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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Projective dimension of module over regular ring is always finite? [duplicate]

Let $R$ be a regular ring, i.e. a commutative Noetherian ring all whose localizations at every prime ideal is a regular local ring. So localization of $R$ at every prime ideal has finite global ...
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Composition of morphisms in Quotient category

I am having trouble understanding the composition of morphisms in the quotient category of an abelian category, following Gabriel's thesis on abelian categories. Let $\mathcal{A}$ be an abelian ...
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Regarding the construction of quotient category

Let $\mathcal{A}$ be an abelian category and $\mathcal{T}$ be a thick subcategory (i.e., closed under taking subquotient and extensions) of $\mathcal{A}$. Then we construct the quotient category $\...
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Understanding homology as a functor from $_R\mathbf{Comp}$ to $_R\textbf{Mod}$

I have some trouble understanding the definition of homology. As the below figure says, $H_n$ is a functor for every $n\in\Bbb Z$. For example, $H_4$ is a functor from $_R\mathbf{Comp}$ to $_R\textbf{...
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Valuation ring, of infinite global dimension, with principal maximal ideal

Does there exist a Valuation ring $(R, \mathfrak m)$ , with principal maximal ideal, of infinite global dimension ? Corollary 2 of the following paper by Osofsky has an example of Valuation ring of ...
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Equivariant projective modules and skew group algebras

This is a question related to the two dimensional McKay correspondence. Let $R = \mathbb{C}[x,y]$, and $G$ a finite group acting on $R$. Recall that a $G$-equivariant $R$-module is an $R$-module with ...
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Understanding the morphism category (arrow category)

$\require{AMScd}$ I want to understand how the morphism category (arrow category) of $\operatorname{Mod}(A)$ works. The material I have found online have not really helped me that much with proper ...
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Why is $f^{-1}f(U)=U+\ker f$?

We are working in some abelian category. Given subobjects $Z, Z'\hookrightarrow X$ of $X$, we can define $Z\cap Z'= Z\times_X Z'$ as the pullback over $X$ and $Z+Z'= Z\amalg_{Z\cap Z'} Z'$ as the ...
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1answer
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Extending scalars from $\mathbb{Z}[G]$ to $\mathbb{Q}[G]$

Let $G$ be a finite group. Let $M$ and $N$ be finitely generated $\mathbb{Z}[G]$-modules such that $M$ is free as a $\mathbb{Z}$-module. Suppose that $\mathbb{Q}\otimes_\mathbb{Z}M$ and $\mathbb{Q}\...
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Internal $Hom^{\bullet}$ of filtered complexes

Gelfand and Manin give the definition of an internal $Hom^{\bullet}(A^{\bullet}, B^{\bullet})$ complex for two cochain complexes $A^{\bullet}, B^{\bullet}$: $$Hom^n (A^{\bullet}, B^{\bullet})=\prod_{...
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How to calculate the 1st simplicial homology group of torus

Consider we want to "calculate" the 1st simplicial homology group of torus without using any imagination or intuition related to chains and cycles and only by applying the algebra definitions involved ...
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Understanding the Yoneda product defined in terms of morphisms of projective resolutions.

On the wikipedia page for the Ext functor, they say that one can equip the graded abelian group $\operatorname{Ext}^*:=\bigoplus_{i=0}^{\infty}\operatorname{Ext}^i(A,A)$ with the structure of a ring (...
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Automorphisms and Extensions of Cyclic Groups

I can't figure out what question III.4.2 from Hilton & Stammbach's book A Course in Homological Algebra is asking: Classify the extension classes $[E]$ given by: $$ \mathbb Z_m \...
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Right derived functors of the $I$-torsion functor and $\varinjlim \mathrm{Ext}^i_R(R/I^n,-)$ are naturally isomorphic?

Let $R$ be a commutative ring with unity and let $I$ be a proper ideal. (I'm not assuming $R$ is Noetherian.) For every $M \in R$-Mod, let $\Gamma_I(M):=\{m \in M : I^n m=0$ for some $n\ge 1\}$. If ...
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Isomorphism between injective resolution

When reading homological theory, I confront with a statement as follow. In an exact sequence of $A$-modules: $0\to M\to U_0\to U_1\to \dots\to U_{n-1}\to C\to0$ with all $U_i$ injective, and let $...
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1answer
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Sign convention for total complex

Let $C^{\bullet,\bullet}$ be a double complex with differentials $d,e$ both of degree $+1$.. I use the convention that the squares commute. I am mainly thinking of $C^{i,j}=A^i\otimes B^j$ or $C^{i,j}=...
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Vanishing of $\mathrm{Ext}$ and finite projective dimension

Does it follow from "for all finitely generated $N$, there exists $n_0$ such that for $n\geq n_0, \mathrm{Ext}^n(M,N)=0$" that $M$ has finite projective dimension, for a finitely generated $M$ ? ...
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Modifying long exact sequences

