Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
21 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 ...
Alex's user avatar
  • 365
3 votes
1 answer
133 views

Minimality of the Koszul resolution

Let $R = \mathbb{C}[x,y]$ and $V = \mathbb{C}x\oplus\mathbb{C}y$. Then, the Koszul resolution of $R$ (as an $R$-bimodule) is given by \begin{align*} 0\to R\otimes_{\mathbb{C}}\wedge^2V\otimes_{\mathbb{...
Qwert Otto's user avatar
0 votes
0 answers
60 views

Matrix of the minimal projective presentation of a $\tau$-rigid module

Let $A$ be a finite dimensional algebra over an algebraically closed field $\mathbb{K}$ of characteristics zero. Suppose $A$ is given by the bound quiver $(Q,I)$. We will use $P_l$ to denote the ...
It'sMe's user avatar
  • 767
3 votes
1 answer
86 views

Thick subcategory containment in bounded derived category vs. singularity category

Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
Alex's user avatar
  • 365
3 votes
0 answers
64 views

Connection between certain finite groups and Frobenius algebras

This question is maybe not so precise yet, but contains some ideas that maybe just search for the right definition. Let $G=F/U$ be a finite group with presentation $F/U$ (here the presentation really ...
Mare's user avatar
  • 26.2k
2 votes
0 answers
78 views

Koszul cohomology associated with a regular sequence

Let $A$ be a local Noetherian ring and $M$ be an $A$-module. Let $\mathfrak{a}$ be an ideal of $A$ generated by a regular $M$-sequence $s_1,\cdots,s_r$. Let $K_\bullet(s_1,\cdots,s_r;M)$ be the Koszul ...
Li Li's user avatar
  • 403
18 votes
0 answers
260 views
+50

Are these comparison morphisms between Čech and Grothendieck cohomology the same?

For better or for worse, there is more than one approach to comparing Čech cohomology $\smash{\check{H}}^\bullet(\mathfrak{U},X;\mathscr{F})$ of a sheaf $\mathscr{F}$ on a space $X$ w.r.t the cover $\...
FShrike's user avatar
  • 821
2 votes
0 answers
41 views

$K_0$-basis modules with a unique extension related to parking functions

Let $B=B_n$ be the quiver algebra of type $A_n$ (with some orietnation) with $n$ points. A $B$-module $M$ is a basis of the Grothendieck group $K_0$ if $M$ has $n$ indecomposable direct summands and ...
Mare's user avatar
  • 26.2k
6 votes
1 answer
269 views

Factoring through projective modules is an equivalence relation

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PHom{PHom}$I'm reading about stable module categories, and I have a question about the definition of the maps. Let $R$ be a ring, and take (left) ...
StuckInTheFridge's user avatar
16 votes
3 answers
928 views

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 ...
Mare's user avatar
  • 26.2k
3 votes
0 answers
149 views

Derived $\infty$-category of quasi-coherent sheaves on schemes

Let $X$ be a scheme. On the one hand, we have the derived $\infty$-category constructed from the abelian category of quasi-coherent sheaves on $X$. On the other hand, we can define the stable $\infty$-...
Y.M's user avatar
  • 71
4 votes
1 answer
109 views

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 ...
Rocky Smith's user avatar
6 votes
0 answers
245 views

Applications of Banach space homology

There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
Andromeda's user avatar
  • 249
4 votes
1 answer
242 views

$\mathbb{Z}$-homomorphism and $\mathbb{Z}_p$-homomorphism

$\newcommand{\cts}{\mathrm{cts}}$Thanks for your reading. Let $A,B$ be two $\mathbb{Z}_p$-modules, where $\mathbb{Z}_p$ is the $p$-adic integer ring. I have two questions. Is $\mathrm{Hom}_{\mathbb{Z}...
Rellw's user avatar
  • 61
2 votes
0 answers
111 views

Quasi-isomorphisms of P-algebras

In the paper "Homotopy algebras are homotopy algebras" from Markl a notion of strong homotopy morphism between strong homotopy P-algebras is defined. The author restricts to the case where $...
groupoid's user avatar
  • 215
2 votes
0 answers
48 views

Relationship between the homology of two types of tensor products of $\mathbb{Z}/ 2 \mathbb{Z}$-graded objects?

