All Questions
6,055 questions
47
votes
2
answers
5k
views
Why do we care whether a PID admits some crazy Euclidean norm?
An integral domain $R$ is said to be Euclidean if it admits some Euclidean norm: i.e., a function $N: R \rightarrow \mathbb{N} = \mathbb{Z}^{\geq 0}$ such that: for all $x, y \in R$ with $N(y) > 0$,...
- 65.4k
14
votes
2
answers
921
views
Why is the symmetric monoidal structure on invertible modules strict?
Let $N$ be an object in a symmetric monoidal category. Then the braid map $N\otimes N\to N\otimes N$ is almost never the identity, and this is the obstruction to making a symmetric monoidal category ...
- 31.2k
0
votes
1
answer
2k
views
Dual of Zorn's Lemma? [closed]
It seems to me that the dual of Zorn's Lemma should be true: if $S$ is a non-empty partially ordered set and every chain of $S$ has a lower bound in $S$, then $S$ has at least one minimal element.
...
- 1
6
votes
1
answer
858
views
Exotic isomorphism of matrix rings
Let R and S be commutative rings with a 1 different from zero. Let m and n be positive integers. Assume the ring of m-by-m matrices over R is isomorphic to the ring of n-by-n matrices over S. Does ...
user5810
2
votes
1
answer
286
views
Linear independence in the algebraic closure of $\mathbb{C}(z)$
Fix $N>0$. Let $b_i=(b_{i,1}, b_{i,2}, b_{i,3}, b_{i,4})$, $i=1,\ldots, m$, be distinct 4-tuples of integers with with all $0\leq b_{i,j}< N$. (The zero tuple is disallowed.)
Define $w_i=(\...
- 454
3
votes
0
answers
180
views
Generic Rank of R^{1/p}
Suppose $R$ is a local Noetherian domain of dimension $d$ in characteristic $p>0$. Suppose $R^{1/p}$ is a finitely generated $R$-module, and suppose $k$ is the residue field of $R$. Is the ...
- 103
6
votes
0
answers
238
views
Moduli space of modules with fixed length
Let $R$ be a (commutative) local Artinian ring, with an algebraically closed residue field $k$. I am interested in the set $L_n(R)$ of isomorphism classes of $R$-modules of length $n$.
If $R$ is a $k$...
- 30.5k
15
votes
5
answers
1k
views
Monoids with infinite products
Say a monoid $M$ has infinite products if, for any (possibly infinite) sequence $(m_1,m_2,\ldots)$ of elements of $M$, there exists an element $m_1m_2\cdots\in M$, satisfying some good properties. ...
- 8,659
15
votes
3
answers
758
views
Locally square implies square
Does there exist a (noetherian) commutative ring $R$ and an element $a \in R$ such that $a$ is a square in every localization of $R$ but $a$ itself is not a square?
- 3,009
0
votes
1
answer
164
views
How to design or create or generate a bijective ring map? [closed]
How to design or create or generate a bijective ring map?
- 11
3
votes
1
answer
461
views
Are valuation rings regular?
This question is short, and to the point:
Valuation rings are certainly integrally closed, but are they regular?
The motivation is that I'm trying to understand the resolution of singularities of ...
- 5,929
14
votes
3
answers
2k
views
Projective dimension of zero module
Is there any consensus on what the projective dimension of the zero module should be? Here are three statements one commonly encounters in textbooks, sometimes with or without the condition $M\neq 0$:
...
- 2,857
3
votes
1
answer
171
views
If $B \subset C \subset B_g$, is $\mathrm{Spec} C \to \mathrm{Spec} B$ necessarily an open immersion?
Let $B \subset C$ be Noetherian integral domains, and $g \in B$. Thus, $\mathrm{Spec} B \to \mathrm{Spec} B_g$ is an open immersion.
If furthermore $C \subset B_g$, does it follow that $\mathrm{Spec}...
- 7,338
8
votes
1
answer
1k
views
Torsion submodule
$A$ a commutative Noetherian domain, $M$ a finitely generated $A$-module. How can I show that the kernel of the natural map $M\rightarrow M^{**}$, where $ M^{ * *}$ is the double dual (with respect to ...
- 2,857
4
votes
4
answers
444
views
Lower bounds on the degrees of representatives of $u^n$ as $n \to \infty$
Let $k$ be an algebraically closed field and $A$ a finitely generated $k$-algebra, together with a specified surjective morphism $\phi \colon k[x_1, \dotsc, x_n] \to A$. For $f \in A$, define $\...
- 7,338
1
vote
1
answer
375
views
Any implemented algorithm to compute the closure of an affine variety in a product of projective spaces?
Let $I$ be an ideal of $k[x_1, \ldots, x_m, y_1, \ldots, y_n]$, $k$ being a field. Does any of the computer algebra systems implement any algorithm to calculate the generators of the 'bi-...