Let $$\dots A_i\stackrel {f_i}\to B_i \stackrel {g_i}\to C_i \stackrel {h_i}\to A_{i+1}\to \dots$$ be a long exact sequence of Abelian groups. Is it true that if there are maps $k_i:D_i\to E_i$ such ...
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When The FP-injective cover of module is surjective

Definition of cover of module is: Let $R$ be a ring and let $\mathcal{S}$ be any class of $R$-modules. Then for any $R$-module $M$, the homomorphism $\beta:S\longrightarrow M$ is called an $\mathcal{...
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On the first and second Ext modules over some valuation rings

Let $(R, \mathfrak m)$ be a valuation ring of finite Krull dimension, with non-principal maximal ideal. So that $\mathfrak m^2=\mathfrak m$. If $M$ is an $R$-module with $Supp M=\{ \mathfrak m \}$ , ...
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if $C$ is a filtered coalgebra, does Gr($B\Omega C)\backsimeq B\Omega ($Gr $C)$ hold?

I have heard that under some assumptions, the functor 'Gr' from filtered graded objects with exhaustive filtration to graded objects $X\rightarrow$ Gr$(X)$ commutes with direct sums (this seems to be ...
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How should I understand “a finitely generated module is too small to be injective”?

I am reading Huybrechts' Fourier-Mukai Transforms in Algebraic Geometry, which including the following sentence: The category of coherent sheaves does not have enough injectives for the simple ...
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The first derived limit of an inverse system of groups

It is well-known that any left exact functor $F : \mathbf{A} \to \mathbf{B}$ between abelian categories yields a sequence of derived functors $F^n : \mathbf{A} \to \mathbf{B}$ such that for any short ...
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“If $g$ is semisimple, It is not too hard to see that $H^2(g,a)=0$. With a little supplementary argument…”

This is a statement made in Knapp, Lie groups, Lie algebras, Cohomology Chpt 4 last paragraph of Sec 2. $H^i(g,a)$ is the $i-$th cohomology group of complex $Hom(\wedge^i g,a)$ with $a$ abelian lie ...
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Abelian group structure of the additive category

Let $A$ be an additive category. We have that for all $X, Y$ objects in $A, Hom_A(X, Y)$ is an abelian group. However, what is the abelian group structure? If $f, g: X \rightarrow Y$ are two morphisms ...
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Chain complexes, homology and homotopy type.

Are they examples of easy chain complexes... that have the same homotopy type but are not isomorphic? That have the same homology groups but haven't got the same homotopy type?
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A consequence of Schanuel's lemma

In Carlson's Cohomology and representation theory, the author states Schanuel's lemma, and then derives a consequence that I cannot understand. They define, for a $kG$ module $M$, $\Omega (M)$ to be ...
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How to construct birelative Hochschild homology?

I am working through Lodays book "Cyclic Homology". For an unital $k$-Algebra $A $ ($k$ being some ring) and a two-sided ideal $I$ of $A$ he defines the relative Hochschild homology $HH_\ast(A,I)$ as ...
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Why is $Tor_0^R(M,N)=0$? [duplicate]

Suppose we have the following free resolution of a module $N$ $$0\dots\to F_2\to F_1\to F_0\to N\to 0 $$ By definition, this free resolution is exact. Now we tensor this by a module $M$. We get $$\...
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On the natural isomorphism between $I$-torsion functor and direct limit of $\mathrm{Hom}$ functor

Let $R$ be a commutative ring with unity with and let $I$ be a proper ideal. (I'm not assuming $R$ is Noetherian.) For every $M \in R$-Mod, let $\Gamma_I(M):=\{m \in M : I^n m=0$ for some $n\ge 1\}$....
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1answer
57 views

Direct limit of directed system of modules commutes with right derived functors of additive, covariant, left exact functor?

Let $R$ be a commutative ring with unity. Let $T: R$-Mod $\to R$-Mod be an additive, covariant, left exact functor which commutes with direct limits indexed by directed sets. Let $R^i T$ be the right ...
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Can someone please explain the map $S:C_n(X) \to C_n(X)$ from Hatcher's AT page 122?

Barycentric Subdivision of General Chains. Define $S:C_n(X) \to C_n(X)$ by setting $S\sigma = \sigma_\# S\Delta^n$ for a singular $n$-simplex $\sigma: \Delta^n \to X$. How are we using $S$ in the ...
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group cohomology with coefficients in a complex

I am reading Brown's "cohomology of groups" when he introduces the group homology and cohomology with coefficients in a chain complex $C_*$. (pp 168) . The homology is defined as $H_*(G, C_*) = H(...
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Creating my own “Category Diagram Database” with query language.

Neo4j and other graph database software out there for one don't support subgraph isomorphism search out-of-the-box which is what I need and I'd also like full expressivity of a CFG on label matchings ...
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Left derived functors vanish on a projective.

