All Questions
Tagged with homological-algebra reference-request
271 questions
3
votes
1
answer
163
views
Theory of $n$-truncated $A_\infty$ categories/functors?
One can define certain higher categorical truncations. For example, a discussion from the quasi-categories point of view can be found in HTT 5.5.6.
On the other hand, as a model of linear $\infty$-...
- 169
2
votes
0
answers
124
views
Generalization of Koszul cohomology for wedge product with a fixed q-form: literature and references?
Let $A$ be a commutative ring. For an odd positive integer $q$ and an integer $p$ in the range $1 \leq p \leq q$, consider the cochain complex:
$0 \to \Lambda^p(A^N) \to \Lambda^{p+q}(A^N) \to \cdots \...
- 21
2
votes
1
answer
149
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Baer sums of extensions
Apologies in advance if this question is too basic, I looked briefly through Weibel and the stacks project and couldn't find any relevant reference.
Let $\mathcal{A}$ denote an abelian category, and ...
- 2,907
3
votes
0
answers
133
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Grothendieck spectral sequence (cohomology version) for posets with functor coefficient
In this paper, Quillen mentioned a spectral sequence as follows.
Let $f:X\to Y$ is a poset map and $\mathcal{F}:X\to Ab$, where $Ab$ is the category of abelian groups, a functor which is contravariant ...
- 139
4
votes
1
answer
227
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Literature Request: The derived category is Krull-Schmidt
I am looking for literature where it is proven that the derived category of bounded complexes over a finite-dimensional algebra is Krull-Schmidt. I found this question
Literature request: $K^b(\text{...
2
votes
0
answers
66
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Projective cover (minimal) for (derived)complete modules over Noetherian local rings exist?
Let $(R,\mathfrak m)$ be a commutative Noetherian local ring. Let $M$ be an $R$-module which is $\mathfrak m$-adically derived complete. Then, does there exist a free $R$-module $F$ and a surjective $...
- 412
5
votes
0
answers
112
views
Finitely generated projective modules over Noetherian endomorphism ring
Let $\mathcal A$ be a locally Noetherian Grothendieck abelian category and $M\in \mathcal A$ be a Noetherian object. Set $B:=\text{End}_{\mathcal A}(M)$. Let $B$-mod be the category of finitely ...
- 413
4
votes
0
answers
124
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Minimal model for $A_\infty$-categories
Is there a reference for existence and construction of the minimal model of an $A_\infty$-category? Most references I found ultimately refer to Lefèvre-Hasegawa's thesis but there doesn't seem to be a ...
- 489
5
votes
1
answer
367
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Reference request: locally erasable delta-functor is universal
It is well-documented that an erasable delta-functor $(F^n,\delta)$ is universal. However, in 'small' abelian categories (in the technical sense or otherwise), there aren't always enough objects to ...
- 28.7k
2
votes
1
answer
112
views
Example of non injective module over Noetherian local ring with trivial vanishing against residue field?
Is there an example of a module $M$ over a commutative Noetherian local ring $(R,\mathfrak m, k)$ such that $\text{Ext}_R^{>0}(k, M)=0$ but $M$ is not an injective $R$-module?
I know that for such ...
- 480
2
votes
0
answers
93
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Minimal injective resolution and change of rings
Let $R$ be a commutative Noetherian ring. For an $R$-module $M$, let $0\to E^0_R(M)\to E^1_R(M)\to \ldots $ denote the minimal injective resolution of $M$. I have two questions:
(1) If $I$ is an ...
- 480
0
votes
0
answers
47
views
Generalized edge map in spectral sequence of double complex
suppose we have a cohomologically indexed double complex $C^{\bullet,\bullet}$ with its spectral sequence
$$E_2^{p,q}=H^p_{vert}(H^q_{horiz}(C))\Rightarrow H^{p+q}(C)$$
and suppose that the horizontal ...
- 2,044
1
vote
0
answers
54
views
contravariant finiteness and limit closure: is there dual to a result of Crawley-Boevey?