Let's consider a $2$-periodic complex $F$ of free $R$-modules, which is just a $\mathbb{Z} / 2 \mathbb{Z}$-graded complex $$F_1 \xrightarrow{d_1} F_0 \xrightarrow{d_0} F_1$$ (really the arrow $d_0$ ...
Rellek's user avatar
  • 401
3 votes
1 answer
223 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 $\...
Jannik Pitt's user avatar
  • 1,181
1 vote
0 answers
96 views

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_\...
Benjamin Steinberg's user avatar
6 votes
1 answer
161 views

Known posets of tilting modules for finite dimensional algebras

Question: For which classes of finite dimensional algebras $A$ is the poset of tilting $A$-modules known? Here two famous examples: -For the path algebra of a linear oriented quiver of Dynkin type $...
Mare's user avatar
  • 26.2k
2 votes
1 answer
187 views

Regular sequence in cohomology of Grassmannians

$\DeclareMathOperator\Gr{Gr}$Consider the polynomial ring $\mathbb{Z}[x_1,\dots,x_m, y_1,\dots,y_n]$, I want to prove that the sequence $$x_1 + y_1, x_2 + x_1y_1 + y_2, \dots, x_my_{n-1} + x_{m-1}y_n, ...
atticusw's user avatar
1 vote
0 answers
77 views

Is $(I(R:_{Q(R)} I))^n$ generated by $(fI)^n$ as $f$ varies over $(R:_{Q(R)} I)$?

Let $(R, \mathfrak m)$ be a Noetherian local domain of dimension $1$ which is not a UFD. Let $Q(R)$ be the fraction field of $R$. If $I\subsetneq \mathfrak m$ is a non-zero, non-principal ideal of $R$ ...
Alex's user avatar
  • 365
2 votes
0 answers
44 views

Depth and codepth of an algebra

Let $A$ be a finite dimensional $K$-algebra over a field $K$ and $0 \rightarrow A \rightarrow I^0 \rightarrow I^1 \rightarrow \cdots$ a minimal injective coresolution of the regular module $A$. The ...
Mare's user avatar
  • 26.2k
3 votes
0 answers
138 views

Is taking Freyd envelopes adjoint to taking stable module categories?

Let $T$ be an (idempotent-complete) triangulated category. Then the Freyd envelope $mod(T)$ is an abelian category, the universal recipient of a homological functor $T \to mod(T)$. The Freyd envelope ...
Tim Campion's user avatar
  • 61.7k
6 votes
1 answer
371 views

Do acyclic amenable groups exist?

Is there an example of a nontrivial discrete amenable group with vanishing integral homology? To put the question in contrapositive. Given arbitrary acyclic group $Q$, is there some reason for the ...
Denis T's user avatar
  • 4,436
2 votes
0 answers
62 views

Bounds for sum of the homological dimensions in the incidence algebra of a Boolean lattice

Let $A$ be a finite dimensional algebra. Define $\varphi_A:= \sup \{ \operatorname{pd} M + \operatorname{id} M \mid M \in \operatorname{ind}(A) \}$, where $\operatorname{pd} M$ denotes the projective ...
Mare's user avatar
  • 26.2k
2 votes
1 answer
139 views

Are projective tensor products left-exact if one considers only maps of norm at most 1?

Consider the category $\mathrm{Ban}$ of Banach spaces and bounded linear maps and the category $\mathrm{Ban}_1$ of Banach spaces and bounded linear maps of operator norm at most 1. Let $\otimes_\pi$ ...
Stephan Mescher's user avatar
2 votes
2 answers
125 views

Infinite radical ideal cubed equals zero for tame hereditary Artin algebras

Let $A$ be a tame hereditary Artin algbera and mod$A$ the category of finitely generated (left) $A$-modules. Further, let rad$_A$ be the radical ideal of mod$A$, which is the smallest ideal containing ...
kevkev1695's user avatar
  • 1,023
4 votes
1 answer
308 views

Gluing objects of derived category of sheaves

Let $X$ be a locally compact topological space (may be assumed to be a stratified space with finite stratification). Let $\{U_i\}$ be an open finite covering. Assume that over each $U_i$ we are given ...
asv's user avatar
  • 21.1k
4 votes
1 answer
190 views

Does hereditary and connected imply that the underlying ring $k$ of a $k$-algebra is a field?