In Weibel's, it says that: If $F: \mathcal{A} \to \mathcal{B}$ is a right exact functor of categories, and if $A$ is projective in $\mathcal{A}$, then $L^iF(A)=0$ for all $i\ne0$. I guess we need to ...
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About semi-orthogonal decomposition of triangulated categories

I was reading Huybrechts' Fourier-Mukai transform in algebraic geometry and trying to solve Exercise 1.63: Suppose $\mathcal{D_1},\mathcal{D_2}\subset \mathcal{D}$ is a semi-orthogonal ...
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On derived functors of certain $\operatorname{Tor}_1$ and $\operatorname{Ext}^1$

Let $R$ be a commutative ring with unity. (1) Let $M$ be an $R$-module having a flat resolution of length $1$ . Then for every $R$-module $N$, we have $\operatorname{Tor}_i^R(M,N)=0, \forall i \ge 2$...
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Snake lemma without elements – exactness

$\newcommand{\coker}{\operatorname{coker}}$ $\newcommand{\im}{\operatorname{im}}$ Consider the setup of the snake lemma with objects and morphisms as follows: As mentioned in this answer, the ...
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1answer
51 views

Regarding condition $AB5$

My question is regarding the condition $AB5$ for an abelian category $\mathcal{A}$ i.e. direct sums exists and filtered colimits are exact. Now taking colimit is right exact in an abelian category ...
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1answer
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On $\mathrm{Ext}_R^n(V,M)$ and $\mathrm{Tor}^R_n (V,M)$, where $M$ is an $R$-module with non-zero annihilator and $V$ is a $Q(R)$-vector space

Let $R$ be an integral domain with fraction field $Q(R)$. Let $M$ be an $R$-module such that $\mathrm{Ann}_R (M)\ne \{0\}$. If $V$ is a $Q(R)$-vector space (hence also an $R$-module), then how to show ...
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Describing $\mathrm{Ext}^1_R (R/J, R/J )$

Let $J$ be an ideal of a commutative ring with unity $R$. Is it true that $\mathrm{Ext}^1_R (R/J, R/J ) \cong \mathrm{Hom}_R(J/J^2, R/J)$ ? Since $\mathrm{Tor}_1^R (R/J, R/J) \cong J/J^2$, ...
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Clarifying a step in an answer about exterior products of coherent sheaves

The question is about the accepted answer here. I decided not to ask in a comment there since the original asker is no longer active. In "Step 2" of the answer given there, Roland claims, "This is ...
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Finding a mistake using Mayer-Vietoris

I was computing the homology of $S^3-\coprod_{i=1}^4 I_i$, where $I_i=[0,1]$ for all $i$ (they are being identified with an embedding). Intuitively, this should be homotopy equivalent to $S^1$, since ...
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1answer
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Bimodule structure on $\mathrm{Ext}_R^n(M,N)$

Throughout the question, $\mathbb{k}$ is a commutative ring, and $R$, $S$, and $T$ are $\mathbb{k}$-algebras. Let $M$ be a left $R$-module. The functor $\mathrm{Hom}_R(M,-):R\text{-}\mathrm{Mod}\to \...
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1answer
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The twisted tensor product $BA\otimes_{\tau} A$ as the non-unital Hochschild complex

The twisting universal morphism $\tau: BA\rightarrow A$ induces a differential $\partial_{\tau}$ on $BA\otimes_{\tau}A$, we have: $$\partial_{\tau}(x\otimes y)=\partial x\otimes y+(-1)^{\lvert x\rvert}...
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$(A[1])^{\otimes n}\backsimeq (A^{\otimes n})[n]$?

When $A$ is a $\mathbb{Z}$-graded module, $A[1]$ is the shift or suspension of $A$ (i.e $(A[1])^{i}=A^{i+1}$). May the $n$th power tensor of the shift be identified in this way?. Am I missing anything?...
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Generators for Ext groups/ring

We know that in an abelian category $\mathcal C$ the sets $\newcommand{\Ext}{\operatorname{Ext}}\Ext^n(N, M)$ of $n$-extensions by $N$ form an abelian group, and by the Yoneda- or cup-product, it even ...
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1answer
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FP-injective , injective and pure R-module

prove that : FP-injective R-modules, which are pure-injective, are injective ? I know that an R-module M is called FP-injective if Ext1(N,M) = 0 for all finitely presented R-modules N and R-submodule ...
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Exercise in Homological Algebra

I'm totally stuck with this problem that I found in an Algebra course. It is the following: Let $F:\mathcal{A} \to \mathcal{B}$ be a left exact functor between two abelian cathegories. Let $\...
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An exact sequence of abelian groups

Consider an exact sequence of abelian groups $$ 0 \to A \to \mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \to B \to 0, $$ where we make no assumption on the map $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{...