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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Short exact sequence of abelian groups how to cancel $\mathbb Z$ in the middle term.

Now I want to calculate homology group of some topological space. Using Mayer-Vietoris sequence I end up with the following short exact sequence: $k$ :a knot $$0\to H_0(Torus)\to H_0(\mathbb R^3\...
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A statement for flat modules analogous to the Baer's Criterion [duplicate]

Recently I came across the following statement about the criterion of flat modules which looks somewhat like Baer's Criterion for injective modules: Statement. Let $R$ be a commutative ring. An $R$-...
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Quotient of a chain complex by two quasi-isomorphic chain subcomplexes

If you take the two quotients of a chain complex by two quasi-isomorphic chain subcomplexes, are the results quasi-isomorphic as well? I think it can be proved by making use of long exact sequences ...
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Comparing two obstructions of splitness in $\textrm{Ext}^1$ [duplicate]

Suppose $$\xi: 0\to B\to X\to A\to 0$$ is a short exact sequence of R-modules, where $R$ is a commutative ring. Applying the Ext functors $\textrm{Ext}^*(A,-)$ to $\xi$, we get an exact sequence $$ \...
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Proving that if $M$ is a finitely generated and projective $R$ module, then $M$ is also finitely presented.

Since $M$ is finitely generated $R$ module, we have an epimorphism $\pi : F \to M$, such that $F= R^{(n)}$ where $n$ is a positive integer. On the other hand, since $M$ is a projective $R$ module we ...
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a zero morphism on t-structures

Good morning to everyone, I am writing here because I need to understand better some topics about t-structures on triangulated categories. Consider this statement: take a, b in $\mathbb{Z}$, $(\...
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Is $B^{\infty}E_r^{p,q}$ always a subobject of $Z^{\infty}E_r^{p,q}$?

I'm reading about spectral sequences from various places (e.g. "The heart of cohomology" by Goro Kato, the Stack Project, Wikipedia), and I have a doubt. We consider a bigraded cohomological ...
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Using cluster categories to show that seed of acyclic cluster algebra is determined by the cluster

In corollary 2 of the article "From triangulated categories to cluster algebras II" by Caldero and Keller, it is stated that a seed $(u,Q)$ (where $u$ is a cluster and $Q$ a quiver) of an ...
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How to compute $\ker(p_1\otimes p_2)$ using only the universal property?

Suppose $M$ and $N$ are $R$-modules (in which $R$ is a commutative ring with unity) and $A\subset M$ and $B\subset N$ are submodules. Then, there are natural projections $p_1:M\rightarrow M/A$ and $...
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Projective resolution of $\mathbb{R}$ as an abelian group

I had a sense that this should be easy to find but googling many different versions of the question I couldn't find anything. My question is how does some projective resolution of the additive group ...
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The map $H_0(S^n) \to H_0(E_-^n)$ is surjective

Let $E_-^n := \{(x_1,\ldots,x_{n+1}) \in S^n \mid x_{n+1} \le 0\}$ be defining the lower hemisphere and $x \in E_-^n$. I try to understand why we have that the map of homology group $H_0(S^n) \to H_0(...
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Show that $H_q(S^n,x) \xrightarrow{\cong} H_q(S^n,E_n^+) \;\;\forall q$

I am new into the topic of algebraic topology and try to understand the argumentation of the following statement in the chapter of homology of CW-complexes. The goal is to show that: $$H_q(S^n,x) \...
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Is there some practical intuition when working with a cooperad given by cogenerators and corelations?

In the case of algebras and operads, a description by generators and relations is common practice and I have a good understanding of this. A non-symmetric operad $\mathcal{P}$ given by a linear space ...
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Why is $\operatorname{Tor}(\mathbb{Z}_m,\mathbb{Z}_n) \cong \mathbb{Z}_{\operatorname{gcd}(m,n)}$? [duplicate]

The definition of $\operatorname{Tor}$ I am using is: Let $K \to F\to A\to 0$ be a free resolution of $A$ and $B$ an abelian group, then $\operatorname{Tor}(A,B) := \ker (f \otimes 1_B)$ if $f$ is the ...
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Definition of an induced homotopy in Hilton-Stammbach's book A Course in homological algebra

Studying Hilton-Stammbach's (2nd Ed), on page 170, it starts from a pair of chain maps, $\varphi, \varphi^{'}: C \rightarrow C' $, for which there is a homotopy, $\Gamma$. Next, a map $\Gamma_{\sharp}:...
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How can i show intersection of injective modules?

