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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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Inverting quasi-equivalences between DG categories

I am recently trying to learn the language of DG categories and I have a question concerning the notion of quasi-equivalence. According to the definition, which you can find for instance on Keller's ...
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Relationship between $HH$ and $THH$

In general, for an $R$-algebra $A$, what can we say about the relationship between $HH_{\ast}^{R}(A)$ and $THH_{\ast}^{HR}(HA)$? Is there some spectral sequence with $E_{2}$-page involving $HH_{\ast}^{...
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Successive maps in exact sequence leads to $0$. Celluar Homology

This comes from Hatcher's Algebraic Topology book on page 139 He says that the map in the diagram $$0 \to H_n(X^n) \stackrel{j_n}\to H_n(X^n, X^{n-1}) \stackrel{\partial_n}\to H_{n-1}(X^{n-1})\to?$$ ...
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Computing projective resolutions over quotients of polynomial rings

I'd like to find projective resolutions for $k$ considered as an $R$-module, where $k$ is a field and (i) $R=\frac{k[x]}{(x^n)}$; (ii) $R=k[x,y]$ (iii) $R=\frac{k[x,y]}{(x^n,y^m)}$; (iv) $ R=\frac{...
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Generalized Schanuel Lemma

This is on page 128, ex 3.15, of Rotman's AIHA, (Schanuel) Let $B$ be a left $R$-module over some ring $R$ consider two exact sequences, $$ 0 \rightarrow K \rightarrow P_n \rightarrow \...
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1answer
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Embedding $\mathbb{Z}$-modules into injective $\mathbb{Z}$-modules

I want to show that any $\mathbb{Z}$-module $M$ can be embedded into an injective $\mathbb{Z}$-module (I'm in the process of showing this can be done for more general rings, but starting with this ...
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Definition of an exact sequence

I am a beginner at homology, and I am trying to learn it from this text: http://www.seas.upenn.edu/~jean/sheaves-cohomology.pdf I see the following definitions for exact sequences and short exact ...
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Dual of quasi-isomorphism of chain complexes.

Let $C_*$, $D_*$ be chain complexes of modules over a ring $R$. Suppose that $f\colon C_* \rightarrow D_*$ is a quasi-isomorphism (i.e. an isomorphism in Homology). I am wondering what conditions ...
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Computing $\mathrm{Hom}_{\mathbb Z}(\mathbb Z_n,G)$ by left-exactness

In chapter 2 of Rotman's An introduction to homological algebra, we are set the exercise to show, $$\mathrm{Hom}_{\mathbb Z}(\mathbb Z_n, G) = \{g\in G : ng = 0\}$$ where $G$ is an abelian group, ...
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Computing cohomology of dihedral group in detail

So I tried to compute the cohomology of $D_{2n}$, for n odd , $H^{k}(D_{2n}, \Bbb Z)$. using Lyndon SS. I have obtained a few obstacles: My computation, using the fact that there is a $C_2$ action ...
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“Restriction” map in group homology, what was meant? Rotman

Def 1: $\alpha:G \rightarrow G'$, group homomorphism. If $A'$ is a $G'$ module $f:A \rightarrow A'$ is a $\Bbb Z$ map we call $(\alpha, f)$ a compatible pair if $f:A \rightarrow _{\alpha}A'$ is a $G$-...
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homotopy equivalence induces homotopy equivalence of Hom complexes?

Let $f\colon A\to B$ be a homotopy equivalence of chain complexes (with $g$ the weak inverse and chain homotopies $\phi\colon {\rm id}_D\to f\circ g$ and $\psi\colon g\circ f\to{\rm id}_C$). Then ...
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Any chain map of Moore complex null homotopic? Can't find mistake

Suppose that $(A_n)_{n\ge 0}$ and $(B_n)_{n\ge 0}$ are cosimplicial abelian groups. Let $f, g: A\to B$ be two maps of cosimplicial abelian groups. Let $s(A)$ and $s(B)$ denote the Moore (co)chain ...
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Induced map on cohomology being zero impies null-homotopic?

Let $f:A^\bullet\to B^\bullet$ be a morphism of chain complexes (of any give abelian category). We know that if $f$ is homotopic to the zero map, then $f$ will induce zero map on cohomology. I want to ...
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Injective module and transfinite induction

There is a proposition that: Each $R$-module $A$ can be embedded into an exact sequence $$0\to A\to Q\to N\to 0$$ where $Q$ is injective. The proof eventually requires transfinite induction. The idea ...
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Injective resolution of $\mathbb Z / p$ but resulting objects not injective?

