All Questions
Tagged with ra.rings-and-algebras ct.category-theory
172 questions
2
votes
1
answer
149
views
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
4
votes
1
answer
251
views
Connected Frobenius algebras non-semisimple as an object
A Frobenius algebra object $A$ in a tensor category $\mathcal C$ is said to be connected if $\text{Hom}_{\mathcal C}(\mathbb{1}, A)$ is a one dimensional vector space, where $\mathbb {1} $ denotes the ...
- 43
5
votes
1
answer
179
views
Semisimplicity of algebras in fusion categories
Let $\mathcal{C}$ be a fusion category and $A \in \mathcal{C}$ be an algebra object. We say that $A$ is semisimple if its category of (right) modules $\mathsf{mod}_A(\mathcal{C})$ is a semisimple ...
- 61
2
votes
1
answer
307
views
Serre functors and global dimensions
Let $k$ be a field.
Let $\mathcal{C}$ be an abelian category (over $k$).
We say that $\mathcal{C}$ has a finite global dimension if there exists integer $n > 0$ such that
$$
\operatorname{Ext}^i(M,...
- 889
5
votes
1
answer
367
views
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
5
votes
1
answer
156
views
The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
3
votes
0
answers
152
views
My category is rigid: what this implies for representation theory?
I am studying a subcategory $\mathcal{C}$ of modules for an associative noncommutative algebra $A$ (which is in fact also a Hopf algebra).
It is clear from our definition of $\mathcal{C}$ that it is ...
- 3,318
0
votes
0
answers
67
views
Automorphism groups for simple objects in abelian linear categories
Let $\mathcal{A}$ be an abelian category that is also $k$-linear, where $k$ is some algebraically closed field. Let $X$ be a simple object in $\mathcal{A}$. What can we say about $\mathrm{Aut}(X)$? I ...
2
votes
1
answer
178
views
In a monoidal category with duals is the coevaluation map determined by the evaluation?
For a monoidal category $(\mathcal{M},\otimes,1)$ and an object $X$ with a left dual $X^*$. Let $(e,c)$ be a pair of (co)evaluation maps for the pair $(X,X^*)$. Is it possible to have another map $c': ...
0
votes
1
answer
294
views
Hopf algebras actions
Can you write down a general type of Hopf algebra action? How do you justify the name "action", when it is already used for group actions?
There must be a common core, if the same term is ...
user525442
3
votes
2
answers
227
views
The evaluation and coevaluation maps for an object isomorphic to a dualisable object
Let $X$ be an object in a monoidal category with dual $X^*$ and evaluation and coevaluation maps $e$ and $c$. Now if we have an isomorphism $\sigma:X \to Y$, for some other object $Y$, then $Y$ must ...
0
votes
0
answers
56
views
Names for product-like algebras involving a "duo of directed pseudoforests"
I am looking for the names (and/or for any information regarding) two algebras, one "free" and one "restricted" by an equivalence class.
In both cases, there is an (infix) binary ...
3
votes
1
answer
223
views
Kernels and cokernels in a quotient of an abelian category
I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
- 323
2
votes
1
answer
78
views
Is uniform dimension monotonic in quotients when there is a unique indecomposable injective?
The notion of uniform or Goldie dimension is something I’ve only seen discussed for categories of modules, but I believe the theory works the same way in any Grothendieck category $\mathcal C$. Recall ...
- 63.9k
3
votes
0
answers
202
views
Coevaluation for linear categories
For a field $k$ and an associative $k$-algebra $R$, the $k$-linear category $R\operatorname{-Mod}$ is self dual inside $\operatorname{DGCat}_k$, with the counit map sending $k$ to $R$ regarded as a ...
- 834
5
votes
2
answers
513
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 ...
- 3,446
5
votes
0
answers
285
views
Serre subcategories of the category of chain complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R$ be a commutative $k$-algebra.
We denote by $\operatorname{Mod}(R), C(R), $ and $ D(R)$ the category of $R$-modules, the category ...
- 889
5
votes
1
answer
208
views
Equivalences of categories of complexes of modules
Let $k$ be an algebraically closed field of characteristic $0$.
Let $R, S$ be two commutative $k$-algebras.
Let $\operatorname{Ch}(R), \operatorname{Ch}(S)$ be the categories of complexes of $R$-...
- 889
2
votes
0
answers
162
views
Can End(F) be viewed as a pro-object in the category of finite dimensional algebras?
In EGNO 1.10, we have essentially the following setup: given a $\mathbb{C}$-linear abelian category $\mathcal{A}$ and an exact faithful functor $F: \mathcal{A} \to Vec$ to the category of finite ...
- 264
6
votes
1
answer
859
views
Are algebroids "just matrices"?
