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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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Bar resolution and the morphisms define on $B_n$(free module)

Given a group extension $0\to K\to G\to Q\to 1$, we define $B_n$ as the free $\Bbb Z[Q]$-module on $Q^n$. And then we want to make a exact sequence $\cdots\to B_3\to B_2\to B_1\to B_0$, where the ...
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Example of non-hereditary cotorsion pair?

$\DeclareMathOperator{\Ext}{Ext}$ Let $\mathcal{A}$ be an abelian category and $(\mathcal{D},\mathcal{E})$ be a cotorsion pair, i.e. classes of objects of $\mathcal{A}$, such that $D\in \mathcal{D}$ ...
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Prove that there is a long exact sequence $\ldots → H_k(X) \overset{2}\to H_k(X) → H_k(X;\Bbb Z/2) → H_ {k−1} (X) → · · ·$

Consider the exact sequence $0 \to \Bbb Z \overset{2}{\to} \Bbb Z → \Bbb Z/2 → 0$. Prove that there is a long exact sequence $\ldots → H_k(X) \overset{2}{\to} H_k(X) → H_k(X; \Bbb Z/2) → H_ {k−1}...
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A particular complex of integral group ring is exact: proof of Jacobson

Let $A$ be a $G$-module. Let $C_n=\mathbb{Z}G\otimes \cdots \otimes \mathbb{Z}G$ ($n+1$ copies). It is free $\mathbb{Z}$-module with basis $g_0\otimes g_1\otimes \cdots \otimes g_n$, $g_i\in G$. ...
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34 views

Projective modules in long exact sequences

Let $A$ be a commutative ring (with unit), and let $(P_i)_i$ be projective $A$-modules sitting in a long exact sequence of $A$-modules: $$0 \longrightarrow P_1 \stackrel{f_1}{\longrightarrow} P_2\...
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Reference request for bigraded vector spaces

I'm currently reading up on Spectral Sequences in Algebraic Topology, and often times authors refer to graded vector spaces and bigraded vector spaces freely without defining them. I've found ...
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If $K$ is a field then $\frac{K[x_1,x_2,…,x_n]}{(x_1-\alpha_1,…,x_i-\alpha_i)}\cong K[x_{i+1},…,x_n]$

If $K$ is a field and $\alpha_1, \alpha_2, ..., \alpha_p \in K$ then $$\frac{K[x_1,x_2,...,x_n]}{(x_1-\alpha_1,...,x_i-\alpha_i)}\cong K[x_{i+1},...,x_n] $$ My Attempt I simply tried to use Hilbert'...
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If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$? [closed]

Is the following statement true? If $R$ is a PID then $\mathrm{Spec}(R)\!=\!\mathrm{Max}(R)\!\cup\!\{0\}$ I know the reverse is false.
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In general, what's the relation between $\text{Ext}^n_R(A,B)$ and $\text{Ext}^n_R(B,A)$ (if any)?

Here $A,B$ are arbitrary $R$-modules. Similarly, what's the relation between $\text{Tor}^n_R(A,B)$ and $\text{Tor}^n_R(B,A)$? I am on 17.1 Dummit and Foote and couldn't find any remarks on this.
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Rank of a free Abelian subgroup gives a lower bound for vcd

Let $G$ be any group. The cohomological dimension (cd) of $G$ is the smallest integer $n$ such that $\mathbb{Z}$ admits a projective resolution of length $n$ over the group ring $\mathbb{Z}G$. Serre ...
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The relation between group extension and factor set

I'm new to homological algebra. I try to state what I know, and ask the question at the end. I know that given a group extension $0\to K\to G\to Q\to 1$, it may have many liftings. Each lifting ...
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Computing Ext functor when no projective modules are known.

I have a $p^3$-dimensional (not semisimple) algebra $\mathcal{U}_{q}(sl_2)$ over $\mathbb{C}$ and i know how all its simple modules look like (there are $p$ of them, each $M_i$ has dimension $i$ for $...
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Pushout of cdgas and the quotient by an acyclic ideal.

