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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Uniqueness of the connecting morphism in the snake lemma
Consider the following commutative diagram in an abelian category:$$\require{AMScd}
\begin{CD}
@. A @>{f}>> B @>{g}>> C @>>> 0\\
@. @VV{a}V @VV{b}V @VV{c}V \\
0@>>>...
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The induced morphism $B/A\to C/A$ is monic/epi if the morphism $B \to C$ is monic/epi in abelian categories
Let $f: A \to B$ and $g: A \to C$ be two monomorphisms in an abelian category. By definition, $B/A:= \operatorname{coker}(f)$, $C/A:= \operatorname{coker}(g)$.
If $h:B\to C$ is another morphism such ...
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CW complex such that action induces action of group ring on cellular chain complex.
Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given ...
user286640
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Example for a functor which preserves direct sums but doesn't preserve split exact sequnces.
Is there a functor from the category of abelian gorups to itself such that $F0=0$ on objects and morphisms and
$$
F(M\oplus N)\cong F(M)\oplus F(N)
$$
for all abelian groups $M$ and $N$, but such that ...
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The homotopy category of complexes
I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688.
The solution is sketched at pag. 754 at the end of the book. The ...
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Isomorphism involving Eilenberg-Maclane space, Tors.
Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. Does there exist an isomorphism between $H_*(K(\pi, 1); ...
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Show $\mathbb{Q}$ is not a projective $\mathbb{Z}$ module.
I have seen a few similar questions to this one, namely this one: Prove that $\operatorname{Hom}_{\Bbb{Z}}(\Bbb{Q},\Bbb{Z}) = 0$ and show that $\Bbb{Q}$ is not a projective $\Bbb{Z}$-module., but I ...
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How do you break up an exact sequence of any length to a "succession of short exact sequences"?
Note that if $\text{Hom}_R(D,-)$ functor takes short exact sequences to short exact sequences then it takes exact sequences of any length to exact sequences since any exact sequence can be broken up ...
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Tensor products of simple modules over algebras
Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules.
We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and $...
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Show that for Abelian groups $G$ and $H, \bigl[K(G, n), K(H, n)\bigr] \cong \operatorname{Hom}(G, H).$
Show that for Abelian groups $G$ and $H, \bigl[K(G,n), K(H,n))\bigr] \cong \operatorname{Hom}(G, H).$
I was given a hint to use the following:
But I have many stupid questions:
1-First, how I am ...
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Give an example of cochain map which is bijective? [closed]
I am looking for a injective cochain map $\psi: C^*\rightarrow D^*$ such that the map $\Psi^i$ from $C^i$ to $D^i$ is injective, but the map $\psi^*$ from $H^i(C^*)$ to $H^i(D^*)$ is not injective ...
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Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra
I am taking a course next term in homological algebra (using Weibel's classic text) and am having a hard time seeing some of the big picture of the idea behind homological algebra.
Now, this sort of ...
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Abstract nonsense proof of snake lemma
During my studies, I always wanted to see a "purely category-theoretical" proof of the Snake Lemma, i.e. a proof that constructs all morphisms (including the snake) and proves exactness via universal ...
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What are exact sequences, metaphysically speaking?
Why is it natural or useful to organize objects (of some appropriate category) into exact sequences? Exact sequences are ubiquitous - and I've encountered them enough to know that they can provide a ...
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Intuition behind Snake Lemma
I've been struggling with this for some time. I can prove the Snake Lemma, but I don't really “understand” it. By that I mean if no one told me Snake Lemma existed, I would not even ...
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Applying Freyd-Mitchell's embedding theorem on large categories
One commonly reads that the Freyd-Mitchell's embedding theorem allows proof by diagram chasing in any abelian category.
This is not immediately clear, since only small abelian categories can be ...
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Distinguished triangle induced by short exact sequence
I'm reading about derived categories of abelian categories from Huybrecht's book Fourier-Mukai transforms in algebraic geometry. I'm having a lot of trouble with one of the exercises. In fact, at this ...
