Questions tagged [homological-algebra]
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
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Invertible bimodules and projectivity
Let $A$ be a noncommutative algebra over a field, say $\mathbb{C}$ or $\mathbb{R}$, and let $L$ be a bimodule over $A$. If $L$ is invertible, that is, if the dual right $A$-module $L^*$ satisfies
$$
L^...
4
votes
1
answer
114
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Postprojective components of quiver algebras
Let $A=kQ/I$ be a quiver algebra with acyclic quiver $Q$.
An indecomposable module $M$ is called postprojective in case $M \cong \tau^{-1}(P)$ for an indecomposble projective module $P$. A component ...
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2
votes
1
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Residual Finiteness for 3-Manifolds Hempel
Hempel, in his 1987 article "Residual Finiteness for 3-Manifolds", shows that if $M$ is a compact Haken 3-manifold, then $\pi_1(M)$ is residually finite. In the proof he starts by reducing the case to ...
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2
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Is Hilbert basis theorem true for positive graded ring?
Let $R=\oplus_{I\geq 0}R_i$ be a positive graded ring(maybe not commutative), where $R_0$ is a commutative Noetherian ring. If $R$ is finite generated $R_0$-algebra, is $R$ Noetherian?
In here, Is ...
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Is being derived equivalent independent of the field?
Let $Q_1, Q_2$ be (connected) acyclic quivers and $I_1, I_2$ admissible ideals (in which the relations have only coefficients 1 or -1).
Let $K$ and $F$ be two fields.
Question 1: Is $KQ_1/I_1$ ...
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1
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0
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76
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Singular chain complex of balanced products
Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...
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38
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Regarding linear splitting of lie algebra morphism and their CE complexes
The main question here is to ask if anyone has ever seen/researched the map $\alpha$ below and get a reference regarding it. Also, if anyone tells me the equivalent conditions to $\alpha=0$, I would ...
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6
votes
1
answer
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Derived invariant for Gorenstein algebras?
Let $A$ be a finite dimensional algebra with simple modules $S_i$ and projective indecomposable modules $P_l$ and global dimension $g< \infty$ (and $n$ is the number of simple modules).
I noted ...
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1
vote
1
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110
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Augmented algebras over semisimple ring
Let $A$ be a non-negatively graded algebra such that $A_0 = k$. We say that $A$ is Koszul if $k$ has a projective resolution by projective modules such that the i-th piece is generated in degree $i$. ...
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Global dimension 2 Nakayama algebras , 321-avoiding permutations and Fibonacci combinatorics
A $n$-LNakayama algebra (for $n \geq 2$) is a list $[c_0,c_1,...,c_{n-1}]$ with $c_{n-1}=1$, $c_i \geq 2$ for $i \neq n-1$ and $c_i -1 \leq c_{i+1}$ for all $i$.
They are in bijection with Dyck paths, ...
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2
votes
0
answers
65
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Projective dimension of the functions with compact support
Let $X$ be a locally compact Hausdorff space. And $C(X)$ the ring of all continous real-valued functions and $J(X)$ the ideal of such functions with compact support.
It is known that $X$ is ...
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3
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Is $C^{\infty}(\mathbb{R}^{m+n})$ a flat module over $C^{\infty}(\mathbb{R}^{m})$?
For $m>0$ we consider the ring $C^{\infty}(\mathbb{R}^{m})$ of smooth functions on $\mathbb{R}^{m}$. For $n>0$ we consider the projection $\mathbb{R}^{m+n}\to \mathbb{R}^{m}$ hence $C^{\infty}(\...
- 8,707
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Quiver algebras with finite global dimension
Given a fixed connected quiver $Q$.
Are there only finitely many quiver algebra $KQ/I$ (I an admissible ideal) up to isomorphism which have finite representation type and finite global dimension?
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1
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Ordered sequence of elements of poset relevant to some filtration -- highest weight category
Let $\mathcal{C}$ be a highest-weight category with $\Lambda$ as a interval-finite poset -- I'm using a definition of the highest-weight category given by Cline, Parshall and Scott and it is presented ...
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Characterisation of algebras with Euler trivial modules
Let $A$ be an algebra of finite global dimension.
The Euler form on $A$ for an indecomposable module $M$ is defined as $\psi(M)=\sum\limits_{k=0}^{\infty}{(-1)^k dim( \operatorname{Ext}_A^k(M,M)) }$.
...
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8
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Bringing cohomology recipes from algebra to topology?
In algebra, cohomology theories are often defined very roughly like this. Start with a scheme $X$ you want to study. Form a category-with-extra-structure (a site) $\mathcal{C}$ whose objects are, ...