Let $A$ be a cdga over $\mathbb{R}$, $B\subset A$ a dg-subalgebra and $I\subset B$ a dg-ideal. Consider the pushout of cdga's $\require{AMScd}$ \begin{CD} B @>\pi>> B/I\\ @V i V V @VV ...
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Hilbert Syzygy Theorem for non-graded modules

The statement of Hilbert Syzygy Theorem is as follows: Let $R = k[x_1 , \ldots , x_n]$ be a polynomial ring over a field $k$ and $M$ be a finitely generated graded $R$-module. Then $\text{pd }M \leq n$...
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For an equivalent functor $\Sigma:T \to T$, if $T(v \circ u)=0$ then $v \circ u=0$.

Im beginnig selfstudy of triangulated categories, and Im working with an additive category $T$ and an additive covariant and equivalent functor $\Sigma:T \to T$, as equivalent the notes say that ...
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Spectral sequence of filtered complex.

I am trying to understand the construction of a spectral sequence of a filtered complex. After reading through the entry in the nLab I came up with an example, that I don't understand: Consider the ...
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Weibel Exercise 2.4.3, dimension shift

Exercise 2.4.3., pg 47 If $0 \rightarrow M \rightarrow P \rightarrow A \rightarrow 0$ is exact with $P$ projective (or $F$-acyclic), then $L_iF(A) \cong L_{i-1}F(M)$ for $i \ge 2$ and that $L_1F(A)$ ...
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Betti numbers of finitely generated module over Noetherian local ring, after going modulo a regular element

For a finitely generated module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ , let $b_i^R(M):= \dim_k \operatorname{Tor}_i^R (k,M)$. It is known that this $i$-th Betti numbers ...
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Invariance of Yoneda product

Are Yoneda products (also known as cup product) on Hochschild cohomology of two quasi-isomorphic DGAs equivalent?
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Understanding the definition of stabilizing automorphisms of groups

I feel weird when I saw the definition of stabilizing automorphisms of groups. If the diagram is required to be commutative, then isn't the only case of $\varphi$ is that it is nothing but an identity ...
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What do $0$ and $1$ mean in a extension (exact sequence) of group?

I'm self-studying homological algebra but stuck in a little place. What does $0$ and $1$ mean by the author? (I didn't read the book from start to end.) Does it mean the trivial group? Then why we ...
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Prove that $2 \otimes 1 \neq 0$ in $2\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ the tensor is over $\Bbb Z$.

Prove that $2 \otimes 1 $ is zero in $\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ but not a zero in $2\Bbb Z \otimes {\Bbb Z/2\Bbb Z}$ the tensor is over $\Bbb Z$. It is easy to show the first part that $2 \...
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How to understand $k \otimes_R \alpha \cong k \oplus k$?

Let $R = k[x, y]$, where $k$ is field. Then we have a projective resolution of $k$ : $$0 \longrightarrow R \stackrel{f}\longrightarrow R\oplus R \stackrel{g}\longrightarrow R \longrightarrow k \to ...
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1answer
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Showing that the Composition of Boundary Maps of the Koszul Complex is Zero

I am following the discussion of the Koszul complex in Lang's "Algebra". Suppose $ R $ is a commutative ring, $ S $ is an $ R-$module, and $ f_{1},\dots,f_{r} \in R. $ Define: $ K_{0}(f) = R. $ $...
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Derived equivalence in Residues and Duality

Let $A'$ be a serre subcategry of $A$, let $A'$ has enough injectives and every injective object of $A'$ is also injective in $A$.Then the natural functor $c:D^+(A')\rightarrow D_{A'}^+(A)$ is ...
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Explaination of a justification: additive functors preserve limits

Lemma: For any collection $\{ M_i\}_{i\in I}$ of $R$-modules, and $R$-module $N$, there is a natural isomorphism $${\rm Hom}_R(\oplus_i M_i, N)\cong \prod_i {\rm Hom}_R(M_i,N).$$ Proof: Additive ...
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1answer
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Do we define left derived functor from $_R\textbf{Mod}$ to “$\textbf{Ab}$”?

The definition of the left derived functor I know is from Rotman's Advanced Modern Algebra II, where he gave the hypothesis that the original functor $T$ that another functor $L_nT$ can derive from is ...
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Can we deduce the exactness of the pervious modules sequence if the localized exact modules sequence is exact?

