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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Tensor Product of Complexes and the definition of the differentials

Suppose we have the following complexes, $$0 \rightarrow R \xrightarrow{x_1} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_2} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_3} R \rightarrow ...
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2answers
200 views

Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a ...
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31 views

Some exercises from Introduction to Homological Algebra by J.J. Rotman (category) [on hold]

Please give solutions for these problems: Give an example of a covariant functor that does not preserve coproducts. Prove that every left exact covariant functor $T$: $_RMod$ → $Ab$ preserves ...
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1answer
34 views

Show that an $R$-module homomorphism $\alpha:A \to B$ is injective.

I am working on an exercise on injective modules: Show that an $R$-module homomorphism $\alpha:A \to B$ is injective if the induced map Hom$_R(B,Q)\to$ Hom$_R(A,Q)$ is surjective for all injective ...
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30 views

On the origins of Homological algebra

In Martin Krieger's book "Doing Mathematics: Convention Subject, Calculation, Analogy" (2003) I find the following statement (apparently, a quote from somone else) : "Homological algebra starts from ...
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1answer
43 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 ...
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1answer
23 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.
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1answer
22 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
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1answer
32 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 ...
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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 ...
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1answer
42 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 ...
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279 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. ...
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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 ...
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1answer
42 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 ...
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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
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1answer
59 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 ...
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14 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 ...
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27 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 ...
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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 ...
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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 ...
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1answer
57 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 ...
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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 ...
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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 ...
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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 ...
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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}, ...
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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 ...
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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 ...
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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 $i>0$?
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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1answer
26 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 ...
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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 $$\cdots \rightarrow P_2\rightarrow P_1\rightarrow P_0\rightarrow 0\rightarrow \cdots$$ ...
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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 ...
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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 ...
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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 : $$ ...
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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 ...
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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 ...
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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) ...
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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?
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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, ... , ...
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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 ...
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2answers
53 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?
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2answers
46 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 ...
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$H^n(G,M)$ 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 ...
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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 $V$ ...
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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 ...