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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its related to mathematical logics
Which of the following are true statements?
(1) (∃x ∈ N)(∃y ∈ N)(x + y ̸ = x · y).
(2) (∃x ∈ N)(∃y ∈ N)(∃z ∈ N)(x − (y − z) ̸ = (x − y) − z).
(3) (∀x ∈ {0, 1})(∀y ∈ {0, 1})(x2 + y2 + xy = (x + y)2).
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Show that if $\operatorname{Hom}_{R}(-,D)$ is exact then the original sequence $0 \to L \to M \to N \to 0$ is *split* exact.
I want to
show that if for all modules $D$ the sequence $$0 \to \operatorname{Hom}_{R}(N,D) \xrightarrow{\varphi^*} \operatorname{Hom}_{R}(M,D) \xrightarrow{\psi^*} \operatorname{Hom}_{R}(L,D) \to 0$$...
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Projective dimension of the residue field
For $R$ a ring and $M$ a module, denote the projective dimension of $M$ over $R$ by $pd(M)$. We have the following:
Lemma (Matsumura Commutative ring theory section 19 lemma 1). Let $(R,m,k)$ be a ...
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Projective-injective modules over $k[x]$
I am looking at graded $k[x]$-modules. I am also interested in the subcategory of modules $M$ all whose graded components are finite dimensional, but that's an optional restriction. In either case, I ...
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Graded Betti numbers of monomial ideals and inclusions
Let $I$ and $J$ be two monomial ideals in some polynomial ring $S=k[X_1,...,X_n]$. Furthermore, assume that $G(I)\subseteq G(J)$, where $G(I)$ denotes the minimal set of monomial generators of $I$, ...
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Graded-commutativity of cup product: Non-commuative coefficient ring
For $R$ a commutative ring and $X$ a topological space the cup product on the singular cohomology $H^{\ast}(X)$ is graded commutative. I have a question about the proof of this claim.
Some definitions ...
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A question about tensor products of fin. gen. proj. modules and module maps
Consider non-zero bimodules $M,N$, and $P$, over a ring $R$, that ar additionally assumed to be finitely-generated and projective as left $R$-modules. Take a bimodule map $f:N \to P$, does it hold ...
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Each $(R,S)$-bimodule is a left $R \otimes_k S^{op}$-module
I am trying to understand and fill the gaps of Rotman's proof (in his homological algebra text). This approach is different from this post ("Post 1") and is more complete than this one (&...
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What are some examples of injective sheaves?
Injective objects in the category of abelian groups are precisely the divisible groups. However, despite using injective resolutions of sheaves everywhere, I realized that I did not know a single ...
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$\mathbb{Z}[u,v]/(u^2+u-v^3+v)$ is torsion free over $\mathbb{Z}$?
Literally,
$B:=\mathbb{Z}[u,v]/(u^2+u-v^3+v)$ is torsion free over $\mathbb{Z}$?
Note that https://en.wikipedia.org/wiki/Torsion-free_module : from contents in the 'Examples of torsion-free modules' ...
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Injective object in category $\mathcal{K}$, which has an object $a$ and $\text{Hom}(a,a) = S $ is a semigroup with unit.
We consider the category $\mathcal{K}$ with an element $a$ and $\text{Hom}(a,a) = S$, where $S$ is a semigroup with unit.
Is $a$ an injective object of $\mathcal{K}$, if $S = \{ 1,\alpha, \alpha^2\}$ ...
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Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
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Products in the category R-Mod
I have already shown that Ab is an Abelian category, in particular it has finite products. Using this, I want to show that R-Mod has finite products. I feel that what I have done makes sense, but I am ...
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Ring with IBN and isomorphic free modules
This is problem 2.26(iii) in Rotman's Homological Algebra.
1. Problem 2.26(iii) (R a ring with IBN)
2. Partial Proof
Let $W \subset X$ be finite and define
$$W' = \{b \in B : \text{supp}(b) \subset W ...
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Duality in cohomology of bialgebras
Let $A$ be a finite dimensional bialgebra over a field $k$. Then $A^\vee=\mathrm{Hom}_k(A,k)$ is a bialgebra. Assume that $A$ and $A^\vee$ are local rings with commutative residue fields isomorphic to ...
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Geometric interpretation of a generalized Euler sequence
Although similar arguments work for $\mathbb P^n$ for any $n$, let us deal with $X=\mathbb P^1_k$ for simplicity. Recall there is a so-called Euler sequence on $X$ (twisted by $\mathcal O_X(1)$):
$$ 0 ...
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$Hom_\mathbb Z$ and direct products
This is problem 2.25(ii) from Rotman's Homological Algebra Text.