Let $\mathcal A$ be a locally finitely presented category. Theorem 4.2 of https://doi.org/10.1080/00927879408824927 says that given a full additive subcategory $\mathcal D$ of finitely presented ...
- 480
16
votes
3
answers
1k
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Conjectures in the representation theory of the symmetric group
Question: What are current open conjectures about the representation theory of the symmetric group?
I am interested mostly in the characteristic 0 case, but conjectures for the modular case can also ...
- 26.5k
4
votes
1
answer
133
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Second cohomology group of the contact Lie algebra $K_3$
Let $F$ be a field of characteristic zero and, for all $n>0$, consider the contact Lie algebra $K_{2n+1}$. It follows from Corollary 3 of the paper [V. Guillemin - S. Shnider: Some stable results ...
- 630
3
votes
1
answer
385
views
Concrete examples of derived categories
What examples of abelian categories $\mathcal{A}$ are there such that the derived category $\mathcal{D}(\mathcal{A})$ can be described concretely? For example, is there a concrete way of describing $\...
- 1,474
1
vote
0
answers
124
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Computing the induced homomorphisms of derived functors using acyclic resolutions
Let's suppose that $F\colon \mathcal A\to \mathcal B$ is a right exact additive functor between abelian categories such that $\mathcal A$ has enough projectives. Standard references shows that if $Q_\...
- 38.6k
4
votes
1
answer
230
views
Pontryagin product on the homology of cyclic groups
Consider the cyclic group $C_{p^N}$ of order $p^N$, and let $k$ be a field of characteristic $p$. I would like to know what the algebra structure on the homology $H_*(C_{p^N};k)$ induced by the ...
- 103
1
vote
1
answer
107
views
When do faithfully semiinjective complexes exist?
Question:
For which (perhaps noncommutative but always unital and associative) rings $R$ do faithfully semiinjective complexes of right or left $R$-modules exist?
Hopefully the answer is: "for ...
- 1,021
4
votes
0
answers
93
views
Vershik's conjecture about generic quadratic algebras
Is it still unknown whether very general (lying in a countable intersection of some Zariski opens in corresponding Grassmannian) quadratic algebras $R$ with $\operatorname{dim} R_2 < \frac{3}{4}(\...
- 4,600
1
vote
0
answers
106
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Uniqueness of indecomposable decomposition (Krull–Schmidt) for finitely generated modules over commutative Noetherian standard graded rings
Let $R_0=\mathbb C$ and $R=\bigoplus_{i\geq 0} R_i$ be a commutative Noetherian graded ring such that the grading is standard, i.e., $R=R_0[R_1]$. Let $M$ be a finitely generated $R$-module. Evidently,...
- 412
8
votes
1
answer
356
views
Homological conjectures for finite dimensional commutative algebras
$\DeclareMathOperator\Ext{Ext}\DeclareMathOperator\Hom{Hom}$>Question: What are some (open) homological conjectures that are also relevent for finite dimensional commutative algebras over a field $...
- 26.5k
9
votes
0
answers
194
views
Group cohomology of ${\rm SL}_{n}(\mathbb{F}_p)$ acting on trace zero matrices over $\mathbb{F}_p$
Let $p>5$ be a prime. For any integer $n\geq 2$, let $M_{n}^{0}(\mathbb{F}_p)$ denote the $n\times n$ matrices with trace $0$ over a finite field $\mathbb{F}_p$ of order $p$. Then we have an action ...
- 667
1
vote
0
answers
120
views
Multiplicativity of Euler–Poincaré characteristics of cohomology of pro-$p$ groups
While reading a paper, I found a mentioning that for an extension $1 \rightarrow H \rightarrow G \rightarrow N \rightarrow 1$ of pro-$p$ groups, the Euler–Poincaré characteristics $\chi(H)$, $\chi(G)$,...
- 1,013
1
vote
0
answers
108
views
On Serre's condition and singular locus of determinantal rings
Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...