Let $E=\mathbb{Z}_4\oplus\mathbb{Z}_4$, regarded as a module over $\mathbb{Z}_4$. Let $M= \{(0,0),(2,2)\}\subseteq E$ be a submodule. Then how can I show that $M$ is the intersection of $2$ injective ...
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Conditions for null-homotopic chain map in a specific example

I am trying to calculate the ring $Ext^{\bullet}_R(k,k)$ where $R=k[x,y]/(xy)$ and $k$ is regarded as an $R$-module via $x$ and $y$ acting as zero. I thought I was done but then to my demise I found a ...
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Rotman's Algebraic Topology Lemma 9.11

This is the Lemma 9.11 of Rotman's "An Introduction to Algebraic Topology". The topic where I found this simple lemma of homological algebra is the Theorem of Acyclic Models. So we are ...
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Exact sequence $0\rightarrow H^i(A_{\bullet})\rightarrow\text{coker}f^{i-1}\rightarrow\text{im}f^i\rightarrow0$ - abstract nonsense proof?

I'm a little stuck on an exercise about cohomology in an abelian category. Given a complex $A^{\bullet}$, where $f^i:A^i\rightarrow A^{i+1}$ (so $f^{i+1}f^i=0$), let $H^i(A^{\bullet}):=\text{coker}(\...
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Quotients preserve monomorphisms in abelian categories [duplicate]

We work in an abelian category. I use $f_k$ and $f^k$ to denote the morphisms given by the kernel and cokernel of the map $f$. Here's a diagram Suppose we have monos $f:A\rightarrow B$, $g:A\...
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To understand whether a module is flat, injective or projective through a explicit example

Suppose a ring $R = \mathbb{C}[x, y]$, An ideal $I = (y^2 − x^3 + x^7)$ and $M = (R/I )_I \oplus (R/I )$. We can easily check that $R/I$ is an integral domain thus $I$ is a Prime Ideal. Hence ...
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Finding $\text{Ext}_R^n(M,R)$ and $\text{Ext}_R^n(M,N)$ for a particular case

Suppose we have the ring $R=\mathbb{Z}[x_1,x_2]$ and ideals $I=(2x_2)$, $J=(x_1x_2)$. Now, consider $M:=R/I$ and $N:=R/J$. I am trying to determine the following: (1) $\text{Ext}_R^n(M,R)$ for all $n\...
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What chain complex is denoted by a single $R$?

I was reading about Koszul Complex and I am fairly new to the topic. So lemme start with a ring $R$, not necesarilly commutative and let $x\in R$ be central. Denote $K(x)$ to be the chain complex $0\...
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Reference Request for Orbits of Group Representation

Recently I have been considering a problem that has had me venture into some mathematical territory that is new to me, specifically group representations, words, and character theory. I would ...
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48 views

Sheaf homology defined in terms of Tor

By the general philosophy of cohomology, cohomology is essentially derived $\operatorname{Hom}$ (i.e. $\operatorname{Ext}$), and homology should be derived tensor product (i.e. $\operatorname{Tor}$). ...
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if every finitely generated $R$-module has Finite Free Resolution, then every finitely generated $R[x]$-module also has it

I am reading the Rotman's book "An Introduction to Homological Algebra" and I do not understand the part of the proof of theorem 8.47 in page 481. Theorem 8.47 Let R be a commutative ...
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Singular homology with coefficients in a ring versus in an abelian group

As described here and here the singular homology of a topological space $X$ with coefficients in a ring $R$ is given by a bunch of $R$-modules $H_n(X,R)$. However, sometimes I see people talking about ...
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Equivalence of homology theories via acyclic models

I'm looking for a proof of "any two homology theories are equivalent" (obviously with some other hypothesis) via the Acyclic Models Theorem. I know that this is an application of the Acyclic ...
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Direct product totalization in the definition of hypercohomology

Let $X$ be a topological space, and $\mathcal{F}^\bullet$ be a cochain complex of sheaves on $X$. The hypercohomology $\mathbb{H}^i(X,\,\mathcal{F}^\bullet)$ is defined as $$\mathbb{H}^i(X,\,\mathcal{...
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Coefficients in homology

The singular homology of a space $X$ is defined to be the homology of the chain complex $${\displaystyle \ldots {\stackrel {}{\longrightarrow }}\mathbb Z[Sing_2(X)]{\stackrel {}{\longrightarrow }}\...
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Direct limit of the sequence $E_{0} \hookrightarrow E_{1} \hookrightarrow \cdots$ in the category of Banach spaces

Recently I have been reading the paper The categorical origins of Lebesgue integration by Tom Leinster (https://arxiv.org/pdf/2011.00412.pdf). In this paper, he said that: For $n \geq 0$, let $E_{n}$ ...
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Show that the homology groups of different projective resolutions of the same R-module are isomorphic to one another

I am taking a course on commutative algebra, and we just defined the Tor functor using projective resolutions of a module. The definition we have is: Let $R$ be a commutative ring with unity. Let $M$ ...
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Products, fibred products and kernels as projective limits

In SGA 4-1, it says that products, fibred products and kernels are projective limits of functors. I have no idea to understand this. Note that kernels mean equalizers. Fix a universe $\mathscr{U}$. ...
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Confusion in understanding new initial terms of Tor sequence

Cartan and Eilenberg mention in their book on Homological Algebra the following: Consider exact sequence of right $\Lambda$-modules $$(1)\hskip1cm 0\rightarrow A'\rightarrow A\rightarrow A''\...
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Equivalent definition of Abelian Category, exercise.