$\DeclareMathOperator\Vect{Vect}\DeclareMathOperator\Mat{Mat}$This question was originally asked on MSE but may be better here.
Algebroids are particularly interesting structures: they are basically ...
- 4,897
4
votes
1
answer
186
views
Exact sequences with two FL-modules
Let $R$ be a ring. An $R$-module $M$ is called FL (FP) if it has a finite resolution consisiting of finitely generated free (projective) modules.
Given an exact sequence of $R$-modules, $0\to M_1\to ...
- 1,418
3
votes
0
answers
96
views
Cohn's localization for rings with enough idempotents
I am in the following situation: I have a non-unitary (associative) ring $R$ with enough idempotents or, if you prefer, a small pre-additive category. Actually, I even know that $R$ is right coherent (...
- 2,452
5
votes
0
answers
242
views
Which properties of the pullback/restriction functor imply surjectivity of a ring homomorphism
Let $f:R\to S$ be a homomorhpism of (noncommutative) algebras over some field k (say the complex numbers) and let $F:Rep(S)\to Rep(R)$ be the corresponding functor between the categories of finite-...
- 1,846
5
votes
2
answers
321
views
Cubical vs. simplicial Hochschild cohomology
Simplicial Hochschild cohomology.
$\newcommand{\Hom}{\mathrm{Hom}}\newcommand{\B}{\mathrm{B}}\newcommand{\Obj}{\mathrm{Obj}}\newcommand{\HH}{\mathrm{HH}}\newcommand{\Mod}{\mathsf{Mod}}$One way to ...
- 11.8k
5
votes
0
answers
181
views
Bar constructions and pushouts
Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout.
Is there any hope of ...
- 1,554
17
votes
3
answers
2k
views
Grothendieck's relative point of view and Yoneda lemma
I asked this question on M.SE, but didn't get any answers.
Occasionally I hear people saying that one of Grothendieck's big insights was that often when interested in an object $X$ it's better to ...
- 173
8
votes
2
answers
225
views
Lifting isomorphisms between linear categories
Let $C$ be a $\mathbb{Z}$-linear category, such that $C(x,y)$ is a free abelian group with finite rank, for every $x,y\in\mathrm
{Ob}(C)$. Given a commutative ring with identity $R$, let $RC$ denote ...
- 245
0
votes
0
answers
254
views
The coevaluation map for a projective module and its dual
$\DeclareMathOperator\coev{coev}$Let $R$ be a noncommutative ring and let $P$ be a bimodule over $R$, that is finitely generated and projective as a left module. It is "well-known" that any ...
- 145
2
votes
0
answers
148
views
Product with special requirements on conjugation
Being a non-specialist in algebra, I stumbled upon the following question:
For a real or complex associative algebra $A$, let $A^\times$ denote its group of units. Is there a natural semigroup ...
- 21
25
votes
2
answers
2k
views
What is the name of this relative semidirect product of groups?
We have two well known definitions of the semidirect product $N \rtimes H$ of groups:
(Internal semidirect product) We write $G = N \rtimes H$ if $N$ is a normal subgroup of $G$, $H$ is another ...
- 114k
0
votes
0
answers
145
views
Is it correct to use extensionality axiom in an algebraic theory? Is "extensionality theory" appropriate name for the identity theory plus this axiom?
There is something about extensionality axiom which makes debatable its use in any theory, not only in an algebraic one -- this law looks more like a definition than a statement when written like this:...
- 889
2
votes
1
answer
192
views
Is the theory of the adjunction operation, used for algebraization of hereditarily finite set theory, algebraizable?
The operation used for algebraization of hereditarily finite set theory is named adjunction in 〈A. Tarski, S. Givant, A Formalization of Set Theory without Variables, Providence, RI: American ...
- 889
7
votes
1
answer
377
views
This is not a tensor: tensoring abelian groups over groups
$\newcommand{\Cats}{\mathsf{Cats}}\newcommand{\MonCats}{\mathsf{MonCats}}\newcommand{\BrMonCats}{\mathsf{BrMonCats}}\newcommand{\SymMonCats}{\mathsf{SymMonCats}}\newcommand{\CMon}{\mathsf{CMon}}\...
- 11.8k
4
votes
2
answers
299
views
Relation of the first Hochschild cohomology and the outer automorphism group
Let $R$ be a ring.
Qeustion: Is it true that the first Hochschild cohomology of $R$ is zero if and only if the outer automorphism group of $R$ is finite?
(It is not true, by the two answers. Is it ...
- 26.5k
2
votes
0
answers
141
views
Actions of rings (and other algebraic structures) on abelian categories
On the project I am currently working on, there are abelian, Krull-Schmidt categories $\mathcal{C}$ where it seems natural to equip $\mathcal{C}$ with the action of a ring $R$ (in some cases a ...
