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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.

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How to prove that for $K$ $\mathbb{Z}$-free & $Q$ projective, $K\otimes_{\mathbb{Z}}Q$ is a projective?

Let $K$ be $\mathbb{Z}$-free and $Q$ a projective $\mathbb{Z}G$ module then $K\otimes_{\mathbb{Z}}Q$ is a projective $\mathbb{Z}G$-module. I believe that this follows from the adjoint isomorphism ...
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Is the kernel of a map from a $\mathbb{Z}$-free module to an abelian group $\mathbb{Z}$-free? [duplicate]

Is the kernel of a map from a $\mathbb{Z}$-free module to an abelian group $\mathbb{Z}$-free? It seems like the answer should be yes as the kernel is contained inside the free $\mathbb{Z}$-module but ...
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Why is this a filtered category? Localization of rings.

I was reading this post I am trying to show that $S^{-1}R=\operatorname{colim}F(s)$, where $S$ is a multiplicative closed set in a commutative ring $R$ and $F$ is a functor from a filtered category ...
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Homology groups of the set with Mayer-Veitoris

i need some help please my professor asked for homology groups of $S^2\cup d$ which $d={\{ (0,0,t) \mid -1 \leq t \leq 1}\} $ . We just find homology groups with some points and Mayor-...
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Existance of certain spectral sequences implies the existence of a long exact sequence of derived functors associated to a short exact sequence

I saw this question asked in some form somewhere, but couldn't quite prove the statement that I wanted to, so I am hoping that someone can help. Essentially I want to prove that a spectral sequence is ...
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+100

proving property ( and Brouwer's fixed point theorem)

Let be $ K:= \overline{K_1(0)} $ a closed unit disk in $ \mathbb{R^2} $. I want to show that: (i) There does not exist a continuous function $ f: K \rightarrow \partial K$, so that $f(x)=x ,\forall ...
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What is $Ext^1(\overline{\mathbb{Q}}^\times, \overline{\mathbb{Q}})$ in abelian groups?

I want to find a way to describe all the extensions of $\overline{\mathbb{Q}}^\times$ by $\overline{\mathbb{Q}}$, i.e., all the abelian groups $A$ (and the maps $\alpha$ and $\beta$) that fit into the ...
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When can I have “short” projective resolution?

When we use universal coefficient theorem, we only need to compute Tor$(A,B)$. But suddenly, later on, like in more advanced homological algebra. We start to construct Tor$_n$, Ext$_n$... Can't we ...
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Equivalent form of the Whitehead problem

Let $M$ be an $R$-module and consider the following statement. $M$ is projective whenever the obvious group map $\tau: \text{Hom}_R(M, R) \to \text{Hom}_R(M, {R}/{I})$ is surjective for any ideal ...
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(Solved) Exact functors commute with homology

I want to show that for any exact functor $F\colon {}_R\mathrm{Mod}\rightarrow {}_S\mathrm{Mod}$ there is a natural isomorphism $$F\circ H_n \cong H_n \circ \mathbf{Ch}(F)$$ where $\mathbf{Ch}(F)$ ...
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$ \operatorname{Ext}_{k[x]}^n(k,k)$ for a field $k$ [on hold]

Consider the polynomial ring $k[x]$ for a field $k$ and the $k[x]$-module $k$, letting $x$ act trivially on $k$. What is $ \operatorname{Ext}_{k[x]}^n(k,k)$ for $n\geq0$?
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Free resolution for finitely generated module over integral group ring $ZG$

Let $M$ be a finitely generated $ZG$-module for a finite group G. I want to show that there exists a $ZG$-resolution $F$ of $M$ such that each $P_n$ is a finitely generated free $ZG$-module. Since $M$...
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Show that $(1_M)^*=1_{A^1(M)}$

I had a problem in manifold which states: Show that $A^1:Man^{op} \to Mod$ is a functor, where $M$ is a manifold, $A^1(M)$ is the $1$-form on $M$. and for $\phi:M \to N$, $A^1(\phi)=\phi^*$. This ...
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A natural isomomorphism of $G$-modules

Let $A$ be a $G$ module, $\Bbb Z$ regarded as a trivial $\Bbb ZG$ module . Then $$ \Bbb Z \otimes _G A \rightarrow A/\mathcal{G}A, \quad m\otimes a \mapsto ma+ \mathcal{G}A $$ is an isomorphism. ...
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How do a double complex on second quadrant viewed as a single complex?

