Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

1
vote
1answer
46 views

Show that $\bigoplus_{i \text{ even}}C_i=\bigoplus_{i\text{ odd}}C_i$

Let $C_*$ be a chain complex such that each $C_i$ is a torsion-free, finite-range abelian group with $C_i=0$ for all $i<0$. Suppose that $C_i=0$ for all $i$ is sufficiently large and that for all $...
0
votes
0answers
13 views

On covariant linear functors $T: R$-Mod $\to R$-Mod which preserves direct-limits or inverse-limits

Let $R$ be a commutative ring with unity. Let $R$-Mod denote the category of $R$-modules, and $Ab$ denote the category of Abelian groups. Now, it is known that a covariant additive functor $T: R$-...
1
vote
1answer
25 views

Does every covariant, additive, faithfully-exact functor $T:R$-Mod $\to Ab$ preserve either direct sum or direct product?

Let $R$ be a commutative Noetherian ring. Let $Ab$ denote the category of abelian groups. Let $T:R$-Mod $\to Ab$ be a covariant, additive functor such that for any sequence of $R$-modules, $A \...
1
vote
0answers
24 views

On faithfully flat and faithfully projective modules

Let $R$ be a commutative Noetherian ring. Let $P,Q$ be some $R$-modules such that $-\otimes_R P $ and $ Hom_R(Q,-) $ are faithfully exact functors i.e., for any sequence of modules $A \xrightarrow{f}...
2
votes
1answer
18 views

Computing the homology of a simple chain complex

Let $R$ be a ring and $x\in R$ be a central element. Consider the complex $$0 \rightarrow R \xrightarrow{x} R \rightarrow 0$$ concentrated in degrees 1 and 0. Compute the homology of this complex. ...
0
votes
0answers
14 views

Exact functor and relationship between Ext functors

Let $F:\mathcal{A}\to\mathcal{B}$ be an exact functor betwen two abelian categories. Let $r\geq 1$ be an integer. I wonder what is the relationship between $Ext^r_{\mathcal{A}}(X,Y)$ and $Ext^r_{\...
3
votes
0answers
49 views

Two questions about the Hom functors

Given a locally small category $\mathcal{C}$, Wikipedia defines the Hom functors as At my lectures (for the more specific case of $\mathcal{C}=\operatorname{R-Mod}$ -- the category of R-modules for ...
1
vote
0answers
35 views

Given $\phi \in \mathrm{End}(M)$, when does $\phi$ injective imply $\phi^*$ surjective and $\phi^*$ injective imply $\phi$ surjective?

Let $M$ be a module over a commutative ring (with unity) $R$. Let $\phi : M \to M$ be an $R$-module homomorphism. Then we have a dual map $\phi^* : M^* \to M^*$ given by $\phi^*(f)=f\circ \phi, \...
0
votes
0answers
42 views

Role of d^2 = 0 in chain complex

What is the motivation for requiring that the square of a differential be 0 for a complex, aside from enabling us to speak of the homology of a complex? Other homological notions like chain maps, ...
1
vote
0answers
22 views

Prove that there is an isomorphism $\phi_n:H_n(C_*)\to\bigoplus_{\alpha\in\Lambda}H_n(C_*^{\alpha}) $

Let $\Lambda $ be a fixed set. For each $\alpha\in\Lambda $ is $\{C_n^{\alpha}\}_{n\in\mathbb{Z}}$ a complex of chains with homomorphism border $\partial^{\alpha}$. Prove that there is an isomorphism ...
1
vote
0answers
19 views

Prove that $\{C_n\}_{n\in\mathbb{Z}}$ is a chain complex with homomorphism border $\partial:=\bigoplus_{\alpha\in\Lambda}\partial^{\alpha} $.

Let $\Lambda $ be a fixed set. For each $\alpha\in\Lambda $ is $\{C_n^{\alpha}\}_{n\in\mathbb{Z}}$ a complex of chains with homomorphism border $\partial^{\alpha}$. For each $n\in\mathbb{Z}$ we ...
2
votes
0answers
37 views

Why does this exact sequence exist?

