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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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
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The size of $\operatorname{Ext}^1(M,N)$ for $M$, $N$ modules of $\operatorname{SL}_n(\mathbb{F}_q)$

I was just wondering has there been any estimation or exact evaluation on the size (cardinality) of $\operatorname{Ext}^1(M,N)$, where $M$, $N$ are modules of $\operatorname{SL}_n(\mathbb{F}_q)$? What ...
user236626's user avatar
2 votes
0 answers
37 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
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3 votes
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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
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6 votes
1 answer
355 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
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2 votes
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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
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2 votes
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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
123 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
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4 votes
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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
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4 votes
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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
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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 $\tau_k,\...
Adrien's user avatar
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1 answer
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$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
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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
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7 votes
1 answer
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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
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216 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
143 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
110 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
78 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
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2 votes
0 answers
39 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
210 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
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3 votes
1 answer
131 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
218 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
329 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
340 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
175 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
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6 votes
1 answer
259 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
346 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
148 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
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2 votes
0 answers
67 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
  • 313
2 votes
1 answer
137 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
  • 73
3 votes
2 answers
211 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
5 votes
0 answers
151 views

Are there known minimal models for the cohomology of semisimple Lie algebras?

My student and I recently found a cute construction of a minimal model for the cohomology of a Lie algebra $\mathfrak{g}$. This is a "minimal model" in the sense that it is a minimal chain-...
user509184's user avatar
6 votes
1 answer
281 views

What conditions on an Abelian category allow members of a direct sum to be determined entirely by their components?

EDIT: In comments, with thanks to Maxime Ramzi, this question has a good answer in that what I want to be true is true when $\mathscr{A}$ satisfies axiom $\mathsf{AB}5$, that $\mathscr{A}$ is ...
FShrike's user avatar
  • 487
2 votes
1 answer
131 views

Is $\mathrm{Hom}_R(M,R)\neq \mathfrak m \mathrm{Hom}_R(M,R)$ if $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$?

Let $(R,\mathfrak m)$ be a commutative Noetherian local ring and $M$ be an $R$-module such that $M \neq \mathfrak m M$ and $\mathrm{Hom}_R(M,R)\neq 0$. Then, is it true that $\mathrm{Hom}_R(M,R)\neq \...
Snake Eyes's user avatar
5 votes
2 answers
391 views

How is the classification of groups extensions by $H^2$ related to Yoneda Ext?

It is well-known that group extensions $$1\to A \to H \to G \to 1$$ where $A$ is abelian with a $G$-action such that the conjugation action of $G$ on $A$ agree with this fixed action are classified ...
Antoine Labelle's user avatar
1 vote
1 answer
133 views

Why is this map a split monomorphism?

I have a question regarding a lemma in the proof of Hopkins-Neeman Correspondence. It is the beginning part of Lemma 1.2 in the The Chromatic Tower for D(R) Let $Y$ be an object of the derived ...
Subham Jaiswal's user avatar
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0 answers
160 views

A complex with homology $=R/p$

Given a Noetherian ring $R$ . I am looking for a bounded complex $X$ of finitel geenerated projectives over $R$ whose homology is $R/p$. Infact I just need $X$ to have $\operatorname{Supp}(H(X)) = \...
Subham Jaiswal's user avatar
3 votes
1 answer
217 views

Vanishing of $\operatorname{Ext}_R^{1}(M,R)$ when $R$ is a Gorenstein local ring of dimension $1$ and $M$ is not finitely generated

Let $(R,\mathfrak m)$ be a Gorenstein local ring of dimension $1$. Let $M$ be an $R$-module (not finitely generated) such that $M\neq \mathfrak m M$ and there exists a non-zero-divisor $x\in \mathfrak ...
Snake Eyes's user avatar
0 votes
1 answer
339 views

The Krull dimension of the tensor product of rings

The Krull dimension of a ring $R$ is defined as the length of the longest chain of prime ideals in it. Let $R_i$, for $i\in\mathbb{N}$ denote a sequence of commutative Noetherian rings of Krull ...
rr314's user avatar
  • 25
1 vote
0 answers
131 views

When is a functor of chain complexes triangulated?

Let $\textsf{A}, \textsf{B}$ be abelian categories. Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...
Jannik Pitt's user avatar
  • 1,103
7 votes
0 answers
262 views

Homotopy theory of differential objects

In Kashiwara and Schapira's wonderful book Categories and Sheaves, they define a category with translation to be a category $\mathsf{C}$ equipped with an auto-equivalence $S: \mathsf{C} \to \mathsf{C}$...
ಠ_ಠ's user avatar
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3 votes
1 answer
154 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 ...
Chase's user avatar
  • 93
3 votes
0 answers
88 views

Confusion about signs in the definition of an $A_\infty$-algebra

We are trying to understand the definition of $A_\infty$-algebras. But we are puzzled by what appear to be two different sign conventions (and we cannot figure out how these two are equivalent) We see ...
ainfg's user avatar
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4 votes
0 answers
518 views

$\mathbb{Z}[T]$-Solidification in light condensed setting

In the lectures to "Analytic Stacks" Scholze and Clausen introduced a new concept of "light" condensed mathematics. In Lecture 7 Clausen introduces the derived $T$-solidification ...
Jonas Heintze's user avatar
3 votes
1 answer
183 views

On infinity-morphisms between algebras over algebraic operads

I posted this question in the "Mathematics" stack exchange, but it hasn't got much attention... I hope it will get more here. Let $P$ be a Koszul operad. In the book of Loday-Vallette "...
groupoid's user avatar
  • 197
1 vote
1 answer
93 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 ...
FShrike's user avatar
  • 487
1 vote
0 answers
104 views

Lengths and additive invariants which preserve positivity

The length of a module is well-known to be an additive invariant of finite-length modules. That is, if $R$ is a ring and $Art(R)$ its category of finite-length modules, then $length : Ob (Art(R)) \to \...
Tim Campion's user avatar
  • 60.5k
1 vote
1 answer
282 views

Why should we study the total complex?

Recall that for every double complex $C_{\bullet,\bullet}$, there is a canonical construction called the total complex $\operatorname{Tot}(C_{\bullet,\bullet})$ associated to it. This complex can be ...
mrtaurho's user avatar
  • 165
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0 answers
83 views

What happens if I take a doubly-free simplicial abelian group?

Suppose that I have a simplicial set $X_\bullet$. I can take the free abelian group generated by $X_\bullet$, $\mathbb{Z}X_\bullet$. But then I can forget that this has an abelian group structure, ...
Inna's user avatar
  • 29
3 votes
1 answer
216 views

Hattori-Stallings trace

Let $R$ be a (possibly non-commutative) unital ring and $M$ be a left $R$-module. If $M$ is finitely generated and projective, the natural map $$\iota:\mathrm{Hom}_R(M,R)\otimes_R M\to \mathrm{Hom}_R(...
Qwert Otto's user avatar

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