- 5,339
78
votes
12
answers
12k
views
Why aren't representations of monoids studied so much?
It seems to me like every book on representation theory leaps into groups right away, even though the underlying ideas, such as representations, convolution algebras, etc. don't really make explicit ...
- 2,392
4
votes
2
answers
544
views
Membership problem in monoids
What is the simplest example of a monoid with undecidable membership problem? In other words, I'm looking for a concrete monoid $S$ such that there is no algorithm which takes elements $s_1,...,s_n$ ...
- 41
2
votes
2
answers
300
views
what conditions can one place on a finitely generated periodic semigroup that will ensure the semigroup is finite?
I am not familiar with much semigroup theory, but this question came up in my research and I've been unable to find much on it.
- 549
4
votes
1
answer
633
views
Determining if a ring satisfies Serre's condition S_{n}
Given a specific ring $R$ (eg, $R=k[x_{1}, \cdots, x_{n}]/I)$ is there a (simple) way to determine whether or not $R$ satisfies Serre's condition $S_{n}$? In particular, is there a way to do this in ...
- 113
3
votes
2
answers
552
views
Is weak normality stable under completion?
I'm curious if anyone knows a reference for the following. It seems like someone must have done this somewhere, but I couldn't find a reference.
Recall that an excellent reduced noetherian ring $R$ ...
- 20.5k
2
votes
3
answers
975
views
Finitely generated monoids are finitely presented?
I saw in the answer of this post that any finitely generated monoids are finitely presented in the sense that there is a coequalizer diagram $P_1\rightrightarrows P_0\rightarrow M$ with $P_1$ and $P_0$...
- 5,052
5
votes
2
answers
2k
views
Is a valuation domain PID when its maximal ideal is principal?
It is known that a valuation domain is a principal ideal ring if and only if its prime ideals are principal. Is it a principal ideal ring when its unique maximal ideal is principal?
- 1,207
21
votes
2
answers
3k
views
Standard reduction to the artinian local case?
Where can I find a clear exposé of the so called "standard reduction to the local artinian (with algebraically closed residue field", a sentence I read everywhere but that is never completely unfold?
...
- 411
1
vote
0
answers
2k
views
Generators of ideals in polynomial rings over commutative rings.
This is my first question; I hope it worthy of this awesome forum and its members.
Let $R$ be a commutative ring, perhaps with unit, perhaps not. As usual let $R[x]$
denote the ring of polynomials ...
- 391
13
votes
2
answers
2k
views
Length of I/I^2 versus Ann(I)/Ann(I)^2 in Artinian rings.
Suppose that $(A,\mathfrak{m})$ is a local Artinian ring.
If $A$ is Gorenstein, then $A$ admits a dualizing functor
on finite length modules defined by $D(M):= Hom_A(M,A)$ which preserves
lengths. If ...
user631
3
votes
0
answers
614
views
nilpotent matrices over polynomial rings
I am looking for an analogue of the Jordan normal form for nilpotent matrices over the
polynomial ring ${\mathbb Z}[x_1, \dots, x_n]$. More precisely, is there a description for the orbits of action ...
- 6,224
6
votes
1
answer
542
views
Abelian varieties over local fields
Let $K$ be a local field of characteristic zero, $k$ its residue field, $R$ its ring of integers and $p$ the characteristic of the residue field $k$. Let $G$ be the Galois group of $K$, $I\subset G$ ...
- 1,673
1
vote
1
answer
274
views
Q-Divisor and Determinant Map on a Maximal Order
Given a smooth projective surface $X$, let $A$ be a sheaf of maximal orders in a division ring.
Let us for simplicity assume $A$ ramifies in one curve $C$ with ramification index $e$. Let $A^*$ be the ...
- 1,391
13
votes
1
answer
908
views
Computational Question about finite local rings:
Let $(A,\mathfrak{m})$ be a local Artinian ring with
finite residue field, which I'm happy to assume is $\mathbf{F}_3$.
(In particular, $A$ has finitely many elements.)
I would like to do some ...
user631
23
votes
1
answer
3k
views
Modules and Square Zero Extensions
Let $R$ be a commutative ring, $RMod$ its category of modules and $CRing$ the category of commutative rings.
There's an embedding $RMod \rightarrow CRing/R$ that sends an $R$-module $M$ to the ring ...
- 1,484
4
votes
0
answers
233
views
When is the ring of invariants of a finite group generated by symplectic reflections a complete intersection ring?
Let V be a finite dimensional symplectic vector space over $\mathbb{C}$. Let $G$ be a finite subgroup of the symplectic group $Sp(V),$ which is
generated by symplectic reflections, i.e. by elements $g\...
- 689
6
votes
0
answers
881
views
Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.5k
3
votes
0
answers
592
views
Basic commutative algebra question.