I'm reading a proof and don't understand a certain part. Let $A^\bullet$ be a (cochain) complex of abelian groups. Let $I^\bullet$ be an injective resolution of an abelian group $B$. Then there is a ...
1
vote
1answer
12 views

Finitely generated projective resolution of a module over a regular local ring

Let $R$ be a regular local ring and let $M$ be an $R$-module. Then there exists a finite projective resolution $P_\bullet\to M\to 0$. However, need there exist a finite projective resolution ...
1
vote
1answer
34 views

Hopfian modules and equivalence of categories of modules

For a ring with unity (not necessarily commutative) $R$, let $R$-$Mod$ denote the category of left $R$-modules. Let $R,S$ be two rings with unity and $T: R$-Mod $\to S$-Mod be an equivalence of ...
4
votes
0answers
47 views

$f_n:C_n\to D_n$ is injective for all $n\in\mathbb{Z}$ but $f_*:H_n(C_*)\to H_n(D_*)$ is not injective for some $n\in \mathbb{Z}$.

Example of two complexes of chains $C_*$ and $D_*$ and a morphism of complexes of chains $f_*:C_*\to D_*$ in such a way that $f_n:C_n\to D_n$ is injective for all $n\in\mathbb{Z}$ but $f_*:H_n(C_*)\to ...
2
votes
1answer
30 views

How does Matlis duality behave w.r.t. Hopfian and Co-hopfian modules?

Let $(R,\mathfrak m, k)$ be a Noetherian, complete, local ring. Let $E$ be an injective hull of $k$. We know that the Matlis duality functor $D(-):= Hom_R(-, E)$ gives an anti-equivalence between the ...
1
vote
0answers
22 views

Torsion-less module over commutative ring whose injective hull is Hopfian

Let $M$ be a module over a commutative ring (with unity) $R$. Let $E_R(M)$ denote the injective Hull of $M$ . If $M$ is torsion-less (i.e. $\cap_{f\in M^*=Hom_R(M,R)} \ker f=(0)$ ) and $E_R(M)$ is ...
6
votes
1answer
66 views

Integral homology group of a 3-torus cut out a donut

I know that the integral homology group of the manifold $M$ is given by $$ H_j(M,\mathbb{Z}) $$ I also have tried that $H_j(T^3,\mathbb{Z})$ is given by $$ H_0(T^3,\mathbb{Z})=\mathbb{Z}, $$ $$ H_1(...
1
vote
1answer
12 views

Homomorphism of chain complexes induces homotopy/isomorphism of chain complexes

I was given the following exercise, but the person who gave me the exercise wasn't sure about some of the details (such as signs). Let $(C, \partial)$ be a chain complex, and $h: (C, \partial) \...
2
votes
0answers
17 views

Derived $p$-completion : injective in the category of diagrams?

I'm reading a set of notes that has an interlude about derived $p$-completions of abelian groups. My first question stems from the fact that this interlude, although interesting, is not very ...
2
votes
0answers
10 views

Difference for even and odd values for $n$ in the equation system $u_{tt}=a^2u_{xx}$ and $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$

This is a follow-up question of What would happen if the boundary value for $u_{tt}=a^2u_{xx}$ is that $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$. In the following one-dimensional wave equation ...
0
votes
1answer
29 views

Chain homotopy between homotopy equivalences [closed]

Are any two homotopy equivalences between two chain complexes always chain homotopic? I guess it’s not always true. What if they are chain complexes of free abelian groups.
3
votes
1answer
53 views

Exact sequence of abelian groups

Let $R= \mathbb Z$ or $\mathbb Q$. Let $A$ be an abelian group. Consider the exact sequence $$0 \to A \to R \to R \to 0.$$ Then $A$ must be zero. It seems correct but can anyone produce a ...
3
votes
0answers
18 views

Reference Request: Proof of $\mathrm{H}(\mathrm{Prim}\,\mathcal{H}) \cong \mathrm{Prim}\,\mathrm{H}(\mathcal{H})$ for cocommutative dg-Hopf algebras

In Loday’s book Cyclic Homology the following theorem appears: A.9 Theorem. On a cocommutative differential graded Hopf algebra $\mathcal{H}$ over a characteristic zero field $k$ the homology and ...
2
votes
0answers
99 views

