24
votes
Accepted
If a polynomial ring is finite free over a subring, is the subring polynomial?
Note that $R$ and $S$ each have only one graded maximal ideal, so they are local in the graded sense, so most of the standard results for ungraded local rings are applicable. There is an obvious ...
- 56.9k
20
votes
Accepted
Classification of subgroups of finitely generated abelian groups
The answer to Question 1 is no.
Let $A=\mathbb{Z}/8\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$
and let $B$ be the subgroup generated by $(2,1)$.
Since $B$ is cyclic of order $4$, if it were contained in a ...
- 35.2k
19
votes
Accepted
When is $A\otimes R$ a free $R$-module?
Here is a 7-line proof of your statement.
Claim. Let $R$ be a commutative ring with identity $1_R$. Let $A \simeq \mathbb{Z}/d_1\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/d_k\mathbb{Z}$ be a ...
- 7,893
19
votes
Accepted
Reference request: a tale of two mathematicians
Here is a recent talk by Ringel (in German):
Algebra und Kombinatorik.
The related part is:
Besonderes Aufsehen hat ein Ergebnis erregt, das meist Satz von Gabriel genannt wird: Ein zusammenhängender ...
- 26.5k
18
votes
Accepted
Is every additive, left exact functor isomorphic to a hom functor?
One can prove the following result: let $F \colon \mathrm{Mod}_R \to \mathrm{Mod}_S$ be left-exact and preserve small products (equivalently, a continuous functor). Then $F$ is of the form $\mathrm{...
- 40.2k
17
votes
Accepted
Tilting Objects in BGG Categories $\mathcal{O}$
Words change their meanings.
The original meaning of “tilting module” is that of Happel and Ringel in the representation theory of finite dimensional algebras, which requires the projective dimension ...
- 35.2k
17
votes
Accepted
Categorical Unification of Jordan Holder Theorems
This does not really involve any category theory, but perhaps it is useful to note the following general setting for the Jordan-Hölder theorem.
For $G$ a group and $\Omega$ a set, a group with ...
- 2,821
17
votes
Accepted
For what modules is the endomorphism ring a division ring?
Such modules are called bricks for finite dimensional algebras and there are in general very many of them.
Having a division ring as the endomorphism ring is equivalent to the condition that every non-...
- 26.5k
17
votes
Accepted
When is the category of finitely presented modules abelian?
Wojowu's idea is right:
Lemma. Let $R$ be a ring, let $\mathbf{Mod}_R$ be the category of (left) $R$-modules, and let $\mathbf{Mod}_R^{\text{fp}}$ be the subcategory of finitely presented modules. ...
- 28.7k
16
votes
Clebsch–Gordan decomposition formula for algebraic groups
Let $G$ denote the group, and suppose one has an enumeration of its irreducible representations $V_{\lambda}$ by some combinatorial objects $\lambda$, like nonnegative integers for $SU_2$ (or for ...
15
votes
How to make Ext and Tor constructive?
Jarl Flaten and I have developed the theory of Ext groups constructively in homotopy type theory, and have formalized most of the results in Coq. We use Yoneda's resolution-free approach to Ext, so ...
- 1,092
15
votes
Dual of a bimodule
As explained in more detail in this blog post linked by Jakob in the comments, every $(A, B)$-bimodule $M$ has two natural duals:
If $M$ is finitely generated projective as a left $A$-module, it has ...
- 118k
14
votes
Categorical Unification of Jordan Holder Theorems
There are various generalizations of the Jordan-Hölder theorem. Beyond groups and groups with operators it holds for any equational theory which contains a Mal'cev operation. This means that from the ...
- 15.5k
14
votes
Accepted
Is the category of chain complexes a reflexive or coreflexive subcategory of the category of functors?
Yes, it's reflective and coreflective, under mild assumptions on the codomain category $\mathcal A.$ The adjoints are given, by definition, by Kan extension along the quotient from the abelian group-...
- 3,411
13
votes
Two abelian groups, each being direct factor of the other
No. A classic result of Corner (On a conjecture of Pierce concerning direct decompositions of Abelian groups. 1964 Proc. Colloq. Abelian Groups (Tihany, 1963) pp. 43–48, MR0169905 (30 #148)) shows ...
- 7,207
12
votes
A question with simple and indecomposable modules
It's not true.
Consider representations of the quiver
$$\bullet\stackrel{\alpha}{\rightarrow}\bullet\stackrel{\beta}{\leftarrow}\bullet.$$
The representation
$k \to k^2
\leftarrow k$, where the ...