I'm trying to find an injective resolution for the $\mathbb Z$-module $\mathbb Z/p$ where $p$ is a prime. I've come across the result that an injective resoltion for a PID $R$ is of the form $$ 0 \...
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1answer
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Dual of a differential graded module

Let $(M, d_M)$ be a differential $\mathbb{Z}$-graded module over a differential graded algebra (over a field) $(A, d_A)$. I am wondering if there is a canonical way of looking at the dual $\hom_A(M,A)...
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Proving that if $A\times A \cong M\times M$ for some A-module M, then $A\times A$ = $N\times M$ for another A-module N

I need to show that if M is an A-module and $A\times A$ is isomorphic to $M\times M$ as A-modules then M is a direct summand of $A\times A$. In other words that $A\times A$ = $N\times M$ for some A-...
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Locally free $\mathcal{O}_{X}$ modules are not projective

Let $(X , \mathcal{O}_{X})$ be a locally ringed space. I know that in general it is not true that locally free $\mathcal{O}_{X}$ modules are projective in the category of $\mathcal{O}_{X}$ modules (...
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1answer
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chain complex of endomorphisms

I am looking for the definition of $End(C_*)$, where $(C_*,d)$ is a chain complex of modules. In degree $n$, $End(C_*)$ is the set of endomorphisms (not chain complexes) $C_* \to C_{*+n}$. It seems ...
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A question on Hartshorne Chapter III Proposition 2.6

When I read Hartshorne, I saw the Proposition 2.6 in Chapter III as follows: Let $(X,\mathcal{O}_X)$ be a ringed space. Then the derived functors of the functor $\Gamma(X,-)$ from the category of $\...
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I am stuck showing a morphism is a monomorphism in a proof about torsion classes and quasitilting modules.

Some notation: For an $A$-module $T$, denote by $\operatorname{Gen}T$ the full subcategory of $\operatorname{Mod}A$ consisting of epimorphic images of modules in $\operatorname{Add}T$. The following ...
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length of minimal projective resolution and projective dimension

Let $K$ be a field, $A$ be a $K$-algebra, and $M$ be a finite generated $A$-module. How do you show that the length of a minimal projective resolution of $M$ is the projective dimension of $M$? (This ...
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Wedge sum of spheres is the quotient $X^n/X^{n-1}$

As in the title, I want to prove that $\bigvee_jS_j^n=X^n/X^{n-1};\ X$ is a $CW$ complex and $X^n$ and $X^{n-1}$ are the $n-$ and $n-1$-skeleta. Below, I present a sketch of an attempt using pushouts, ...
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1answer
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Is this sequence exact?

Let $A$, $B$ and $C$ be abelian groups and let $$0\to A\overset{\varphi}{\longrightarrow} B\overset{\psi}{\longrightarrow} C\to 0$$ be a short exact sequence. Denoting by $T(X)$ the torsion subgroup ...
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1answer
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Definition of “objet en catégories”.

I don't understand the following definition in Illusie, Complexe Cotangent et Déformations II (Springer LNM 283), page 17: Soit $T$ une catégorie possédant des produits fibrés. On appelle objet en ...
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1answer
41 views

$\text{Hom}_S(S\otimes_R M,N)\cong\text{Hom}_R(M,N)$

Given homomorphism $\phi:R\to S$ and $M$ is an $R$-module and $N$ is an $S$-module. We know that an $S$-module can be turn into a an $R$-module naturally by $\phi$. So the role of modules may alter in ...
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Wedderburn Artin theorem and module category

I am reading "An Introduction to Homological Algebra" by Rotman. In the chapter about adjoint functor, he says "The Wedderburn–Artin theorems can be better understood in the context of determining ...
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1answer
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Why can a projective resolution of $A$ be used to calculate $Ext_R^n(A,B)$?

I know the definition of $Ext^n_R(A,B)$ as the $n$th right derived functor of $Hom_R(A,-)$ applied to $B$, which should be calculated by taking an injective resolution $I_\bullet$ of $B$ and taking ...
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On derived categories of exact categories

The excellent overview paper Exact Categories - Bühler discusses exact categories and all basic definitions surrounding them. In particular section 10 discusses the derived categories of exact ...
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1answer
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$\sum_{i=0}^{n} (-1)^i l(H_i)=\sum_{i=0}^{n} (-1)^i l(G_i)$ [duplicate]

Let $$0 \overset{d_{n+1}}{\rightarrow} G_n \overset{d_n}{\rightarrow} G_{n-1}\overset{d_{n-1}}{\rightarrow}\ldots\overset{d_2}{\rightarrow}G_1 \overset{d_1}{\rightarrow} 0$$ be a sequence of ...
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question about semisimple ring.

Wedderburn-Artin's theorem: A ring is left semisimple ring iff it is finite product of $M_{n_i}(D_i)$ for some division ring $D_i$. Due to this theorem,we know that if a ring is left semisimple,...
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1answer
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Weibel - Left Derived Functor proof explanation Theorem 2.4.6

The full proof is freely available online page 46, Theorem 4.2.6 where Weibel proves that $L_*F$ is a a $\delta$ -functor. For an exact exact sequence $$0 \rightarrow A' \rightarrow A \rightarrow A'' \...
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Book Recommendation in Homological Algebra.