- 413
3
votes
1
answer
173
views
$\Omega$ for noetherian semiperfect rings
Let $A$ be a a two-sided noetherian semiperfect ring and assume that the injective dimension of the left and right regular modules are equal to $n \geq 1$.
Let $\Omega^n(mod A)$ be the category of $n$-...
- 26.5k
1
vote
0
answers
175
views
Grothendieck trace formula for schemes with étale fundamental groups that have no dense cyclic subgroup
This question may be more of a philosophical rather than mathematical nature.
Assume I have a scheme $X$ and an endomorphism $F:X\longrightarrow X$. For instance, $X$ might be of finite type over $\...
- 1,805
4
votes
1
answer
109
views
Simplicial enrichment on unbounded algebras over an operad
In his paper "Homological Algebra of Homotopy Algebras" V.Hinich introduced a simplicial structure on algebras in unbounded chain complexes over arbitrary $\Sigma$-split operads. Not to get ...
- 1,374
6
votes
1
answer
515
views
Leray spectral sequence and pullbacks
I am trying to find a reference for the following well-known result on the functoriality of the Leray spectral sequence:
Let $\pi:X\to Y$ and $\pi':X'\to Y'$ be morphisms of schemes and denote by $E_2^...
- 177
1
vote
0
answers
83
views
Hochschild homology computation of certain type
I know that in general Hochschild homology is not very computable. However, this certain type of Hoshschild homology shows up and I feel like there could be existing result.
Let $k$ be a field and $A$ ...
- 449
5
votes
0
answers
154
views
Kähler differentials give a left Quillen functor
Is there a reference for the fact that the functor of Kähler differentials is a left Quillen functor on the category of $\mathrm{CDGA}_k/B$? Here $k$ is a field of characteristic $0$, and $B$ is some ...
- 1,374
1
vote
0
answers
135
views
Explicit proof of Quillen's connectivity theorem
Definition
Let $A$ be a commutative ring. An ideal $I \triangleleft A$ is called quasiregular if $I/I^2$ is flat over $A/I$ and there is a canonical isomorphism of algebras
$$
\Lambda_A I/I^2\...
- 1,374
4
votes
0
answers
97
views
Lifting theorem for modules over a DGA
In the survey paper "Cartan's Constructions, the homology of $K(\pi,n)$'s, and some later developments" by J. C. Moore ( http://www.numdam.org/item/AST_1976__32-33__173_0/ ) he states a ...
2
votes
0
answers
145
views
Is it true that $D(\mathcal{O}_Y\text-\mathrm{mod}_X)\cong D_X(\mathcal{O}_Y\text-\mathrm{mod})$?
Is it true that $D(Y_X)\cong D_X(Y)$, where $Y$ is a (smooth) variety, $X$ is a closed subvariety of $Y$ (possibly singular)? Here $D(Y_X)$ denotes the derived category of the abelian category of ...
- 367
3
votes
1
answer
496
views
Regular ring is smooth when the field is perfect
Take $A$ a (not necessarily local) commutative algebra over a field $k$ which is essentially of finite type (i.e. a localization of a finitely generated algebra). In simple words, I just want to know ...
2
votes
0
answers
138
views
Construction of a certain long exact sequence
Let $A$ be a noetherian ring (not necessarily commutative) or for simplicity even a finite dimensional algebra over a field.
Let $X$ and $U$ be finitely generated $A$-modules and let $add(U)$ be the ...
- 26.5k
10
votes
1
answer
956
views
Reference request: infinity categories for the commutive algebraist/algebraic geometer
In a survey article Algebraic geometry in mixed characteristic, B. Bhatt writes
For instance, given a commutative ring $R$ with a finitely generated ideal $I$,
the assignment carrying $R$ to the $\...
- 785
2
votes
0
answers
128
views
On the generalization of a Cech-to-sheaf type spectral sequence
Let $(X,O_{X})$ be a ringed space and $\mathcal{F^{\bullet}}$ be a bounded below complex of $O_{X}$-modules on $(X,O_{X})$. On [Stacks Project 20.25.1][1] it is shown that there is a weakly convergent ...