Background Dimca's book "Sheaves in topology" Theorem 2.3.29 (Projection Formula). Let $f : X \rightarrow Y $ be a continuous map, $ \mathcal{F^{\cdot}} \in D^{-}(X), \mathcal{G^{\cdot}} \in ...
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1answer
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Chain complex homotopy equivalent to its homology

Let $(C_*,d)$ be a chain complex of vector spaces over a field $F$. Can always be constructed a homotopy equivalence $C_* \rightarrow H(C_*)$ ? (Here $H(C_*)$ is seen as a chain complex with $d=0$) ...
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Complexifying de Rham Complex

In Madsen's Calculus to Cohomology, it is defined $H^*(M; \Bbb C):= H^*(M) \otimes_{\Bbb R} \Bbb C$ I am curious, suppose we start with the de Rham complex, $D$, tensor it by $-\otimes_{\Bbb R} \...
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Bridge between classical and “modern” derived functors

This is a question for a reference. What I would call the classical approach to derived functors, is the following: Let $F:\mathcal{A}\to \mathcal{B}$ be a right exact functor between abelian ...
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Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup: Let $A$ be a finite-dimensional $k$-algebra over some field $k$. Let $\mathcal{B} = Hot^-(Proj A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) ...
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Projective dimension on a short exact sequence

Let $0 \to B \to P \to A \to 0$ be a short exact sequence of $R$-modules with $P$ is projective. By projective dimension of a module, I mean the length of the shortest projective resolution of it. A ...
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spectral sequence example diagonal map confusion

I'm attempting to wrap my head around spectral sequences, so constructed a really basic example to apply the definitions and go through the motions. My filtered chain complexes are: $F_2C_*: 0 \...
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Is a left homotopy inverse of a quasi-isomorphism automatically a right homotopy inverse?

Let $R$ be an associative ring with unit. Let $E$ and $F$ be two chain complexes of $R$-modules and $\phi: E\overset{\sim}{\to} F$ be a quasi-isomorphism between them, i.e. $\phi$ induces ismorphisms ...
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A different approach towards spectral sequences.

I am somewhat acquainted with spectral sequences as in Weibel's book(the usual definition with many indices and pages), but I have found a different approach in the Stacks project.Link Although I can ...
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Why homotopic equivalence is equivalent to quasi iso in K(I)

Let $X,Y \in K(A)$ where $A$ is an abelian category and $X,Y$ are complexes s.t. $X^i$ and $Y^i$ are injecive for every i. How can I prove that if $t : X \to Y$ is a quasi-isomorphism he $t$ is an ...
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1answer
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Two types of Grothendieck groups for rings

For a Noetherian ring $R$, there seem to be two versions of zeroth K-theory one can associate to it: $K_0(R)$ the Grothendieck group of the exact category of projective modules and $G_0(R)$ the ...
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1answer
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Is additivity necessary for a left exact functor to preserve pullbacks?

I'm having a bit of difficulty with exercise 5.16 from Rotman's An Introduction to Homological Algebra (second edition). The exercise (at least the relevant part) reads Prove that every left exact ...
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1answer
37 views

A module $B$ is flat if Tor $= 0$

From Weibel's "An Introduction to Homological Algebra": Exercise 3.2.1: An $R$-module $B$ is flat if Tor$_i^R(A,B) = 0$ for every $R$-module A. It seems to me that the obvious way to do this would ...
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cohomological dimension of groups vs cohomological dimension of subgroups

Let $\Gamma$ be a group and $\Gamma^\prime$ a subgroup. Then, $\text{cd }\Gamma^\prime \leq \text{cd } \Gamma$ because a projective resolution of $\mathbb{Z}$ over $\mathbb{Z}\Gamma$ can also be ...
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A Remark in Weibel's “Introduction to Homological Algebra”

In the section on the derived functors of the inverse limit(with $...3\rightarrow 2 \rightarrow 1 \rightarrow 0$ as index category), Weibel constructs the inverse limit using the map $\Delta$ in the ...
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quasi isomorphism of two dg algebras

I want to construct a chain of morphisms from a dg algebra $A$ to $B$. I assume that $A$ and $B$ is non positive, i.e, $A^n$ vanishes for $n$ greater than zero. What I have is that $H^*(A)$ is ...
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For a group homomorphism $\alpha : G\rightarrow G'$, does the pullback $\alpha^\# : Mod(G')\rightarrow Mod(G)$ send $G'$-projectives to $G$-acyclics?

Let $\alpha : G\rightarrow G'$ be a group homomorphism. There is a natural functor $\alpha^\# : Mod(G')\rightarrow Mod(G)$ sending a $G'$-module $M$ to the $G$-module given by the same underlying ...
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Calculating $ \operatorname{Ext}(\mathbb{Z}/2, \mathbb{Z}/2)$ over $\mathbb{Z}$

Calculating $ \operatorname{Ext}(\mathbb{Z}/2, \mathbb{Z}/2)$ over $\mathbb{Z}$: I just need someone to confirm that I have calculated $ \operatorname{Ext}(\mathbb{Z}/2, \mathbb{Z}/2)$ correctly. I ...
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1answer
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differential on total chain complex

There is the definition of (second) total chain complex of double complex of chains from GTM 004.He says $(\partial b)_{p,q}=\partial'b_{p+1,q}+\partial''b_{p,q+1}$,but I don't have any clues what $b_{...
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1answer
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Can we deduce that $M_0$ is a submodule of the limit of the following diagram?