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0
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When is a chain complex induced up to quasiisomorphism
I have a field extension $F/F'$ and an algebra $L'$ over $F'$. Let $L$ be the induced $F$-algebra $F\otimes_{F'}L'$ and $C_*$ a chain complex over $L$.
Is there a good way to decide whether $C_*$ is ...
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A lemma on a sequence of three morphisms
Few months ago, I proved that when there is three morphisms of modules over a commutative ring with zero composition, i.e., a sequence
$$A \xrightarrow{\alpha} B \xrightarrow{\beta} C \xrightarrow{\...
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1
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Indecomposable objects in bounded derived category of $\mathbb C[x]/x^2$-mod
We know for any principal ideal domain, objects in the bounded derived category are all formal hence we can classify those objects with finitely generated cohomology using structure theorem for ...
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5
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0
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113
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On algebras where all indecomposables have no selfextensions
Let $A$ be a finite dimensional algebra (we can assume it is a connected quiver algebra).
Call $A$ extfree in case for every indecomposable $A$-module $M$ we have $\operatorname{Ext}_A^i(M,M)=0$ for ...
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2
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76
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Bimodule resolutions
I have asked this question on Mathematics Stack exachange but didn't get any reply yet. So, I am asking it here.
Let A be a finite-dimensional algebra. Let M be a left A-module and
N be a right A-...
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"Differential graded homological algebra" by Avramov, Foxby and Halperin
I am looking for "Differential graded homological algebra" by L. Avramov, H. Foxby, and S. Halperin, which is widely cited as a preprint o as a manuscript, e.g. https://scholar.google.com/scholar?q=L....
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Simpicial resolution in Cotangent Complex
I am reading Cotangent complex from Loday's book Cyclic Homology. My doubt is related to the simplicial resolution of $k$- algebra $A$ which is used in the definition of cotangent complex.. Let me ...
- 609
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0
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160
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multiplication in spectral sequence
I am trying to understand this paper. Let $M$ be a compact Kaehler manifold of dimension $n$, $X$ is a holomorphic vector field, $i_X$ the contraction operator, i.e. for $\alpha$ a $p$-form, then $i_X(...
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1
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injective hulls in mixed characteristic
Let $R=\underleftarrow\lim (R/\mathfrak m^i)$ be a complete local ring, with residue field $k=R/\mathfrak m$,
and let's assume that $R$ is Noetherian.
If $R$ is a $k$-algebra, then
I believe that ...
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1
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When is the Jacobson radical reflexive?
Let algebras be Artin algebras.
It is well known that a an algebra has global dimension at most one if and only if the Jacobson radical is projective. As reflexive is a natural generalisation of ...
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3
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0
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Fibered category vs indexed category
In Benabou's paper Fibered categories and the foundations of naive category theory, it is mentioned (end of page 31) that 'an indexed category is just a presentation of a fibered category.'
It seems ...
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Yoneda extension and splittings
Let $X$ be a non-singular algebraic variety and $F$ be a coherent sheaf defined over $X$. Suppose that we have a locally free resolution
$$0 \to L_n \xrightarrow{f_n} L_{n-1} \to ... \to L_0 \to F \to ...
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Resolutions of $\mathbb{Z}_{(p)}$ as $\mathbb{Z}$-module
Are there any interesting canonical (maybe unbounded) projective resolutions of $\mathbb{Z}_{(p)}$ over $\mathbb{Z}$, for instance by tensoring together all the $\mathbb{Z}[x] \stackrel{qx-1}\to \...
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Artinian Tor modules (Reference request)
I am looking for a reference for the following basic fact:
Let $R$ be a noetherian ring, let $M$ be an artinian $R$-module, let $N$ be a finitely generated $R$-module, and let $i\in\mathbb{N}$. ...
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2
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Concrete examples of Freyd-Mitchell embedding
I originally posted this on math.SE (https://math.stackexchange.com/questions/3438528/concrete-examples-of-freyd-mitchell-embedding) but since it's been a few days I figured I would crosspost it here. ...
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Which abelian categories have homological dimension 1?
In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either
a category of representations $\mathrm{Rep}_\mathbf{...
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1
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Computation of $\mathcal{Tor}_i^{O_X}(O_{Z},O_{Z})$ for non-complete intersection $Z$ in $X$
Let $Z \hookrightarrow X$ be a closed subvariety of a smooth projective variety. How do we compute $\mathcal{Tor}_i^{O_X}(O_{Z},O_{Z})$ $(i>0)$ as coherent sheaves on $Z$ where $Z$ is not of ...
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What are the projective dimensions of big fraction fields?