- 35.2k
12
votes
Accepted
An example of a non-geometric $C^\infty(M)$-module
Let $M$ be the unit circle in $\mathbb C$, and consider the algebra homomorphism $C^\infty(M)\to M_2(\mathbb R)$ given by $$ f\mapsto \begin{bmatrix} f(1) & \frac{df}{d\theta}(1) \\ 0 & f(1)\...
- 830
12
votes
Accepted
local ring all whose non-maximal ideals are finitely generated
There exists no such non-noetherian local ring.
Below I assume by contradiction that we have such a ring.
(a) The first observation is a particular case Proposition 1.2(a) in your reference to ...
- 63.9k
12
votes
Accepted
Inverse of the Structure Theorem for Finitely Generated Modules over PID
There seems to be quite some literature about rings with this property, sometimes under the name "FGC domains".
From Googling, not personal knowledge:
In Theorem 14 of
Kaplansky, Irving, Modules ...
- 35.2k
12
votes
Beauville-Laszlo for schemes
Completing the discussion under Will Sawin's answer. The question has been answered completely and affirmatively by Ben-Bassat and Temkin in their paper "Berkovich spaces and tubular descent" (Adv. ...
- 16.1k
12
votes
Accepted
For every ring R, is there a block-diagonal canonical form for a square matrix over R?
Your question is equivalent to whether the category $\mathcal{E}$ of pairs $(V,f)$ consisting of a finitely generated free (right) $R$-module and an endomorphism $f$ of $V$ is a Krull-Schmidt category,...
- 2,928
11
votes
Accepted
Reduced ring with all non-prime ideals finitely generated
Question: Let $R$ be a reduced ring with all
non-prime ideals finitely generated. Then is $R$ Noetherian?
The answer is Yes.
To lessen my typing, let me use the abbreviation
NFG to mean not-finitely-...
- 14.6k
11
votes
Accepted
Basis for free modules over an affine domain
1) In the positive direction: If any entry in $e$ is an $(n-1)!$ power, then $e$ is an element of a basis. (More generally, it's enough to have $e=(z_1^{m_1},\ldots z_n^{m_n})$ with $(n-1)!$ ...
- 23k
11
votes
Accepted
Must the inclusion of an indecomposable module in the direct sum of two copies always split?
Yes, it must be split.
Since $M$ is an indecomposable module for an Artin algebra, its endomorphism ring $E$ is a local ring with nilpotent Jacobson radical $J(E)$. Say $J(E)^n=0$.
Let the ...
- 35.2k
11
votes
Accepted
Is there essentially unique notion of module over monoidal stable $\infty$-categories?
The ∞-categorical analog of the fact you mention can be found in Higher Algebra, corollary 7.3.4.14:
Let $\operatorname{CAlg}$ be the category of $E_\infty$-rings and $A\in \operatorname{CAlg}$. Then ...
- 16.5k
11
votes
Accepted
Trivial group cohomology induces trivial cohomology of subgroups
For any abelian group $A$ we have a canonical isomorphism $\bigwedge^2A\to H_2(A,\mathbb{Z})$, given by the (anti-symmetric) Pontrjagin product $H_1(A,\mathbb{Z})\times H_1(A,\mathbb{Z}) \to H_2(A,\...
- 16.2k
11
votes
Accepted
Symmetric tensor of highest weight modules for $\mathrm{SU}(2)$
The multiplicities of $V_e$ inside $n$-th symmetric power of $V_d$ are given by the Cayley-Sylvester formula
$$
P(n,d;\frac{nd-e}{2}) - P(n,d;\frac{nd-e}{2}-1)
$$
where $P(n,d;k)$ denotes the number ...
- 8,597
10
votes
The projective covers of Artinian module
No: as soon as the (local noetherian) ring itself is not artinian, there exists an artinian module with no projective cover.
First recall (over any ring) that a projective cover of $M$ is $N$ ...
- 63.9k
10
votes
Accepted
Rational Canonical Form over $\mathbb{Z}/p^k\mathbb{Z}$
The problem is open, and not because nobody tried. For instance, it is known that the number of similarity classes in $M_n(\mathbf Z/p^2 \mathbf Z)$ is equal to the number of simultaneous conjugacy ...
- 5,654
10
votes
Accepted
Is an overring of an order reflexive as a module over the order?
The answer to Question 1 is yes by (the proof of) Proposition 2.14 in this paper.
The answer to Question 2 is not. Let $R=k[t^3,t^4,t^5]$ so $\tilde R=k[t]$. I claim that for this ring any immediate ...
- 30.5k
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