Let $M_0$ be an R-module, and suppose $M_{n+1}$ is the pushout of the diagram below as shown, for all $n \in \mathbb{N}$: $$\begin{array}{ccc}M_n&\to& M_{n+1}\\\uparrow &&\uparrow\\A&...
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1answer
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Isomorphism between torsion subgroups of $H_q (M)$ and $H_{n-q-1} (M)$ where $M$ is a compact oriented $n$-manifold

It it the exercise in Massey's book Massey, William S., Singular homology theory, Graduate Texts in Mathematics, 70. New York Heidelberg Berlin: Springer- Verlag. XII, 265 p. DM 49.50; $ 29.20 (1980)...
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1answer
40 views

Non-existence of non-zero chain maps from a complex to its homology

I am trying to solve the following exercise. Let $\mathcal A$ be an Abelian category and consider the category of cochain complexes in $\mathcal A$. Construct an example of a cochain complex $(...
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show that functor $L_0T$ is right exact

Show that $L_0T$ is right exact.Here T is an additive functor from category of $\Lambda$-modules to abelian group. I try to prove it,but I think the condition is too little.Any hints?
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Problem of modules over PID, Exercise $9.13(2)$ in Rotman's Advanced Modern Algebra $2$nd Edition

Rotman's Advanced Modern Algebra ($3$rd Edition), Exercise B-$3.24(2)$, also Exercise $9.13(2)$ in $2$nd Edition. $R$ is PID, $M$ is $P$-primary $R$-module. $P=(p)$ is non-zero prime ideal. $k_P:=R/...
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1answer
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Very basic question about pre-additive category

I am trying to prove whether in a pre-additive category, $0_{Mor(y,z)}\circ f=0_{Mor(x,z)}$ for objects $x,y,z$ and $f\in Mor(x,y)$. Now, by bi-linearity of composition maps $$ 0_{Mor(y,z)}\circ f+ 0_{...
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2answers
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$S^{-1}Ext^n_R(A,B) \cong Ext^n_{S^{-1}R}(S^{-1}A,S^{-1}B)$

I am trying to understand a proof from the book An Introduction to Homological Algebra, Weibel. Let $A$ be a finitely generated module over a noetherian ring $R$. Then for every multiplicative set $...
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1answer
29 views

Co-filtered limits in algebraic categories.

If $R$ is a commutative ring with unity, we know that filtered colimits are exact. We also know that in an algebraic category, filtered colimits commute with finite limits. Are the following ...
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$\mathfrak{m}= (x_1,x_2,x_3,…)$ as an $R$-module ($R= k[x_1,x_2,x_3,…]$, $k$ field)

Let $R= k[x_1,x_2,x_3,...]$, where $k$ is a field. Consider the $R$-module $\mathfrak{m}= (x_1,x_2,x_3,...)$. I would like to check: If $\mathfrak{m}$ is free $R$-module. I have seen that it is not ...
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1answer
71 views

The long exact sequence for left derived functors in Eisenbud's Commutative algebra

Could anyone say any details about 3.17c (author only writes that it's immediate from 3.15 and 3.16)? (The photos below are from "Commutative algebra with a view toward algebraic geometry" by David ...
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1answer
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Under what assumptions on the grading is the fundamental theorem of twisting morphisms true?

In my favorite book Algebraic Operads by Loday and Vallette there is a theorem (2.3.1 in my book) which says that for a twisting morphism $\alpha:C\to A$ from a connected wdga coalgebra to a connected ...
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1answer
64 views

Computing $\operatorname{Tor}_*^{R}(\mathbb{Z},\mathbb{Z})$ for $R=\mathbb Z[C_n]$.

I would like to compute $\operatorname{Tor}_k^{R}(\mathbb{Z},\mathbb{Z})$, where $R = \mathbb{Z}[x]/(x^n -1)=\mathbb Z[X]$. I think I am near to do so, but I can not figure out the last step. I ...
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45 views

chain homotopy equivalence and quasi-isomorphism

Suppose $(C,d)$ and $(D,\delta)$ are two chain complexes over a field and $f:C\to D$ is a chain map. We say $f$ is a quasi-isomorphism if it induces an isomorphism of the homology groups $H(C,d)\to ...
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1answer
24 views

tensoring with flat module factors the kernel

I want to show that if $F$ is a flat $R$-module, then for any $R$-homomorphism $\varphi: M \rightarrow N$, we have $$\ker (1_F \otimes \varphi) \cong F \otimes \ker \varphi $$ The $\supset$ ...
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37 views

Is it true that a family of functors is exact precisely when their direct sum is?

If $\{F_i: i\in I\}$ is a family of functors $F_i: \mathcal{A} \rightarrow \mathcal{B}$ between categories, then $\oplus_I F_i$ is exact if and only if each $F_i$ is exact? What I tried: Given a ...
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1answer
50 views

Why is induced map on zero homology the identity and not negative the identity?

Suppose we have a simplicial map $f$ on a path connected simplicial complex $X$. The answer here: Induced map on zeroth homology is zero claims that the induced map on the $0$-homology given by $f_*: ...
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$\operatorname{Ext}^\bullet_R(R/rR,M)$ and $0 \to A[r] \to B[r] \to C[r] \to A/rA \to B/rB \to C/rC \to 0$

$\newcommand{Ext}{\operatorname{Ext}}\newcommand{Hom}{\operatorname{Hom}_R}$Let $R$ be a commutative ring with unity and $r \in R$. Applying snake lemma to the following diagram: $$\begin{array}{c} ...