Flat $\mathbb{Z}$-modules and algebraic independence

I know that this question is naive, but I really want to find out if this observation is correct. First, let $\alpha_1, \alpha_2, \dotsc, \alpha_k$ be real numbers. Consider the $\mathbb{Z}$-module ...
1
vote
1answer
45 views

Global dimension of tensor product of algebras

I am looking for a reference or proof of the following fact: If $K$ is algebraically closed field and $A$ and $B$ are finite dimensional $K$-algebras then $\text{gl.dim}(A\otimes_KB)= \text{gl.dim}(A)...
1
vote
2answers
49 views

how to prove that $\mathbb Q$ is flat as a $\mathbb Z-$module [duplicate]

I know that $Tor^{\mathbb z}_1(\mathbb Z, N) = 0$ for any $\mathbb Z-$module, because free modules are flat. Then because $Tor_1$ is local, we have $Tor_1^{\mathbb Q}(\mathbb Q, S^{-1}N) = 0$, which ...
1
vote
1answer
39 views

What would happen if the boundary value for $u_{tt}=a^2u_{xx}$ is that $u|_{x=0}=0$ and $u|_{x=l}=\sin\frac{n\pi a}lt$

During the discussion of non-homogenous boundary values for the one-dimensional wave equation $$ u=u(x,t),\;\frac{\partial ^2u}{\partial t^2}=a^2 \frac{\partial ^2u}{\partial x^2} $$ where the ...
2
votes
1answer
47 views

Kernel and Image definition in general category theory.

Let $A \xrightarrow{f} B$ be a morphism of objects $A,B$ of a category $\mathcal{C}$. Then We say that a morphism $K \xrightarrow{\iota_K} A$ of objects of $\mathcal{C}$ is a $\mathit{kernel}$ of $f$ ...
0
votes
1answer
30 views

Confusion over distinguished triangle

According to https://stacks.math.columbia.edu/tag/08J5 we have for every complex $K$ and integer $a$ a distinguished triangle $$\tau_{\leq a}K\rightarrow\tau_{\leq a+1}K\rightarrow H^{a+1}(K)[-a-1]\...
2
votes
0answers
86 views

Tor commutes with direct limits

Weibel has shown that the filtered colimit functor $$ \varinjlim :(RMod)^I \rightarrow (RMod)$$ where $I$ is a filtered category, is exact. In corollary 2.6.16, pg 58 he claims Corollarry 2.6.16. ...
0
votes
1answer
28 views

First group homology for trivial module

Let $G$ be a group and $A$ be a $G$-module. I use $H^i(G,A)$ for group cohomology, and $H_i(G,A)$ for group homology. It is well known that if $A$ is a trivial module, then $$ H^1(G,A) \cong \...
0
votes
1answer
26 views

Invertible modules are locally finite free , proof explanation

I am trying to understand the particular argument in Lemma 15.102.2's proof. It writes that if $M$ is invertible $R$-module, then we have an automorphism $M \rightarrow M$ which factors as $$M \...
2
votes
0answers
30 views

Homology of solvable Lie algebras

Let $\mathfrak{g}$ be a solvable lie algebra and $\lambda\in (\mathfrak{g}/[\mathfrak{g},\mathfrak{g}])^*$ be the character of $\mathfrak{g}$. How to compute homology for $\mathbb{C}_\lambda$, the ...
2
votes
2answers
32 views

An infinite product of fields is not a semisimple ring

I want to show that an infinite product of fields is not a semisimple ring. I know Artin-Wedderburn Theorem, but I wonder can I explain it without using this theorem? Any help will be appreciated. ...
2
votes
1answer
67 views

Left ideal $I$ is a direct summand of $R$ iff $I = Rr$, $r^2 = r$

I am stuck on proving a left ideal $I$ of a ring $R$ is a direct summand of $R$ if and only if $I = Rr$ with $r^2 = r$. Could you help me with that? Any help will be very appreciated. Thanks!
1
vote
0answers
24 views

Snake lemma for derived functors

Assume we are in some abelian category with enough injectives, and we are given a short exact sequence: $$0\longrightarrow A'\longrightarrow A\longrightarrow A''\longrightarrow 0 $$ And a left exact ...
4
votes
1answer
25 views

Module of arbitrary projective dimension

Given any natural number $n$, does there a ring $A$ and an $A$-module $M$ such that projective dimension of $M$ is $n$? I am think this statement should be true but I don’t know how to find such $A$ ...
13
votes
0answers
273 views

Abstract proof that $\lvert H^2(G,A)\rvert$ counts group extensions.