I am attending a lesson in Homological Algebra this semester, in the following special topics. I know that there are similar posts, but in this post I specifically ask to recommend me a combination of ...
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Constructing an injective resolution for a bounded below cochain complex

Let $\mathcal{A}$ be an abelian category with enough injectives. If $X^{\bullet}$ is a bounded below complex, it is a well known fact that you can obtain a bounded below complex $I^{\bullet}$ of ...
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Another question on ideal and tensor product

Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module. Let $I$ be a proper ideal of $R$ and $a,b \in R$ be such that $ab\in I$ and the maps $(I+Rb) \otimes_R M \to R\otimes_R M$ and $(I+ ...
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On ideal and tensor product , retaining injectivity

Let $R$ be a commutative Noetherian ring and $M$ be an $R$-module. Let $I$ be a proper ideal of $R$ and $b\notin I$ be such that the maps $(I+Rb) \otimes_R M \to R\otimes_R M$ and $(I:b) \otimes_R M \...
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When $P\otimes_R M \to R \otimes_R M$ is injective for every (finitely generated) prime ideals $P$ of $R$?

Let $R$ be a commutative ring. If $M$ is an $R$-module such that for every finitely generated prime ideal $P$ of $R$, the map $i\otimes Id: P\otimes_R M \to R \otimes_R M\cong M$ is injective, then ...
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commutative Noetherian ring whose every maximal ideal is projective

Let $R$ be a commutative Noetherian ring. If every maximal ideal of $R$ is projective as an $R$-module, then is $R$ hereditary?
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The $n$th homology of convex subsets of Euclidean space using homotopic maps

I confronted a problem when I was reading the theorem stating that if $f_0$ and $f_1$ are homotopic maps (not to be confused with chain homotopy notion) from $X$ to $Y$, then they induce the same ...
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1answer
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Is there any Kunneth formula for homology group with cofficient in an abelian group

I am reading Hatcher Chapter V on spectral sequence. This is a paragraph after Theorem 5.3: The Kunneth formula and the universal coefficient theorem then combine to give an isomorphism $$H_n(B\...
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1answer
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Commutative hexagonal diagram of Abelian groups; proving a certain equality

I'm trying to prove the following lemma by diagram chasing, but I've had no success, so I decided to ask for help here. Let $A$, $B$, $C$, $D$, $E$, $F$, and $G$ be Abelian groups, and let $a_{1}$, ...
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Order relation between cohomology groups.

We have $\mathbb{Q}$-graded finite dimensional vector space $V=\bigoplus_{i=0}^{n}V_{i}$ and following cochain complex $$0\rightarrow V_{0}\xrightarrow[]{d_{0}} V_{1}\xrightarrow[]{d_{1}}\ldots\...
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25 views

Nullhomotopy of a chain map

Let $(C, \partial), (C',\partial')$ be two chain complexes, and $f: C \rightarrow C'$ be a chain map. In general we know that if $f$ is zero-homotopic (i.e. $\exists$ a collection of maps $s_k: C_k \...
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2answers
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Definition of graded abelian group

I am reading Homology Theory by Vick, and in this book a graded abelian group $G$ is defined to be a “collection of abelian groups {$G_i$} indexed by the integers with component-wise operation”. What ...
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Stable module category and Syzygy functor

I'm trying to understand the construction of the functor $\Omega:{}_{R}\underline{\mathfrak{M}}\longrightarrow {}_{R}\underline{\mathfrak{M}}$ of first syzygia, but I don't understand how it's define: ...
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On cancelling special type of ideals in Noetherian local domain of dimension $1$

Let $(R,\mathfrak m)$ be a local, Noetherian domain of dimension $1$. If $J$ a non-zero ideal of $R$ such that $J^2=\mathfrak m J$, then is it true that $J^2=\mathfrak m^2$ ? If this is not true in ...
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Direct proof for Cech Cohomology stabilizing for a good open cover

I am searching for a proof that the Cech cohomology with values in locally constant functions, $\check{H}^p(\mathcal{U}, \mathbb{R})$, for a good open cover of the space, $X$ (with whatever reasonably ...
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How is functor with “image” unique up to a unique isomorphism defined exactly?

In an abelian category $\mathscr A$ we encounters the notions of kernel, cokernel, chain homology, derived functors, etc. These notions are frequently referred to as functors, and yes, they actually ...
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On cancelling primary ideals in Noetherian, local, unique factorization domain of dimension $2$

Let $(R,\mathfrak m)$ be a local , Noetherian , UFD of dimension $2$. Let $J$ be an $\mathfrak m$-primary ideal ($\sqrt J=\mathfrak m$) of $R$ such that $J^2=\mathfrak m J$. Then, is it true that $J^...