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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Free right resolution?

As I can not find a question it must be particularly obvious but I do not see ... When one talks about free resolution, it's always with left resolution. Why are we not dealing with free right ...
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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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Definition of term-wise split exact sequences of complexes

In Stacks, Section 13.9, one can find the definition of term-wise split exact sequences of complexes: we say that the exact sequence of complexes $ 0 \rightarrow A^\bullet \rightarrow B^\bullet \...
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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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Exact functors and derived functors

Given an exact (additive) functor $F$, i.e. an additive functor preserving exact sequences, it is not hard to show that all derived functors of $F$ vanish. At the same time given a right exact ...
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Exact sequence of double complexes induces exact sequence on total complexes

This is a homework question, so I'd appreciate hints (or perhaps explanations of concepts I've not properly digested) Anyhow: This is exercise 1.3.6 in Weibel's book on homological algebra. Let $0 \...
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Dimensions of vector spaces in an exact sequence

I've read the following formula in wikipedia: Given finite dimensional vector spaces $V_i$ and an exact sequence $\cdots\rightarrow V_i\rightarrow V_{i+1}\rightarrow\cdots$, we have $$ \sum_{n\in 2\...
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Hochschild dimension

I'm curious; if $A$ is a commutative $k$-algebra over a field $k$ of global dimension $n$, then is its $A^e$-projective dimension $2n$ (this is also sometimes called the Hochschild cohomological ...
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Bass' formula (homological algebra)

I am studying homological algebra and I want to be familiar with Bass' formula which associate the injective dimension to the depth of module and ring. I am looking for an online version and I can not ...
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Derived functor for of a projective module.

I'm stuck in the exercise 1 of section 6.6, Derivative Functors of Jacobson's book "Basic Algebra II". It states: If $M$ is a projective module, then $(L_0 F)M = FM$ and $(L_n F) M = 0, \, \forall ...
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Compute $Tor_i^{\mathbb Z/ 6 \mathbb Z}(\mathbb Z/ 2\mathbb Z, \mathbb Z/ 2\mathbb Z)$

I am working on this problem, and here is what I figured out: $\mathbb Z/ 6\mathbb Z \xrightarrow{\times 2} \mathbb Z/ 6\mathbb Z \xrightarrow{\times 3} \mathbb Z/ 6\mathbb Z \xrightarrow{\times2 } \...
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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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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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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 $H_1(X) \cong \frac{\mathbb{Z}^3}{im(f)} \cong \frac{\...
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$\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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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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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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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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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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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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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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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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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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$\mathbb{Z}$ 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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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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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 $...
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$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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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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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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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 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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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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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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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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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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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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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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Short Exact Sequences & Rank Nullity

This is a well known lemma that consistently appears in textbooks, either as a statement without proof, or as an exercise (see for example pp. 146 of Hatcher) If $0 \stackrel{id}{\to} A \stackrel{f}{\...
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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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Infinite Sum Axioms in Tohoku

In his Tohoku paper, section 1.5, Grothendieck states the following axioms that an abelian category might satisfy: AB4)Infinite sums exist, and the direct sum of monomorphisms is a monomorphism. AB5)...
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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 $(...