All rings are assumed to be associative and have a 1. Let $k$ be a commutative artininan ring and $R$ a finitely generated $k$-algebra. Is it true that if $R$ is connected and hereditary, then $k$ is ...
kevkev1695's user avatar
  • 1,023
3 votes
1 answer
127 views

Are the two families of Johnson invariants of the Torelli groups related beyond the first one?

$\newcommand{\sp}{\operatorname{Sp}(H)}$ $\newcommand{\gr}{\operatorname{gr}}$ $\newcommand{\id}{\operatorname{id}}$ $\newcommand{\der}{\operatorname{Der}}$ Johnson has defined two families $\tau_k,\...
Adrien's user avatar
  • 8,369
0 votes
1 answer
99 views

$S/I$-freeness of $I/I^2$ vs $I/I^{(2)}$, where $I$ is a radical ideal of regular local ring $S$

Let $I$ be a radical ideal of a regular local ring $S$. Put $R:=S/I$. Let $I^{(n)}$ be the $n$-th symbolic power of $I$. It is well-known that $I^n \subseteq I^{(n)}$. Is it true that $I/I^2$ is $R$-...
uno's user avatar
  • 280
5 votes
0 answers
184 views

Is there a way to “derive” a (non-exact) functor which preserves images?

Let $F : \mathcal A \to \mathcal B$ be an additive functor between abelian categories. If $F$ is left exact, then under certain conditions $F$ admits right derived functors which “measure” its failure ...
Tim Campion's user avatar
  • 61.7k
7 votes
1 answer
111 views

Semi-simple algebras over operads

I believe people thought about this questions, however I couldn't find any reference. I appreciate if someone could direct me to some detailed discussions about it. The categories of associative ...
Li Guanyu's user avatar
  • 449
9 votes
1 answer
222 views

Formal smoothness of path algebras and connections

Let $k$ be a field of characteristic zero and $A = kQ$ the path algebra associated with a quiver $Q$. The algebra $A$ is said to be formally smooth over $k$ if $$ \Omega^1_kA = \operatorname{Ker}(\...
Qwert Otto's user avatar
2 votes
0 answers
150 views

Connection on relative topological periodic cyclic homology

I have been looking Bhatt-Morrow-Scholze's paper: https://arxiv.org/pdf/1802.03261.pdf and came to a naive question. Let $C$ be a dg-category (with assumptions?) over $\mathbb{F}_p[[z]]$ and view this ...
Daniel Pomerleano's user avatar
2 votes
1 answer
112 views

Projective dimension and subrings

$\DeclareMathOperator\pd{pd}$Suppose that $R$ is a commutative ring and $R'$ is a subring of $R$ such that $R$ is a free $R'$-module of finite rank. Assume that both $R$ and $R'$ are regular local ...
Ahmed Matar's user avatar
2 votes
0 answers
80 views

Splitting of $\mathbb{Z}/p\to E\to (\mathbb{Z}/p)^n$ in cohomological terms

Let $d>1$ be an odd integer. Given a simplicial set $X$ and $[\gamma]\in H^2(X,\mathbb{Z}/d)$, there exists a fibration $N\mathbb{Z}/d\to E\to X$, with $$E= X_\gamma:=N\mathbb{Z}/d\times_{\gamma} X....
Antoine's user avatar
  • 143
2 votes
0 answers
43 views

When can GKZ setup encompass HMS?

