All Questions
6,057 questions
16
votes
2
answers
1k
views
Is the support of a flat sheaf flat?
Note: in the following, all scheme/algebra morphisms should be assumed essentially of finite type.
Geometric version: Let $X$ be a scheme flat over $S$ (both noetherian), and let $\mathscr{F}$ be a ...
- 7,338
5
votes
2
answers
2k
views
how to compute the henselization of some simple rings?
Hi,everyone.
I want to know that how to compute the henselization of some simple rings, for example: $k[x]_{(x)}$ and $R[X]_{(X)}$ where $k$ is a field and $R$ is a excellent DVR.
thank you very ...
- 1,921
1
vote
3
answers
585
views
Terminology for certain monoids which are to monoids like fields are to rings
Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "...
40
votes
1
answer
2k
views
Is the radical of an irreducible ideal irreducible?
I originally posted this to math.stackexchange.com
here. I got a partial answer, but I now suspect that the complete answer is much harder than I thought, so I'm posting it here.
Fix a commutative ...
- 401
0
votes
0
answers
87
views
Standard Notation for Monomial Orderings?
Is there a standard way to denote a particular lexicographic (resp. reverse lexicographic) monomial ordering using subscripts or superscripts? For example, I might want to refer to the lexicographic (...
- 326
7
votes
3
answers
2k
views
Does a "composite field" always exist?
Suppose $F$ is a field, and $F_1, F_2$ are two extension fields of $F$. Is it always the case that there is a field $L$, containing three subfields $F, K_1, K_2$ and two ring isomorphisms $\varphi_{i}...
- 18.7k
3
votes
1
answer
537
views
R noetherian is factorial [closed]
This is exercise 2.2.27 / 72 from Cohen Macaulay Rings, Bruns & Herzog
Let R be a Noetherian ring over which every finite module has a finite free resolution. Show R is a factorial domain.
- 287
5
votes
3
answers
676
views
Does every compact Hausdorff ring admit a decomposition into primitive idempotents?
Let $\mathbf{R} = (R,\mathcal{T},+,\cdot,0,1)$ be a compact Hausdorff topological unitary ring, and consider the set $I(\mathbf{R}) := \{ e \in R \mid e \cdot e = e \}$ (of idempotents in $\mathbf{R}$)...
- 1,498
6
votes
1
answer
437
views
"Archimedeanising" an ordered field
If $K$ is an ordered field, let $B$ be the subring comprising the $x \in K$ such that $|x| \le n$ for some $n \in N$, and let $I$ be the ideal of $B$ comprising the infinitesimal elements (i.e. the $x ...
- 879
5
votes
1
answer
1k
views
Free and surface groups cohomology
What is a good reference for results on cohomology of finite rank free groups and surface groups with group ring coefficients?
I am interested in the case when the group acts on its group ring via ...
- 51
4
votes
1
answer
543
views
Generators of a certain ideal
In view of Mariano Suárez-Alvarez's answer I see how badly phrased my question was, and decided to rewrite it. The drawback is that some comments of Martin Brandenburg are now incomprehensible, but I ...
- 4,416
0
votes
1
answer
136
views
pd finite for finite module over local CM ring?
Let (R,m) be a CM ring of dimension d, and M a finite R- module. Is pd M always finite?
- 287
17
votes
1
answer
4k
views
A Relative Algebraic Hartogs Lemma
The Algebraic Hartogs Lemma states that in a Noetherian normal scheme, a rational function that is regular outside a closed subset of codimension at least two, is in fact regular everywhere.
In a ...
- 7,338
2
votes
0
answers
355
views
Is the invariant part of the canonical module by finite group action nonzero?
This is further question to this.
Let $q \in V=(f=0) \subset \mathbb{C}^{n+1}$ be an isolated rational singularity of dimension $n$. Suppose that $ G:=\mathbb{Z}/m \mathbb{Z}$ acts on $V$ freely ...
- 909
4
votes
1
answer
210
views
Explicitly generating 1 in an ideal without prime support
The Question
Let $R$ be a unital commutative ring, and let $a,b_1,b_2\in R$. The following is a basic commutative algebra exercise.
Lemma. If $Ra+Rb_1=R$ and $Ra+Rb_2=R$, then $Ra+Rb_1b_2=R$.
Proof. ...
- 13k
6
votes
2
answers
1k
views
Are there one-dimensional ideals in any local ring
I would like to know if it is always possible to find a one-dimensional ideal in a local commutative ring... actually I am interested in finding a curve through a point on a scheme (locally). If the ...
- 227
1
vote
1
answer
482
views
Injective hulls of residue fields of a local ring and its ring invariants by finite group action
Let $R$ be a local ring, $m$ its maximal ideal and $k:= R/m$ its residue field.
Suppose that a finite cyclic group $G= \mathbb{Z}/ m \mathbb{Z}$ has a linear nontrivial action on $R$.
Let $R^G$ be a ...