My question comes from a proposition: if M is flat, will $S^{-1}M$ be flat? Where $S = R - \mathfrak{p}$, $\mathfrak{p}$ is a prime ideal. Since localization keeps the exactness, I find that what I ...
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Understanding How We Get Koszul Complexes From Regular Sequences

Let $ R $ be a ring, and $ M $ be an $ R $-module. We say that a sequence $ x_{1},\dots,x_{r} $ of elements of $ R $ is regular if: 1) $ (f_{1},\dots,f_{r})M \neq M, $ and 2) $ f_{i} $ is a non-zero ...
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Understanding the definition of left derived functor

I'm studying homological algebra. But I stuck in a place in Rotman's Advanced Algebra II, 3rd ed. What does $T\hat{f}$ mean in the paragraph? The point made me confused is that $T$ is a functor from $...
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Noetherian rings whose prime ideals have projective dimension bounded above

For a module $M$ over a commutative Noetherian ring $R$, let $pd_R (M)$ denote the projective dimension of $M$ as an $R$-module. Now let $R$ be a commutative Noetherian ring such that $\sup \{ pd_R (Q)...
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Closure of category of sheaves under inverse limits.

How do I show that the category of sheaves on a space $X$ taking values in , a category $K$ admitting inverse limits, admits inverse limits? The limit is a presheaf, I tried to show that it is also ...
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Categorical product of non-unital associative differential graded coalgebras

Given two non-unital associative dg coalgebras $D$ and $C$, I want to give an explicit construction of the product $C\prod D$, this may follow from the dual construction (coproduct of non-unital ...
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38 views

How to decide what kind of resolutions should we use to calculate Ext and Tor?

In general, when we calculate groups like Ext and Tor, how should we choose between free, projective and injective resolutions? For example, why does Hatcher 3.1 only consider free resolution but not ...
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On a special type of Noetherian regular rings

Let $R$ be a commutative Noetherian ring having the property that for every $R$-module $M$ that has finite projective dimension, every submodule of $M$ also has finite projective dimension. Then $R$ ...
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Index of products in the Brauer group.

Let $k$ be a field of characteristic 0. Let us consider two elements $A,B \in$ Br$(k)$, which are not inverse to each other. What is the index of the product $A \otimes B$ in Br$(k)$? I assume that ...
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81 views

Does every short exact sequence split?

If we have a short exact sequence $0 \to A \to B \to C \to 0$, then the map $A \to B$ is injective and the map $B \to C$ is surjective. Therefore, there always exists a left inverse for $i$ and a ...
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The normalized chain complex of a simplicial set

The paper The Geometry of Rewriting Systems, by K. Brown, mentions the notion of a normalized chain complex associated to a simplicial set. How is this complex defined? (and perharps the non-...
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Submodules of modules of finite projective dimension over regular ring

Let $R$ be a regular ring i.e. a commutative Noetherian ring whose localisations at every prime ideal is regular local ring. Then every finitely generated $R$-module has finite projective dimension, ...
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Does every covariant functor on module category preserve inclusion?

Does every covariant functor $F : ~_R\mathcal{M} \rightarrow \mathcal{C}$ on module category $_R\mathcal{M}$ preserve inclusion? I have proceeded in the following way. Suppose $A \subseteq B$ in $_R\...
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1answer
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Projective Resolution of exterior algebra as a module over divided polynomial algebra

Let $\Lambda_\mathbb{Z}[x]$ be an exterior algebra on one generator with $|x|=n$, let $\Gamma_\mathbb{Z}[x]$ be a divided polynomial algebra with $|x_k|= kn$, and suppose that $\Lambda_\mathbb{Z}[x]$ ...
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Cohomology Via Resolutions of “Flat” Objects

Let $A$ and $B$ be abelian categories and let $F : A \rightarrow B$ be a left exact functor. We can form the right derived functors $R^i F$ of $F$ by taking an injective resolution $X \rightarrow I_0 \...
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What is meant by $\operatorname{Tor}(P,K)$ for $P$ a left $R$-module?

Let $R$ be a finite dimensional algebra over a field $K$. Suppose there is a two-sided ideal $I$ of $R$ such that (a) $R=K\oplus I$; (b) $I$ is nilpotent. Question If $P$ is a left $R$-module, ...
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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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1answer
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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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1answer
37 views

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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1answer
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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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1answer
65 views

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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1answer
83 views

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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1answer
66 views

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 ...