- 482
4
votes
0
answers
215
views
Does the tropical semiring admit a universal property?
Each of the rings $\mathbf{Q}$, $\mathbf{Z}_p$, $\mathbf{Q}_p$, $\mathbf{R}$ admits a universal construction: the rationals are the field of fractions of the integers: $\mathbf{Q}:=\mathrm{Frac}(\...
- 11.8k
10
votes
0
answers
446
views
In search of lost graded rings
$\newcommand{\la}[1]{\kern-1.5ex\leftarrow\phantom{}\kern-1.5ex}\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\...
- 11.8k
4
votes
0
answers
190
views
Are PD-, $\lambda$-, $\psi$-, and $\delta$-rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
- 11.8k
9
votes
1
answer
326
views
Are differential rings monoids in a monoidal category?
$\newcommand{\pt}{\mathrm{pt}}\newcommand{\N}{\mathbb{N}}\newcommand{\Z}{\mathbb{Z}}$A number of algebraic structures can be defined as monoids in some appropriate monoidal category:
A monoid ...
- 11.8k
1
vote
2
answers
310
views
Pullback of Lie algebras [closed]
Let $k$ be a field of characteristic 0 and let $\varphi:\mathfrak{g}\rightarrow\mathfrak{f}$ and $\psi:\mathfrak{h}\rightarrow\mathfrak{f}$ be maps of Lie algebras. Is there a reference showing that ...
- 85
2
votes
0
answers
76
views
Diagrammatic model for free product in monad infinity category
$\newcommand{\C}{\mathcal{C}}$ Suppose $M$ is a monad in an $\infty$-category $\C,$ and $A, B$ are two algebras over $M$. I'm willing to assume any reasonable "niceness" conditions on $\C$, $...
- 7,952
29
votes
3
answers
2k
views
Categorification of determinant
The notion of trace of a matrix can be generalized to trace of an endomorphism of a dualizable objects in a symmetric monoidal category. (See Ponto & Shulman for a nice description.)
Is there a ...
- 290
1
vote
0
answers
142
views
How can I build free unital magmas?
N. Bourbaki formally defines the free magma $M(X)$ over a set $X$. However, it does not define the free unital magma over $X$, which I am denoting by $M^{\ast}(X)$ (maybe you know some more common ...
- 353
6
votes
1
answer
215
views
What is the smallest group for which Broué's abelian defect group conjecture has not yet been verified?
Let $G$ be a finite group. Let $p$ be a prime dividing $|G|$. Let $k:=\overline{\mathbb{F}_p}$.
Let $b$ be a $p$-block of $kG$ with abelian defect group $D$. Let $H:=N_G(D)$. Let $c$ be the Brauer ...
- 1,755
10
votes
2
answers
473
views
Varieties where every algebra is projective?
Is it possible to classify all varieties (in the sense of universal algebra) where every algebra is projective?
Several years ago I asked a similar question, with "free" in place of "...
- 63.9k
5
votes
0
answers
135
views
Can a braided fusion category have an order-2 Morita equivalence class which cannot be simultaneously connected and isomorphic to its opposite?
Let $\mathcal{B}$ be a braided fusion category over $\mathbb{C}$. Let me write $\mathrm{Alg}(\mathcal{B})$ for the set of isomorphism classes of unital associative algebra objects in $\mathcal{B}$, ...
- 54.6k
5
votes
0
answers
241
views
Growth rate of cohomology
Fix a finite dimensional graded vector space $V$, a differential $d$ on $V \otimes V $ $(i.e.\,\, d^2=0\,\, and \,\, deg\, d =1)$ such that $(d_{1,2} \otimes id_3)$ anti-commutes with $((-1)^{deg}_1 \...
- 189
7
votes
1
answer
969
views
When is the category of finitely presented modules abelian?
Let $R$ be an associative ring with identity and $\mathrm{mod}R$ be the category of finitely presented $R$-modules. I would like to know when the category $\mathrm{mod}R$ is abelian.
I know that if $R$...
- 75
4
votes
1
answer
372
views
On the tree-ishness of magmas and the stringiness of groups
Let me start off by saying that I suspect the answer to my question might fall under the domain of universal algebra, which is why I'm giving it that tag. However, I know only the very basics of ...
- 309
19
votes
1
answer
473
views
Vopěnka's principle and contravariant full embeddings between module categories
I was recently reminded about this old question on math.stackexchange.
Let $\operatorname{Mod}R$ be the category of (right) modules for a ring $R$. The questioner mistakenly thought that the Freyd-...
- 35.2k