1. Problem
Let $p$ be a prime, $B_n$ be a cycle group of order $p^n$, and $A = \bigoplus_{n=1}^\infty B_n$. Show that
$$ \mathcal S := ...
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Example about endomorphism ring of an indecomposable object, which have non-trivial idempotent element in an additive category.
Recently I read some about Krull-Remak-Schmidt category.
If $A$ is an additive category in which every idempotent splits, every object is the biproduct of finitely many indecomposable objects and the ...
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Special case of Snake's Lemma - Neukirch Proposition 12.9
I was wondering how Neukirch gets in his Proposition 12.9 (see http://www.math.toronto.edu/~ila/Neukirch_Algebraic_number_theory.pdf) from the snake-lemma long exact sequence to his claimed sequence.
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Enveloping algebra of an algebra essentially of finite type (Weibel 9.4.5)
Let $k$ be a commutative Noetherian ring and $R$ be an algebra essentially of finite type (that is, $R$ is a commutative $k$-algebra and it is a localization of a finitely generated $k$-algebra). In ...
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Beginner's guide on homological algebra for commutative ring settings [duplicate]
At the moment, I'm a masters student, and I find commutative algebra to be pretty fascinating. The first seven chapters of "Introduction to Commutative Algebra" by Atiyah & Macdonald, &...
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Third cohomology of a group
I'm familiar with two primary ways to discuss the n$^{th}$ cohomology of groups with coefficients in an abelian group $A$:
(1) Through the exploration of n-fold extensions.
(2) By examining the map $...
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Modules over the ring $\mathbb{F}_p[C_p]\cong \mathbb{F}_p[X]/(x^p-1)$
I would like to understand the category of modules over the group algebra $\mathbb{F}_p[C_p]\cong\mathbb{F}_p[X]/(x^p-1)$.
I am interested in computing the group cohomology of $C_p$ with coefficient ...
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Webeil's Intro to Homological Algebra: does theorem 10.6.3 implicitly reindex cochain complexes to chain complexes?
Let $R$ be a ring, let $\mathbf{D^-(R-mod)}$ denote the derived category of bounded above cochain complexes of $R$-modules, and consider the total tensor product functor
$$
\otimes_R^\mathbf{L}: \...
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Example 2.2 in Rotman's Homological Algebra Text
1. Example 2.2 (excerpt)
Example 2.2. Let $G$ be a finite group and let $k$ be a commutative ring. The group ring is the set of all functions $\alpha : G \to k$ made into a ring with pointwise ...
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Commutative diagrams in Opposite Category
I am working on the kernels and co-kernels of morphisms in categories with zero objects. I came across the following: $(C,j:B\to C)$ is a cokernel of $f\in \text{Hom}_{\mathcal{C}}(A,B)$ $\iff$ $(C,j^{...
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mapping cone and derived functor
Let $F:\mathcal{C}\rightarrow \mathcal{D}$ be a left exact functor between abelian categories. If $f:A\to B$ is a morphism in $D^+(\mathcal{C})$, do we have
$$\operatorname{Cone}(\operatorname{R}F(A)\...
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Trouble understanding torsion
I'm reading this paper. In definition 6.1.2 it is mentioned that $H_n(G;A) \cong \text{Tor}_n^{\mathbb{Z}G}(\mathbb{Z}, A)$ (for all $n=0, 1, 2, ...$). What exactly is meant by the right-hand side? I'...
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$\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ on finite abelian groups is not right exact?
In his answer here Martin Brandenburg claims that the Tor functor $\operatorname{Tor}^{\mathbb{Z}}_1(-,-)$ in the category of finite abelian groups is not right exact in neither argument. Since Tor is ...
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Stallings' Theorem
I am trying to understand Stallings' Theorem for lower central series. Here is the statement:
Say we have groups $A, B$ with lower central series $A=A_1, A_2, ...$ and $B=B_1, B_2, ...$ respectively. ...
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Category Theory and Cardinality
This is a very basic question from Rotman's Homological Algebra text. I think the issue is that I don't know how to rigorously talk about cardinality.
Problem
If $\mathcal A$ is a small category (...
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Acyclicity of co-chain complexes in terms of quasi-isomorphisms.
Weibel gives equivalent conditions for acyclicity of chain complexes, one of which is that the chain complex map $0\to C_.$ is a quasi-isomorphism (i.e. $H_n(0)=0\to H_n(C_.)$ is an isomorphism). ...
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Corollary to Yoneda Lemma
I'm working from Rotman's Homological Algebra text. There's just one detail of the Corollary that I don't follow.
1. Yoneda Lemma
Theorem 1.17 (Yoneda Lemma). Let $\mathcal{C}$ be a category, let $A \...