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Rank-nullity theorem for free $\mathbb Z$-modules
From linear algebra we know that given vector spaces $V$, $W$ over a field $k$ and a linear map $f\colon V\to W$ we have
$$\dim V = \dim \operatorname{im} f + \dim \ker f.$$
Is this still true when ...
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Induced short exact sequence on wedge product
Let$$0\to E \to F \to L\to0$$be a exact sequence on coherent sheaves and $L$ be a line bundle, then it induces a short exact sequence on wedge product $$0\to \Lambda^p E \to \Lambda^p F \to \Lambda^{p-...
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"The Yoneda embedding reflects exactness" is a direct consequence of Yoneda?
Let $A,B,C$ be objects of a category of modules over a ring. It is not hard to see that the Yoneda embedding "reflects exactness" (as Weibel puts it, on p. 28), i.e. if
$\hom(X,A)\stackrel{f_*}{\to}\...
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Tangent space in a point and First Ext group
Let $X$ be an abelian variety over an algebraically closed field $k$.
I have read that one has for an arbitrary closed point $x$ on $X$ a canonical identification
$$T_x(X)\simeq \operatorname{Ext}^1(...
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On equivalent definitions of Ext
Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way:
Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$
Where $X[0]$ is the complex with all zeros except in degree 0 ...
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Can we use the internal logic of a category to do diagram chases "as in $\mathbf{Ab}$" ?
Disclaimer: I'm not yet really comfortable with internal logics, though I know the basics, so the best answer would not be the most technical one.
Suppose you want to prove a diagram lemma (say the ...
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Signs in the tensor product and internal hom of chain complexes
Let $R$ be a commutative ring and $\text{Ch}(R)$ the category of chain complexes of $R$-modules. $\text{Ch}(R)$ is first of all an abelian category, but it can also be equipped with the structure of a ...
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Minimal free resolution
I'm studying on the book "Cohen-Macaulay rings" of Bruns-Herzog
(Here's a link and an image of the page in question for those unable to use Google Books.)
At page 17 it talks about minimal free ...
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Arbitrary products of quasi-coherent sheaves?
I have a short question:
Does the category of quasi-coherent sheaves on a scheme have arbitrary products? I know that it does if the scheme is affine and I know that they will not be isomorphic to ...
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How *should* we have known to invent homological algebra?
Previously I asked How did we know to invent homological algebra?, because I was under the misapprehension that concrete examples of long exact sequences had been a major motivation for developing ...
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Is the image of a tensor product equal to the tensor product of the images?
Let $S$ be a commutative ring with unity, and let $A,B,A',B'$ be $S$-modules. If $\phi:A\rightarrow A'$ and $\psi:B\rightarrow B'$ are $S$-module homomorphisms, is it true that
$$\operatorname{im}(\...
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Description of $\mathrm{Ext}^1(R/I,R/J)$
Let $R$ be a commutative ring with unit and $I$ and $J$ are nonzero ideals of $R$. Do we have a nice description for $\mathrm{Ext}^1_R(R/I,R/J)$?
What do I mean by a nice description? For example $$\...
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Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$?
Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$? Or maybe $\prod_p\mathbb{Q}_p$?
I know $\mathbb{Q}/\mathbb{Z}\cong\bigoplus_p \mathbb{Z}_{p^\infty}$, ...
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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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Singular $\simeq$ Cellular homology?
Given an arbitrary CW-complex, are the singular chain complex $S_\ast(X)$ and cellular chain complex $C_\ast(X)$ homotopy equivalent or just quasi-isomorphic (some chain map induces isomorphisms on ...
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Does the splitting lemma hold without the axiom of choice?
In part of the proof of the splitting lemma (a left-split short exact sequence of abelian groups is right-split) it seems necessary to invoke the axiom of choice. That is, if $0\to A\overset{f}{\to} B\...
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Different definitions of projective objects
There are various characterizations for an $R$-module to be projective. Two of them can be generalized to any category:
i) $P$ is an object such that given morphisms $\alpha: A \rightarrow B$ and $\...
