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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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Shifting the nonzero homology

Suppose there is a finite chain complex $$C_{n} \rightarrow \cdots \rightarrow C_{d}\rightarrow \cdots \rightarrow C_{0}\,,$$ such that $H_{i}(C_{\bullet})$ is vanished except for $i=d$. Are there ...
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How to construct base change of cohomology map $H^n(Y,\mathcal{F})\to H^n(X,f^*\mathcal{F})$?

Let $f:X\to Y$ be a morphism of schemes, let $\mathcal{F}$ be an abelian sheaf on $Y$, in some topology $\tau$, safely speaking we assume it is the Zariski topology but it shouldn't really matter. We ...
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Is $H_i(Tot(C))\neq 0$($i\geq 0$) with $C$ double complex on upper half plane every arrows are $Z_4\to Z_4$ by 2 multiplication

This is related to Weibel Exercise 2.7.1 $C$ is periodic upper half plan complex $C_{pq}=Z_4$ for all $p\in Z,q\geq 0$ all differentials are multiplication by 2. It is easy to apply acyclic assembly ...
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Not understanding the shifting in the index of balancing Tor's proof in Weibel

This is related to Weibel's Homological Algebra Chpt 2, Sec 7, Balancing Tor and Ext. I do not think the question I am asking is related to $P,Q$ being $A,B$'s projective resolution as it is related ...
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34 views

Show that there exists a pullback square for modules.

Let $R$ be a commutative ring with identity and let $$\matrix{&&X\\&&\downarrow\\Y&\to&M}$$ be homomorphisms of $R$-modules. Show that it can be embedded into some pullback ...
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Injective resolution of a complex is unique up to homotopy type?

I was reading about how we can construct an injective resolution $I^\bullet$ of a bounded-below complex $A^\bullet$ in a category $\mathcal{A}$ with enough injective in this MSE post. But I'm ...
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S.E.S given by: $0 \rightarrow \mathbb{Z} \stackrel{f}{\rightarrow} \mathbb{Z}^3 \stackrel{h}{\rightarrow} H_1(X) \rightarrow 0$

$0 \rightarrow \mathbb{Z} \stackrel{f}{\rightarrow} \mathbb{Z}^3 \stackrel{h}{\rightarrow} H_1(X) \rightarrow 0$ Where $f(1) = (1,0,2)$ Then $H_1(X) \cong \frac{\mathbb{Z}^3}{im(f)} \cong \frac{\...
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1answer
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Homology with coefficients for $S^n$.

It is claimed in page 154 Lemma 2.49 that if there is a group homomoprhism $\varphi:\Bbb Z \rightarrow G$, then we have the following commutativity $$\require{AMScd}\begin{CD}\Bbb Z @> {\varphi}&...
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Module over restriction of scalars

Here is an exercise of the book commutative algebra by Atiyah and MacDonald (Ex 2.13): Let $ f : A \rightarrow B$ be ring homomorphism and $N$ be a $B$ module. Regarding $N$ as a $A$ module by ...
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$\text{Tor}$ and $IJ=I\cap J$: Eisenbud exercice A3.17

$I$ and $J$ are ideal of a ring $R$. From the short exact sequence $$ 0\to I\to R\to R/I\to 0 $$ we have $$ 0\to \text{Tor}_1^R(R/I,R/J)\to I/(IJ)\to R/J\to R/(I+J) \to 0 $$ So $$ \text{Tor}_1^R(R/I,...
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Functor $L:\mathcal{A}\to\mathcal{B}$ is left adjoint to $R$ functor, then $L$ preserves all colimit.

This is a statement made in Weibel Chpt 2, Sec 6, Adjoint Functors and Left/Right Exactness. 2.6.10 Let functor $L:\mathcal{A}\to\mathcal{B}$ be left adjoint to a functor $R:\mathcal{B}\to\mathcal{A}...
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group homomorphism $\Phi: \pi_1(X,x_0)^{ab}\to H_1(X,\mathbb{Z})$ where $\pi_1(X,x_0)^{ab}$ is the abelianization of the group $\pi_1(X,x_0)$.