$\DeclareMathOperator{\Hom}{Hom}$ $\DeclareMathOperator{\im}{im}$ $\DeclareMathOperator{\id}{id}$ $\DeclareMathOperator{\ext}{Ext}$ $\newcommand{\Z}{\mathbb{Z}}$ Let $G$ be a group, let $A$ be a $G$-...
1
vote
1answer
19 views

Projective dimension over restriction of scalars

Let $A$ and $B$ be two rings with a ring homomorphism $f: A\to B$ such that $B$ has finite projective dimension over $A$. Is it true that any module which has finite projective dimension as $B$-module ...
4
votes
1answer
80 views

Does equivalence of derived categories preserve boundedness?

Let $\mathcal{C}$ be an abelian category and consider the derived category $D(C)$. Suppose that $F: D(\mathcal{C}) \to D(\mathcal{C})$ is an auto-equivalence. My question: must $F$ preserve the ...
2
votes
1answer
17 views

Homology of $H_1(X_m)$.

Let $X_m$ be a space obtained from $S^1$ by attaching $D^2$ through the map $f(z)=z^m$ around the boundary. I have computed the homology group of it by exact sequence $$\mathbb{Z} \cong H_1(S^1)\...
0
votes
1answer
30 views

Reference for Universal Coefficient Theorem

I am looking for a proof of the following fact: let $C$ be a chain complex of real vector spaces, $C^*$ the dual cochain complex. Then $H^n(C^*) \cong H_n(C)^*$. That is, taking homology commutes with ...
2
votes
1answer
87 views

Computing algebraic de Rham cohomology

Let $R=\mathbb C[x,y]/(y(x-a)(x-b)-1)$ where $a,b$ are distinct complex numbers. Show that the cohomology of the de Rham complex $$0\to R\to \Omega_{R/\mathbb C}\to 0$$ is $\mathbb C$ in degree zero ...
0
votes
0answers
13 views

Any $f': A \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ an $R$ module homo can be lifted to $F': B \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ [duplicate]

I have posted a question here: Any $f': A \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ can be lifted to $F': B \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ but I didn't have many views and after 36 hours I got ...
0
votes
1answer
35 views

The projective dimension of modules in a short exact sequence

Let $0\to K\to P\to A\to 0$ be a short exact sequence of right modules with $P$ projective and $A$ not projective. Suppose $\text{pd}A<\infty$ and $\text{pd}K<\infty$, where $\text{pd}$ is the ...
1
vote
1answer
43 views

Unique $\mathbb{R}$-linear map between tensor products

I am reviewing materials in tensor products and I got stuck on this one, and I am never comfortable with "showing there exists a unique linear map" type of question. Let $V$ be a real vector space ...
1
vote
2answers
84 views

Any $f': A \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$ can be lifted to $F': B \to \mathrm{Hom}_{\mathbb{Z}}(R,M)$

let $\psi: A \to B$ be an injective $R$ module homomorphism, and it is given that any $f: A \to M$ $\mathbb{Z}$ module homomorphism can be lifted to a $\mathbb{Z}$ module homomorphism $F: B \to M$ s.t ...
3
votes
1answer
98 views

Characterizing a module of Kahler differentials

Consider the $\mathbb C$-algebra $R=\mathbb C[x,y,z]/(z(y^2-x^3)-1)$. How to prove that the module of Kahler differentials $\Omega_{R/\mathbb C}$ of $R$ over $\mathbb C$ is a free $R$-module of rank 2?...
1
vote
0answers
29 views

Fiber integration is independent of the operations involved.

In Definition 2 of Fiber Integration nlab post, the author claimed that the operation is indpendent of the choices involved. How is this so? The post itself is quite long. I think it is easier ...