So I came across this post earlier today. I tried to understand it but I am stuck at a seemingly easy point. Apologies in advance as I am really new at this type of stuff! My question is that the OP ...
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$B$ a chain complex of free abelian groups and $C$ a chain complex such that $H_n(C)=0$. Then any chain map from $B$ to $C$ homotopic to the zero map

Let $B$ be a chain complex of free abelian groups and let $C$ be a chain complex such that $H_n(C)=0$. Then any chain map from $B$ to $C$ is homotopic to the zero map. So, $B$ is split and $C$ is ...
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Being direct summand of free module implies having dual basis.

I need to provet equality of two definitions of projective module: being direct summand of free module (or equally: having embedding into free module) and having dual basis. Lets use Wikipedia ...
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Isomorphic cokernels giving isomorphic kernels

$\require{AMScd}$ This is an issue with a proof related to a characterization of flatness by Eisenbud (in his "Commutative Algebra with a View Toward Algebraic Geometry", p. 162). If we have ...
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spaces with the same cohomology groups but not homology groups [duplicate]

Suppose we are given spaces $X$ and $Y$. If they have the same homology groups (i.e. $H_i(X,\mathbb{Z})=H_i(Y,\mathbb{Z})$), then by universal coefficient theorem $$0\to\mathrm{Ext}^1(H_{n-1}(C_\...
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Hyper-derived functors and Cartan-Eilenberg resolutions

I'm confused by the significance of Cartan-Eilenberg resolutions when constructing hyper-derived functors. Let $F$ be a right-exact functor, and let $A^\bullet$ be a chain complex. According to this ...
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Equivalence of cross products Kunneth formula

In Hatcher Theorem 3.19, he writes that the following cup product for $i+j=n$: $$H^i(\mathbb{R}^n,\mathbb{R}^n-\mathbb{R}^j)\times H^j(\mathbb{R}^n,\mathbb{R}^n-\mathbb{R}^i)\xrightarrow{\cup} H^n(\...
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Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$.

Let $R$ be an integral domain and let $X$ be a torsion module over $R$. Then $Tor_n(X,Y)$ is a torsion module for every $n≥0$. I attempted proving this fact using the definition of what it means to ...
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Translating a diagram chase into an element-free proof

One part of the four lemma says that: Consider the following diagram with exact rows in an abelian category $\mathsf{A}$: If $m$ and $p$ are monomorphisms and $l$ is an epimorphism, then $n$ is a ...
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Three ways to to prove that projective modules are flat

I am trying to show that projective modules are flat using their defining property that $Hom(P,-)$ is an exact functor when $P$ is projective. The two ways I know of come down to the fact that ...
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Compute the Derived functor $RF^i(A)$ for $i\ge 1$

Let $\mathscr{A}$ be a category with enough injective,$F$ be the left exact functor ,we can compute the right derived functor as follows: First since enough injective,we can construct a injective ...
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Localization_of_triangulated_category

I am writing here because I need to understand better some topics about homological algebra. I am reading these notes http://www.anmath.ulg.ac.be/fp/rec/dea.pdf about homological algebra in quasi-...
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57 views

$R^iF(M)=0$ for $i>\dim A$ where $F$ is left exact functor?

Let $A$ be a commutative ring s.t. the Krull dimension $\dim A<\infty$, $M$ be a $A$-module, and $F$ be left (resp. right) exact functor. Now for $i>\dim A$, the right derived functor $R^iF(M)=0$...
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Higher order homology operations for diagrams of chain complexes (reference request)

Let $R$ be a small category enriched in $\mathcal{A}b$, the category of abelian groups (my adviser enjoys calling these "many-object rings"). Given a diagram $M:R\to\mathcal{A}b$, taking ...
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Chain complexes indexed over the reals

One way to think of chain complexes is as a certain subcategory of a functor category: each chain complex is a functor from $\mathbb{Z}$ regarded as a poset category to, say, abelian groups, where ...
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35 views

When was Grothendieck attending the Seminaire Cartan?

If I recall correctly, Grothendieck was at the Seminaire Cartan prior to writing his Tohoku article, and the homological algebra there influenced his work. (I also recall a quote about Grothendieck ...

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