Let's suppose $$\phi: \pi_1(X,x_0)\to H_1(X,\mathbb{Z})$$ $$[f]\to [[f]]$$ I already prove that this function is well defined and is a group homomorphism. I have to show that this $\phi$ function ...
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Extending maps from a submodule

I want to prove the following statement: Let $M, L$ be $R$ - modules. If for every $(N, \phi: N \to L)$, where $N$ is a finitely generated submodule of $M$, $\phi$ can be extended to a map from $M \...
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Prove $F[x]/(x^n)$ is an injective module

Let $F$ be a field and $n\geq1$ (1) Prove $R=F[x]/(x^n)$ is an injective $R$-module. (2) Give a projective resolution and an injective resolution of the $R$-submodule $M=(x)/(x^n)$ For part (1), I ...
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1answer
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Show that any monomorphism and a homomorphism can be embedded into a commutative diagram with exact rows.

Let $R$ be a commutative ring with identity. Show that the diagram of $R$-module homomorphisms with the row exact \begin{matrix} 0&\to&M&\mathop{\to}\limits^{f}&X\\ &&\...
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Why do we want to compute derived functors via forgetful functors though easier?

This is a statement made in Weibel, Homological Algebra, Chpt 2, right after Exercise 2.4.2. Assume $F$ is right exact functor and thus it allows left derived functors. "Forgetful functors such as $...
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Finite representation-infinite rings

Rings are not necessarily commutative, but associate and unital here. Recall that representation-infinite means that there are infinite non-isomorphic indecomposable modules. For a natural number $m$ ...
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1answer
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Prime ideals and maximal ideals of the Pullback of rings

Let $A,B,C$ be commutative Noetherian rings with given surjective ring homomorphisms $f:A\twoheadrightarrow C $ and $g: B \twoheadrightarrow C$. Let $A\times_C B:=\{(a,b)\in A \times B : f(a)=g(b)\}$ (...
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1answer
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Serre classes and the Serre spectral sequence

Let $C$ be a Serre class which satisfies the additional axioms about $\otimes, \mathrm{Tor}, K(A,1)$'s. It is then easy to check that if $F\to X\to B$ is a Serre fibration with $\pi_1(B)$ acting ...
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1answer
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Why does $H_{n}(\operatorname{Hom}_{R}(M,I_{\ast}(N))=H_{n}(\operatorname{Hom}_{R}(P_{\ast}(M),N))$?

I'm trying to show that the two definitions for the $\operatorname{Ext}$ functor are the same whether obtained via injective or projective resolutions. I understand I need to show the following but ...
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1answer
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$\mathbb{Z}$ has cohomological dimension one

My lecture notes state the following with no further explanation: If $R=\mathbb{Z}$, then every R-module admits a resolution of length 1. This implies that $Tor_i^{\mathbb{Z}}$ and $Ext_{\mathbb{Z}}...
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$f$ is a cycle since it is a loop, and since $H_1(point) = 0$, $f$ must then be a boundary.

I am reading Hatcher's book (algebraic topology, p.166) and I can not understand what he says in the book: I know that $H_1(point)=0$, but I do not know why this implies that "$f$ must then be a ...
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2answers
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When $\textrm{Tor}_n^A(-,A/radA)\neq 0$? ($A$ a finite dimensional $K$-algebra)

This question arrise from a proof in paper: Unbounded derived categories and finitistic dimension conjecture - Jeremy Rickard, more spefically Theorem 4.3. Let $A$ be a finite dimensional algebra ...
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1answer
24 views

Quotient of category of f.g. modules by subcategory

Let $\mathcal A$ be the category of finitely generated modules over $A[t]$ and $\mathcal B$ be its subcategory of modules which is annihilated by some power of $t$. Then I want to show that quotient ...
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1answer
62 views

Weibel exercise 1.1.4, taking $A = Z_n$…

Exercise 1.1.4 Show that $\{\text{Hom}_R(A, C_n)\}$ forms a chain complex of abelian groups for every $R$-module $A$ and every $R$-module chain complex $C_{\cdot}$. Taking $A = Z_n$, show that if $...
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What is the “see-saw exact sequence”?