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Condition for a ring on projective and free modules problem
Let $R$ be a ring. Then we know that a free module over $R$ is projective. Moreover, if $R$ is a principal ideal domain then a module over $R$ is free if and only if it is projective or if $R$ is ...
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Is it true that Tensor product of injective modules is injective?
Is it true that if $M$, $N$ are injective modules over a commutative ring $R$ (with identity) then $M\otimes_R N$ is also injective ?
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Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra
How to prove that a chain complex is a projective object in $ {Ch} $ (chain complexes of $R$-modules) iff it is a split exact complex of projectives?
A chain complex of projectives means a chain ...
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Injective resolutions of a complex
Let $\mathcal{A}$ be an abelian category, $M\in\mathcal{A}$. An injective resolution of $M$ is a quasi-isomorphism $M\longrightarrow I$, where $I$ is a complex of injective objects. This can be made ...
user114885
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Computing the homology of genus $g$ surface, using Mayer-Vietoris
Let $\Sigma_g$ be the compact orientable surface of genus $g$. I'm trying to compute the homology groups of $\Sigma_g$ using Mayer-Vietoris sequence. To do this, I first calculated the homology groups ...
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Relating the Künneth Formula to the Leray-Hirsch Theorem
I am reading through Bott & Tu's Differential Forms in Algebraic Topology, which very early on discusses the Künneth formula and the Leray-Hirsch theorem for smooth principal bundles. The proof of ...
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Correspondence between Ext group and extensions (from Weibel's book)
I am trying to understand the proof of Theorem 3.4.3 from Weibel's book Introduction to homological algebra.
The statement is the following.
Let $R$ be a ring. Given $R$-modules $A$ and $B$, an ...
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Can we see directly from the cocycle condition that 2-cocycles are symmetric?
Let $A$ be an abelian group and let $C$ be a cyclic group. All central extensions of $C$ by $A$ are abelian because in any such extension
$$ 1\rightarrow A\rightarrow E\rightarrow C\rightarrow 1$$
...
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Computing $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},\mathbb{Z})$
I'm trying to find an abelian group $B$ such that $\operatorname{Ext}^{1}_{\mathbb{Z}}(\mathbb{Q},B)$ is non-zero. My first guess was just to choose $B=\mathbb{Z}$. Using the following argument, I ...
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More on the homology functor
Following from this question, I am having trouble constructing the Homology functor on chain maps. Let me first sum up what I managed to do.
Let $\textbf{A}$ be an abelian category, and let $C_\bullet,...
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Additive functor over a short split exact sequence.
$0\to A'\stackrel{f}{\longrightarrow} A \stackrel{g}{\longrightarrow} A''\to 0$ is a short split exact sequence, where $A'$, $A$, $A''$ are $R$-modules, and $T$ is an additive functor from $R$-$\...
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Extension group and Baer sum
Suppose $\mathcal{A}$ is an abelian category and $A$ and $B$ are two objects of $\mathcal{A}$, the extensions of $A$ by $B$ consist of isomorphism classes of short exact sequences of the form
\begin{...
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Direct sum of exact sequences?
In Hatcher's Algebraic Topology textbook, he has been referring to "direct sum of exact sequences". As far as I know he's never defined this and I can't find what I'm looking for online.
Without a ...
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Long exact sequence in homology: naturality=functoriality?
In every book I've looked, the "naturality" of the long exact sequence in homology simply says that every arrow between short exact sequences translates into an arrow between the long exact sequences ...
user153312
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Elementary problems that could be solved using homological algebra
I'm in the rather unusual position that I know a bit of homological algebra, like how to compute $\mathsf{Tor}, \mathsf{Ext}$ or local comohology though I barely know more than the basics about groups,...
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When does tensor product have a (exact) left adjoint?
Let $A$ be a commutative Noetherian ring, and let $F$ be a flat $A$-module. We can assume $A$ is local, so $F$ is projective.
Question 1. When does $F\otimes_A-$ preserve injective objects?
...
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