- 909
2
votes
1
answer
877
views
maximal Cohen-Macaulay module [closed]
This is from CM Rings (Bruns & Herzog) p 64 2.1.20. Show that a one dimensional Noetherian local ring has a maximal Cohen Maucaulay module.
- 287
7
votes
1
answer
457
views
Normality for non-noetherian schemes
I am interested to know to what extent the notion of normality makes sense on a non-noetherian scheme.
Specifically, I can ask the following question: let $\pi:X\to Y$ be a formally smooth morphism of ...
- 7,037
7
votes
0
answers
658
views
Invertible elements in generalized fields
Durov's theory of generalized rings also includes generalized fields (5.7.6), which are defined as generalized rings, which are not subtrivial and whose proper strict quotients are subtrivial. For ...
- 63.1k
13
votes
2
answers
3k
views
Elements in a localization - category theoretic approach
This question is about the elements in a localization $S^{-1} A$ of a commutative ring $A$. Is it possible to derive $\frac{a}{1} = 0 \in S^{-1} A \Rightarrow \exists s \in S : sa = 0$ only using the ...
- 63.1k
2
votes
1
answer
272
views
Noether normalization with auxiliary conditions?
Let $k$ be a field and $R$ a $d$-dimensional integral domain over $k$. Noether normalization produces an injective algebra homomorphism $k[x_1,\dots, x_d] \to R$ which is module-finite.
Given a ...
- 13.1k
3
votes
2
answers
928
views
Quartic Space Curves
It is an exercise in Hartshorne to classify nonsingular quartic curves in projective 3-space. I am interested in what happens when we allow singularities. In particular, I am looking for an ...
- 95
2
votes
1
answer
387
views
Simple Question on Injective Hulls
Let $R$ be a noetherian local ring with maximal ideal $\mathcal m$ and denote by $E$ the injective hull of the residue field $k$.
Then, as an $R-$module, what is the support of $E$?
- 659
2
votes
0
answers
629
views
Induced map on algebraic de Rham cohomology
Let $X/k$ and $Y/k$ be two smooth affine varieties over a field $k$ with $\mathrm{char}(k) = 0$ and $\varphi: X \rightarrow Y$ be a morphism. I would like to know under what conditions, the induced ...
- 1,109
6
votes
0
answers
3k
views
Tensor product of two algebras [closed]
Let $A$ and $B$ be algebras over a field $K$. The ideals of the tensor product $A\bigotimes_K B$ are of the form $I\bigotimes_K J$ where $I$ is an ideal of $A$ and $J$ is an ideal of $B$?
- 545
4
votes
1
answer
932
views
Relative integral closure
Definition: Let $R$ be an $A$-algebra, where $R$ and $A$ are both commutative rings with unit. Let $S \subset R$ be the (multiplicatively closed) subset consisting of those nonzerodivisors $s \in R$ ...
- 7,338
11
votes
1
answer
675
views
When is there a deformation of a given singularity to a normal singularity
Question: Given a variety $X_0$ with a singularity (say Cohen-Macaulay), when does this exist as a special fiber of a flat family $X \to C$ mapping to a smooth curve $C$, such that the generic fiber ...
- 20.5k
12
votes
5
answers
2k
views
analysis over non-Archimedean ordered fields
Can anyone suggest any good references for (or any experts on) analysis over non-Archimedean ordered fields, such as the field of rational functions in one variable (ordered at 0, or if you prefer at ...
- 19.7k
6
votes
0
answers
2k
views
Newton Method in $p$-adic case
The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentiable assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic integers,...
- 1,109
8
votes
1
answer
4k
views
Maximal ideals in a polynomial ring over the real numbers.
Let $\mathbf{R}$ be the field of real numbers. What are the generators of the maximal ideals of the polynomial ring $\mathbf{R}[x_1, ... , x_n]$? If instead of $\mathbf{R}$ one considers the field $\...
- 545
2
votes
1
answer
286
views
Modules with small support have big depth - reference wanted
Hello,
I would appreciate an exact reference / proof of the following fact, which I am almost able to prove, but not really:
Let $A$ be a regular Noetherian comm. ring, of finite Krull dimension. ...
- 5,562
2
votes
2
answers
899
views
Number of generators of an ideal in a polynomial ring over a Noetherian ring
Let $R$ be a Noetherian ring. By the Hilbert Basis Theorem the polynomial ring $R[x_1, \ldots , x_n]$ is also a Noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[...
- 545
4
votes
4
answers
3k
views
Subtle examples of morphisms that are finite but not flat
Let $R$ be a ring (commutative noetherian with unit), and let $K(R)$ be its total ring of fractions (obtained by inverting all nonzerodivisors). Thus, $R \hookrightarrow K(R)$. Let $a \in K(R)$ be ...
- 7,338
2
votes
2
answers
1k
views
when tensor complex resolves S/I+J?