- 541
3
votes
0
answers
148
views
An account of "Homologie nicht-additiver Funktoren. Anwendungen"'s results
Is there an account in English of results from "Homologie nicht-additiver Funktoren. Anwendungen" by Dold and Puppe? I am mostly interested in the spectral sequence of cross-effects which ...
- 1,374
3
votes
0
answers
248
views
Explicit computation of hyper Ext in terms of the homologies of the input chain complexes
This question was asked on math.stackexchange and didn't receive any traction, so I'm turning to the MO community. Hello!
Let $C_{\bullet}$ and $D_{\bullet}$ be chain complexes of $R$-modules for some ...
- 301
7
votes
2
answers
917
views
Is this exact sequence known?
$\newcommand{\Tors}{{\rm Tors}}
\newcommand{\tf}{{\rm\, t.f.}}
\newcommand{\Gt}{{\Gamma\!,\,\Tors}}
\newcommand{\Gtf}{{\Gamma\!,\tf}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Z}{{\mathbb Z}}
\...
- 14.1k
2
votes
0
answers
84
views
Representation finite Hopf algebras up to stable equivalence
It is well known that every representation-finite group algebra $KG$ is stable equivalent to a symmetric Nakayama algebra.
Question: Is it true that every representation-finite Hopf algebra is stable ...
- 26.5k
6
votes
1
answer
441
views
Where can I find a definition of $\underline{H}^p(X, \mathscr{F})$?
Let $X$ be a topological space and $\mathscr{F}$ a sheaf on $X$. In the paper Tropical cycle classes for non-archimedean spaces and weight decomposition of de Rham cohomology sheaves by Yifeng Liu, ...
- 1,153
8
votes
2
answers
897
views
Derived functors out of an unbounded derived $\infty$-category
Let $\mathcal A$ be an abelian category. In this lecture, Thomas Nikolaus
Defines the unbounded derived category $\mathcal D(\mathcal A)$ as $\mathcal K(\mathcal A)[W^{-1}]$, where $\mathcal K(\...
- 1,217
7
votes
1
answer
411
views
Under which conditions is the bar construction a conservative functor?
The bar construction is a functor $A\mapsto Bar(A)$ from the category of augmented differential graded algebras over a commutative ring $R$ to the category of chain complexes of $R$-modules. It sends ...
- 2,670
6
votes
0
answers
498
views
“Cohomological equation” in dynamical systems
Let $$\dot{x}=Ax+v_r(x)+v_{r+1}(x)+ \dots$$ with $x \in \mathbb{C}^n$ and $v_r: \mathbb{C}^n \to \mathbb{C}^n$ a homogenous, polynomial function of order $r.$
Then, being able to find a suitable $h$ ...
- 1,514
5
votes
1
answer
207
views
Finite lattices that are Koszul
Let $L$ be a finite lattice and $A=KL$ the incidence algebra of $L$.
It should be true that $L$ is modular if and only if the algebra $KL$ is quadratic (since being modular is equivalent to having no ...
- 26.5k
2
votes
0
answers
70
views
Rigid modules for hereditary algebras
Let $A=KQ$ be a path algebra of a connected quiver. (K algebraically closed if it helps)
Question: Is there an explicit classification of all indecomposable $A$-modules $M$ that are rigid, that is $...
- 26.5k
2
votes
1
answer
194
views
Reference request: example of pointwise free module which is not projective
I am looking for a reference for examples showing the following phenomena:
Let $A$ be a commutative noetherian ring, and let $F$ be an $A$-module such that for all $p \in Spec(A)$ it holds that $F_p$ ...
2
votes
1
answer
448
views
Can we compute the Hochschild cohomology for $k[x]$ through the Hochschild complex?
For an algebra $A$ we can define its Hochschild cohomology $HH^{\bullet}(A,A)$ as in this wikipedia page.
Now let $A=k[x]$ be the polynomial ring where $k$ is a field. It is well-known that $HH^{0}(A,...
- 9,019