Are there any instances when the Landau-Ginzburg superpotential describing the mirror of a smooth projective Fano variety $X_\Sigma$ is encompassed by a GKZ hypergeometric system? In some sense I am ...
locally trivial's user avatar
6 votes
1 answer
231 views

A formula for the projective dimension of finite dimensional algebras

Let $A$ be a finite dimensional ring-indecomposable $K$-algebra that is not selfinjective for $K$ a field and let $I(A)$ denote the injective envelope of the regular module $A_A$. Define the stable $A$...
Mare's user avatar
  • 26.2k
3 votes
1 answer
134 views

Linearity of topological periodic cyclic homology

Let $A$ be an $E_\infty$ ring spectrum, $B$ a ring spectrum. Then if I understand correctly, $TP(A)$ is a ring spectrum by the lax monoidal property of $TP$. Suppose there is a map of ringed spectra ...
onefishtwofish's user avatar
5 votes
1 answer
222 views

Comparing stabilization of stable category modulo injectives and a Verdier localization

Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
Snake Eyes's user avatar
4 votes
1 answer
339 views

How to calculate $\mathrm{TP}(\mathbb{F}_p[t])$?

$\DeclareMathOperator\TP{TP}$I am trying to learn about topological periodic cyclic homology following the notes: https://www.uni-muenster.de/IVV5WS/WebHop/user/nikolaus/Papers/Lectures.pdf https://...
onefishtwofish's user avatar
9 votes
0 answers
347 views

How large must the characteristic of $k$ be, for the cohomology of the Lie algebra $\mathfrak{sl}_n(k)$ to be exterior as in characteristic zero?

$\DeclareMathOperator\SU{SU}$In this question, all Lie algebra cohomology is of the form $H^*(\mathfrak{g}; k)$, with $k$ the trivial one-dimensional representation of $\mathfrak{g}$. All Lie algebra ...
user509184's user avatar
5 votes
1 answer
176 views

Are module finite algebras over semiperfect rings again semiperfect?

Let $S$ be a Noetherian semiperfect ring (https://en.m.wikipedia.org/wiki/Perfect_ring). Let $R$ be a module finite associative $S$-algebra. Then, is $R$ also a semiperfect ring? (Clearly, $R$ is ...
uno's user avatar
  • 280
6 votes
1 answer
272 views

Comparing the Stacks Project Homotopy limit with limits in the $\infty$-category

In the Stacks project Tag 08TC, there is a definition of a homotopy limit in a derived category, and I expect it to compare with a limit in the $\infty$-categorical enhancement. I guess this is also ...
user141099's user avatar
4 votes
1 answer
365 views

Two spectral sequences arising from a simplicial spectrum

Let $X_\bullet$ be a simplicial spectrum, and let $X = |X_\bullet|$ be the geometric realization. Let's assume each $X_n$ is connective. From this situation, we can form two filtrations on $X$: the ...
Brian Shin's user avatar
2 votes
1 answer
161 views

How does the behaviour of a hyperderived functor of many variables change if you use $\prod$-totalisation instead of $\oplus$-totalisation?

$\newcommand{\tot}{\operatorname{Tot}}\newcommand{\A}{\mathscr{A}}\newcommand{\L}{\mathbb{L}}\newcommand{\R}{\mathbb{R}}$Say $T$ is is a functor $\A_1\times\A_2\times\cdots\times\A_n\to\A$ of Abelian ...
FShrike's user avatar
  • 821
2 votes
0 answers
70 views

From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences

Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...
Alex's user avatar
  • 365
2 votes
1 answer
140 views

Using the mapping cone to show that a chain map defines a stable equivalence between two symmetric algebras

This question is about an argument in the proof of Theorem 9.8.8 in Linckelmanns Block Theory of Finite Group Algebras. I need to understand the argument in order to do something similar in my ...
jb2g4's user avatar
  • 75
3 votes
2 answers
217 views

Adjunctions and inverse limits of derived categories

Consider a tower $\dots\to A_{2}\to A_{1}$ of rings. This gives rise to a diagram $\mathbb{N}^{\text{op}}\to\text{Cat}_{\infty}$ of $\infty$-categories (confusing $\mathbb{N}^{\text{op}}$ with its ...
user141099's user avatar

1
2 3 4 5
53