Assume that $I\subset k[x_1,\ldots,x_n]$ and $J\subset k[y_1,\ldots,y_m]$ are monomial ideals in different rings, and the minimal free resolution of $S/I$ and $S/J$, say $F_\cdot$ and $G_\cdot$, are ...
- 43
0
votes
2
answers
1k
views
Generic rank of a coherent sheaf on a projective variety vs. generic rank on the cone
Let $S:=k[X_1,\ldots,X_n]$ be a polynomial ring over a field $k$, with its natural grading, and let $\mathfrak{p}$ be a homogeneous prime ideal of $S$. Also, let $M:=\bigoplus_{i} M_i$ be a finitely ...
- 3,101
3
votes
2
answers
393
views
How to make a function depending on some operation?
Let $S$ be an infinite set and $f:S\times S\rightarrow\mathbb R$ be bounded.
Question: Are there simple hypotheses on $f$ such that there is a commutative and associative operation $\cdot$ on $S$ ...
- 6,034
1
vote
1
answer
771
views
Prime ideals in coordinate rings
Is there a way to characterise prime ideals in affine coordinate rings (i.e. quotients of polynomial rings). To be more specific, how can I say if principal ideals in such rings are prime or not in an ...
- 11
4
votes
2
answers
705
views
Why are canonical modules supported everywhere?
Let $A$ be a local CM ring, and $\omega$ a canonical module of $A$. Here are two properties of $\omega$ from Bruns & Herzog:
$\omega_{\mathfrak{p}}$ is a canonical module of $A_{\mathfrak{p}}$ ...
- 2,857
0
votes
1
answer
426
views
Are maximal Cohen-Macaulay modules supported everywhere?
Let $A$ be a local CM ring, and $M$ a maximal CM $A$-module. Is it true that $\operatorname{Supp}M=\operatorname{Spec}A$ ? This suspicion stems from such statements as:
If $\omega$ is a canonical ...
- 2,857
14
votes
2
answers
1k
views
Economical hard word problem
Can anyone give me an example of a very simple word problem, where by "simple" I mean that it has very few generators and relations, that is nevertheless insoluble. To make the question easier, I am ...
- 29k
22
votes
2
answers
1k
views
Toposes (topoi) as classifying toposes of groupoids
A famous theorem of Joyal and Tierney says that each Grothendieck topos is equivalent to the classifying topos of a localic groupoid. I believe that Butz and Moerdijk have shown that if the topos has ...
- 38.6k
4
votes
1
answer
675
views
Are all (commutative) rngs ideals of (commutative) rings? [closed]
To avoid repeating it endlessly, assume all rings and rngs are commutative. I do not know if this is necessary.
The question then is exactly the title, but I think a stronger statement is true:
...
- 1,979
0
votes
1
answer
327
views
Examples of complexes of modules for wich homomorphisms "homological" implies "homotopic"
Let $V$ and $V^\prime$ - complexes of modules over ring $A$, and $f, g$ - homomorphisms $V\rightarrow V^\prime$.
I am interested in various conditions on $A, V, V^\prime$: ($f$ and $g$ are ...
- 101
3
votes
1
answer
1k
views
A Module with $Ext^i(M,R) = 0$ for all $i > 0$
Let $M$ be a finitely generated module over a noetherian local ring $R$. We can take our ring to be Cohen-Macaulay. Suppose $M$ satisfies the condition $Ext^i(M,R) = 0$ for all $i > 0$. We want to ...
- 200
3
votes
2
answers
653
views
Does the first singular cohomology of an ACM surface vanish?
Hi everybody, I am interested in the following:
Let $I\subset S=\mathbb{C}[x_0,\ldots ,x_n]$ be a graded ideal such that $\operatorname{depth}(S/I)\geq 3$, and let $X^h$ denote the analytic space ...
4
votes
0
answers
331
views
What is the pro-algebraic completion of the free semigroup on one generator?
This question is motivated by an attempt to understand what is going on in Tom's post from a certain point of view.
Let $\mathbb N^+$ be the free semigroup on one generator (so the positive natural ...
- 38.6k
3
votes
1
answer
493
views
Efficient Algorithm for Matrix Version of Waring's Problem
Given an $n \times n$ matrix $A$ with entries in a commutative and associative ring with $1$ (say $Z[x_{1},\dots,x_{n^{2}}]$), the following paper guarantees existence of seven $B_{i}$s such that $A = ...
- 13.9k
2
votes
0
answers
216
views
Modules with first Betti number bigger than the second Betti number
Let $R$ be a commutative noetherian local ring (with 1) and let $M$ be a finitely generated $R$-module. Consider a minimal free presentation of $M$ as follows: $R^{\beta_2}\rightarrow R^{\beta_1}\...
- 3,101
4
votes
2
answers
1k
views
Is the normalisation of an integral noetherien dimension one ring a finite morphism?
This feels like something I should know but I can't find an answer in Liu or in Atiyah-MacDonald, or a counter-example.
To state the question again: let $A$ be an integral Noetherien ring of Krull ...
- 1,347