Let a connected linear algebraic group $G$ acts on an algebraic variety $X$, proper over a filed $k$. In the proof of Proposition 1.5. of Mumford's GIT book, he says "....consider the see-saw exact ...
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1answer
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Distinguished triangle in abelian triangulated categories

I know that an abelian triangulated category is semisimple,i.e.,any exact sequence splits.But Why does any distinguished triangle is isomorphic to a triangle of the form $X \stackrel{f}{\rightarrow} ...
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1answer
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Prove the $R$-module isomorphism $P\oplus P\cong R\oplus R$

Let $$R=\{f:\mathbb{R}\to\mathbb{R}:f\text{ is continuous and }f(x+\pi)=f(x)\}$$ $$P=\{f:\mathbb{R}\to\mathbb{R}:f\text{ is continuous and }f(x+\pi)=-f(x)\}$$ Then under addtition and multiplication $...
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Example of non-zero object of an abelian category that is both injective and projective

In the category of R-modules an object that is both injective and projective is necessarily the zero module. Are there any abelian categories with examples non-zero objects that are both injective ...
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Krull dimension of polynomial rings over Noetherian rings via homological methods

The following problem is an exercise at $11^{th}$ chapter in Atiyah's book on commutative algebra: For any Noetherian ring $R$ we have $\dim R[x] = 1 + \dim R$ where $\dim$ stands for Krull ...
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1answer
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Homology of total complex of a double complex

The following question arose while studying chapter 12.5 of the book Categories and Sheaves by Kashiwara and Schapira, abbreviated in the following by [KS]. Let $X$ be a double complex (with ...
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1answer
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Equivalent condition to left exactness

I'm trying to prove that an additive functor $F:\mathcal{A}^\text{op}\to \text{AbGrp}$ on an abelian category to abelian groups is left exact if and only if for every epimorphism $p:A\to B$ in $\...
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1answer
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Reference request: Construction of torsion pair from a class closed under quotients, extensions and coproducts.

The category I'm working in is $\operatorname{Mod}A$ for some unitary ring $A$. I'm looking for a reference on when certain subcategories of $\operatorname{Mod}A$ give rise to torsion pairs. Let $\...
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1answer
42 views

Cancelling finitely generated projective modules from a tensor product of finitely generated projective modules

Let $R$ be a commutative Noetherian ring (with unity) and $M,N,P$ be finitely generated projective modules over $R$ such that for some $n\ge 1$, we have $M\otimes_R N \cong M \otimes_R P \cong R^n$. ...
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2answers
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How can one express $\mathbb{Q} / \mathbb{Z}$ as a direct sum of $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime?

Consider the $\mathbb{Z}$ modules $\mathbb{Q}$ and $\mathbb{Z} / p$ for $p$ prime. I have a result that says that every injective $\mathbb{Z}$ module is a direct sum of these modules. I also know that ...
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Zeroth left derived functor of right exact functor

I have some difficulty understanding the following proof (source): Claim: If $T: \mathcal{A} \rightarrow \mathcal{B}$ is a right exact functor of two Abelian categories, then $L_0 T$ and $T$ are ...
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1answer
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Tensor product of exact complexes is exact

Let $M_\circ = \dots \to M_n \dots \to M_0 \to 0$ and $N_\circ = \dots \to N_n \dots \to N_0 \to 0$ be exact complexes of modules over a ring $A$ such that each module is flat. Is it then true that $(...
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1answer
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On the depths of symbolic powers of the Stanley-Reisner ideal of a bow-tie complex

Consider the polynomial ring $S=k[x_1,...,x_5]$. Consider the Stanley-Reisner ideal $I$ (i.e. the face ideal) of the simplicial complex which is a bow-tie $\Delta:=\left<x_1x_2x_3,x_3x_4x_5\right&...
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1answer
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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 $...
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1answer
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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 $T: R$-...
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1answer
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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 \...
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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}...
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1answer
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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. ...
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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_{\...
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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 ...
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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, \...
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1answer
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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, ...
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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 ...
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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 ...
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0answers
42 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 ...