Let $A$ be an integral domain, $B$ is its fraction field. Can the projective dimension of the $A$-module $B$ be greater than $1$? This surely cannot happen if the spectrum of $A$ is countable (since ...
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An action of the symmetric group $S_n$ on group cohomology $H^n(G, A)$ of abelian groups
Let $H$, $A$ be discrete abelian groups, and for simplicity suppose $A$ is given the trivial $H$-action.
When considering the second cohomology group $H^2(H,A)$, it is natural to talk about the ...
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Why does K-theory need schemes to be Noetherian?
The definition of K-theory of a scheme $X$ is defined as
$G_i(X):=K_i(\mathrm{Coh}(X))$ or $K_i(X):=K_i(\mathrm{Vec}(X))$.
But usually the schemes are required to be (at least locally) Noetherian, and ...
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'Quotient' of indexed categories and pushforward congruence relations
If $R\subset C$ is a vector subspace of $C$, and $F: C \to D$ is a linear map, then we have a natural linear map $C/R\to D/F(R)$. I was wondering if this can be also generalized to categorical ...
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Geometric meaning of Koszul modules
Assume $C$ is a smooth projective curve and $L$ is a line bundle on it. We say $L$ is $p$-very ample if for any effective divisor $D$ of degree $p+1$, the evaluation map $H^0(C,L)\to H^0(C,L\otimes\...
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Tate cohomology for group algebras
Let $A=kG$ be a group algebra with a finite group $G$ and a field $k$.
Let $T^i(M,N)= \underline{Hom_A}(\Omega^i(M),N)$ be the $i$-th Tate cohomology group. Note $T^i(M,N)= Ext_A^i(M,N)$ in case $i \...
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9
votes
1
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Betti sequence of finite dimensional commutative algebras
Given a finite dimensional commutative local $K$-algebra $A$ for a field $K$. Associated to $A$ is its dimension $d_A$ and the Betti-sequence $c_i=dim(Ext_A^i(S,S))$ where $S$ is the unique simple $A$-...
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Translate a construction into categorical languages
Let $\mathscr C$ be a category whose objects are sets.
Let $\mathscr V$ be another category. It seems that the usual composition in $\mathscr V$ can be somehow 'twisted' by $\mathscr C$. I notice that ...
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Commutative algebras with modules of small complexity
Let $A$ be a finite dimensional commutative algebra. We can assume that it is local.
Question: Which such $A$ have the property that every finite dimensional $A$-module has complexity at most 1? (...
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votes
1
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Functoriality of filtered spectral sequences
What is the appropriate functoriality statement of a filtered chain map between filtered spectral sequences?
Suppose that we have two filtered chain complexes $C,C'$ and a filtered chain map $f\colon ...
6
votes
1
answer
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Relative Ext of Avramov-Martsinkovsky as a derived Hom
Avramov-Martsinkovsky (http://mathserver.neu.edu/~martsinkovsky/Relative.pdf) have defined an exotic version of Ext between two modules over (for simplicity) Gorenstein rings. The basic idea of their ...
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Geometric intuition behind this chain homotopy
My question has to do with the chain homotopy that appears in Lee's Introduction to Topological Manifols and Rotman's Introduction to Algebraic Topology proofs that the inclusion
$$C_\bullet^\mathcal{...
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3
votes
1
answer
543
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Questions about Deligne's tensor product of abelian categories
Let $A$ and $B$ be abelian categories, then $A \times B$ is an abelian category. Also denote $Ab$ to be the category of abelian groups or any abelian category.
If want to study a bi-additive (also ...
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0
answers
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On NCR for finite dimensional algebras
Let $A$ be a finite dimensional algebra.
A NCR of an algebra $A$ is a faithful module $M$ such that $End_A(M)$ has finite global dimension. Call a faithful module PNCR (=pseudo-NCR) in case $End_A(M)$ ...
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11
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1
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Derived categories and classical theorems in homological algebra
So far I have studied fundamental part of derived category theory, for example, the existence of derived functors, the "composition of derived functors", and so on.
Now I came up with some questions ...
- 1,352
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1
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Tor functor and invertible elements
Let $A$ be a commutative ring, $a \subset A$ be an ideal. For $A$-module $M$ let $S \subset A$ be the set of elements, which are invertible in $M$, so $M$ is actually a $S^{-1}A$-module. It is not ...
- 332
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1
answer
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Gaps in the projective dimensions of simple modules
Let $A$ be a finite dimensional algebra with finite global dimension $g$ and $n$ simple modules.
Let $d_1<d_2<...<d_r$ be the sequence of projective dimension of simple $A$-modules in ...
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