology of a free commutative algebra

Let $V$ be a graded vector space over $k$. Let $Com(V)$ be the free commutative algebra over V. Let $HH_*(-,k)$ be the Hochschild homology with coefficients in $k$ functor. My questions are : $$ ...
0
votes
1answer
48 views

A question about the definition of tensor product

Let $M$ and $N$ be modules over a ring $R$. Generally, the tensor product $M\otimes N$ is defined to be an abelian group with a balanced map $j:M\times N\to M\otimes N$ such that for any abelian group ...
0
votes
1answer
17 views

every projective module has a free complement.

I have to prove that every projective module has a free complement. Now Rotman ask it to first do for $R=Z/6Z$ and $P=Z/2Z$, We know $Z/6Z \cong Z/2Z \oplus Z/3Z$. Now $Z/2Z$ is projective as it a ...
3
votes
0answers
33 views

Calculating Hodge numbers by means of locally free resolutions

In this paper the author considers a smooth $3$-fold $X$ in $\Bbb{CP}^6$ with the following locally free resolutions of its structure sheaf and squared ideal sheaf: $$0\to \mathcal O_\Bbb {P^6}(-7) ...
1
vote
1answer
59 views

example of inverse limit and direct limit

Does a direct limit of projective need to be projective? And is the inverse limit of injectives injective? I guess they need not, but I can't find an example. Can you help please?
2
votes
1answer
37 views

Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
1
vote
0answers
68 views

Let $R$ be a domain. Then $\operatorname{Tor}_n^R(A,B)$ is a torsion module

I have some problem to understanding the proof of this problem. This theorem is on page $414$ introduction to homological algebra Rotman. The theorem says: If $R$ is a domain, then ...
2
votes
2answers
52 views

How to show fraction field is flat (without localization)

Here I asked that if one can prove the field of fraction of a domain is flat. The answers used localization, which I am not familiar with. Can anyone prove it without using localization?
1
vote
2answers
45 views

Why field of fractions is flat?

I want to show this lemma: Let $R$ be a domain. If $A$ is a torsion $R$-module, then $\operatorname{Tor}_1^R (K,A)\cong A$ where $\operatorname{Frac}(R)=Q$ and $K=Q/R$. When I was reading ...
1
vote
0answers
24 views

$H^n(G,M)$ with a cyclic $G$

I have to show that $H^n( \text{Gal}(\mathbb{F}_{q^d}/\mathbb{F}_q),\mathbb{F}_{q^d}^{\times})=0$ if $n\geq 1$. I suppose I have to use the fact that if $G$ is cyclic generated by $\phi$ and $M$ is a ...
1
vote
1answer
28 views

torsion free module and injective envelop of this

suppose that $R$ be a domain, $M$ a torsion free $R$-module and $V=E(M)$, the injective envelop of $M$. Is it true that if $M$ is torsion free then $V$ is torsion free? I guess it is true because $V$ ...
3
votes
0answers
41 views

How much information about $R-\mathrm{Mod}$ can be extracted from $\underline{R-\mathrm{Mod}}$ and $K_0(R)$?

The question is in the title, so let me just repeat it: How much information about $R-\mathrm{mod}$ can be extracted from $\underline{R-\mathrm{mod}}$ and $K_0(R)$? Here ...
1
vote
1answer
21 views

Contractible chain complex

Suppose that we have a chain complex $... \to P_n \to ... \to P_1 \to P_0 \to 0$ (denote by $d_k:P_K \to P_{k-1}$ the differential of this complex) and suppose that we have $D_k:P_k \to P_{k+1}$ such ...
4
votes
1answer
35 views

Short exact sequence is split iff contractible

Let $0\rightarrow A\overset{f}{\rightarrow} B \overset{g}{\rightarrow} C\rightarrow 0$ be a short exact sequence in an abelian category. I am trying to prove this SES is contractible iff it is split. ...
0
votes
0answers
36 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!
1
vote
1answer
57 views

How do we get this quotient $\textrm{Ext}^1(N,M)/\textrm{Hom}(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of vector bundles on a surface. Here $M$ and $N$ are line bundles, and so rank $ E$=2. Also, if ...
0
votes
2answers
60 views

Exact sequence - proof

Let $R$ be a ring. Prove that a sequence of left $R$-modules and homomorphisms $$0 \to N_1 \xrightarrow{f} N_2 \xrightarrow{g} N_3$$ is exact if and only if for all left $R$-modules $M$ sequence ...