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Calculating a specific example of a simplicial resolution of an algebra
In the Stacks Project Tag 09D4, there's an explicit description of a simplicial resolution
\begin{align*}
P_{2}\to P_{1}\to P_{0}
\end{align*}
of an $A$-algebra $B$. I am trying to follow this ...
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Basic confusion about $Tor_n$ and projective resolutions
Osborne's text has the below basic example.
1. Example 9
$R = \mathbb Z_4$, $A = B = \mathbb Z_2$ A projective resolution of $\mathbb Z_2$ is
$$ \cdots \rightarrow \mathbb Z_4 \xrightarrow{\times 2} \...
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How to construct $\delta^i$ morphism for right derived functors?
Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left-exact functor between abelian categories. Assume $\mathcal{A}$ has enough injectives. I want to prove the following:
For every short exact sequence $...
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Resolution for exterior power of a quotient
Let us assume that we have exact sequence of vector spaces:
$$0\to U\to V\to W\to 0.$$
We can think of $0\to U\to V$ as a resolution of $W$.
Can we construct some canonical resolution of $\Lambda^n W$...
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Proof regarding when left exact functors are $\delta$-functors
Here is the theorem I am trying to prove:
$\mathcal{A}$ and $\mathcal{B}$ are abelian categories. Suppose $F: \mathcal{A} \to \mathcal{B}$ is a left exact functor. If $\mathcal{A}$ has enough ...
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Tor-dimension of $A/(a)\otimes_k B$, where $A$ and $B$ are Dedekind domains
Let $A$ and $B$ be two Dedekind domains which contain a field $k$ which is algebraically closed in both $A$ and $B$. Let $a$ be a non zero element in $A$.
What is the Tor-dimension of $A\otimes_k B$? ...
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Right derived functors are additive
I am trying to prove the following statement:
Let $F: \mathcal{A} \to \mathcal{B}$ be a left-exact functor between abelian categories. Suppose $\mathcal{A}$ has enough injectives. Then the right ...
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Alexander-Whitney map gives a coalgebra?
Let $R$ be a unital ring with multiplication $\mu\colon R\otimes R \rightarrow R$. Consider the category $\mathcal{Ch}(R-\text{mod})$ of chain complexes of $R$-modules.
This category becomes a ...
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Does the inclusion map $\iota: \mathbb{Z}\rightarrow \mathbb{R}$ induce an inclusion of co/homology groups?
Let $M$ be a smooth $n$-manifold, and $\iota: \mathbb{Z}\hookrightarrow \mathbb{R}$ be the natural inclusion homomorphism of abelian groups. Then, it is easy to check that we obtain a well defined ...
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Question on Proof of Splitting Lemma for Modules
The setup of the question is as follows. Let the following be a short exact sequence of modules:
$$
0 \rightarrow A \xrightarrow{i} B \xrightarrow{p} C \rightarrow 0,
$$
and let $r: B\to A$ be a map ...
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Why is the Yoneda product sometimes called cup product?
In algebraic topology there is the cup product, which endows the direct sum of (singular) cohomology groups with the structure of an associative, graded-commutative, unital ring. Given an associative, ...
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Prove the mod 2 Steenrod algebra has counit
Edit: $\epsilon$ must send $Sq^i$ to $0$ since it is the case where $\mathcal{A}_2$ is a graded algebra. The Wikipedia page presents the answer itself.
I'm trying to show that the mod $2$ Steenrod ...
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Term-wise added regular sequence
Suppose $k$ is a field. Suppose $f_1,\cdots,f_r\in k[x_1,\cdots,x_n]$ is a regular sequence, and $g_1,\cdots,g_r\in k[y_1,\cdots,y_m]$ is a sequence, where $x_i$ and $y_j$ are different symbols.
It ...
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Abelian group as second cohomology group of a pair G,M
I'm currently studying group cohomology and in particular group extension; I'm trying to figure out a solution to the following problem: let A an abelian group, is possible to find a group G and a G-...
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Leray spectral sequence degenerate Case Hartshorne exercise III.8.1
In Hartshorne III.8.1 has the following exercise:
8.1. Let $f: X \rightarrow Y$ be a continuous map of topological spaces. Let $\mathscr{F}$ be a sheaf of abelian groups on $X$, and assume that $R^i ...
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Natural isomorphism on homology implies natural chain homotopy equivalence?
Let $F,G: C \to \mathrm{Ch}(\mathbb{Z}\textrm{-free})$ be two functors, where $C$ is an arbitrary category and $\mathrm{Ch}(\mathbb{Z}-free)$ is the category of chain complexes of free abelian groups.
...
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Homology in a picture? (Is this picture just metaphorical, or a rigorous example that can be formalized?)
A post-doc colleague showed me this picture and said: going from the diagram No.2 to No.3 and to No.4 is taking the homology.
I did not quite understand this comment. For me, if I take simplicial ...
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