Suppose that A is a local ring (commutative with unit), finite over a field k. Let L be the residue field A / m where m is the unique maximal ideal of A.
Does the dimension of L (as a k-vector space) ...
- 467
6
votes
1
answer
2k
views
Hochschild and cyclic homology of smooth varieties
Many of the standard sources which discuss the Hochschild Kostant Rosenberg theorem and cyclic homology for smooth varieties such as Loday and Weibel's paper "The Hodge Filtration and Cyclic Homology" ...
- 3,758
12
votes
2
answers
1k
views
Is completeness of a field an algebraic property?
Pretty straitforward:
If a field has a metric in which it is complete can it have a metric in which it is not complete?
By metric I mean field norm of course
- 700
8
votes
1
answer
371
views
How to construct a ring with global dimension m and weak dimension n?
Given two integers $m,n$ such that $n < m$, it is easy to construct a ring with global dimension $m$ or weak dimension $n$. But I wonder whether there exists a ring satisfying both the conditions?
- 1,207
3
votes
2
answers
376
views
How to compute the ring of invariants of SO_3(k) acting on a polynomial ring
Let $k$ be a field and let $A$ be the polynomial ring over $k$ in $3n$ variables: $A = k[X_{ij} \vert i=1,2,3 \quad j=1,2,\cdots,n]$.
${\rm SO}_3(k)$ acts on $A$ in the following way: Given $g \in {\...
- 137
16
votes
3
answers
3k
views
Is being torsion a local property of module elements?
Say $R$ is a ring, not necessarily a domain, and $M$ is an $R$-module. All rings are commutative with 1. An element $m\in M$ is called torsion if $r.m=0$ for some regular element (non-zerodivisor) $r\...
- 11.3k
0
votes
0
answers
165
views
Support sets along a ring homomorphism.
Let $(R,m)$ and $(S,n)$ be commutative noetherian local rings, and $f: R\rightarrow S$ be a local homomorphism (i.e., $f(m) \subseteq n$) with $S$ flat as $R$-module. If $M$ is a finite generated $R$-...
- 1,207
9
votes
3
answers
3k
views
Why are divisible abelian groups important?
I just quote wikipedia:
"Divisible groups are important in understanding the structure of abelian groups, especially because they are the injective abelian groups."
I am asking for detail ...
2
votes
2
answers
866
views
Modules over a Gorenstein ring
$A$ a Gorenstein ring, $M\neq 0$ a finite $A$-module with finite injective dimension. According to Bruns, this implies that $M$ has finite projective dimension. How do I see that?
- 2,857
3
votes
3
answers
670
views
Algebraic, analytic, formal modules
Consider torsion free modules over the germ of a fixed isolated algebraic hypersurface singularity {$f=0$}$\subset\mathbb{C}^n$.
There are natural functors (using categories of finitely generated ...
- 2,232
0
votes
1
answer
502
views
Finiteness of injective hull of residue field for Artin local ring
$(A,\mathfrak{m})$ an Artin local ring, $E(A/\mathfrak{m})$ the injective hull of $A/\mathfrak{m}$. How do I see that $E(A/\mathfrak{m})$ is a finite $A$-module?
- 2,857
-1
votes
2
answers
671
views
Do Gorenstein rings necessarily have a finite projective dimension (as a module over itself)? [closed]
Do Gorenstein rings necessarily have finite projective dimensions?
- 2,857
9
votes
2
answers
3k
views
Projective & injective dimensions
$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. I'm not quite sure about the relation between finiteness of projective and injective dimensions of $M$. Does the finiteness (or infiniteness)...
- 2,857
3
votes
2
answers
1k
views
Depth and dimension
$A$ a Noetherian local ring, $M\neq 0$ a finite $A$-module. Then is it true that $\mbox{depth }M\le\mbox{depth }A$ just like $\mbox{dim }M\le\mbox{dim }A$? I don't see any relation between an $M$-...
- 2,857
5
votes
2
answers
732
views
What is the completion at a family of ideals?
Let $A$ be a (commutative with unit) noetherian ring. If $I$ is an ideal of $A$, the $I$-adic completion of $A$ is by definition
$$
\widehat{A} := \underset{\leftarrow}\lim A/I^n.
$$
This operation is ...
- 3,704
13
votes
4
answers
4k
views
Size of a Groebner basis
I've spent some time recently looking at some Groebner bases for some specific ideals coming from problems in computer vision. The generators are not sparse, and they all have the same degree (...
- 191
-3
votes
1
answer
1k
views
An elementary question about the Krull dimension of modules [closed]
Let $R$ be a commutative ring. If $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0$ is an exact sequence of modules, we have that $\operatorname{Supp}M=\operatorname{Supp}M'\cup \operatorname{...
- 1,207