All Questions
Tagged with ac.commutative-algebra ag.algebraic-geometry
181 questions
19
votes
3
answers
2k
views
Guises of the Stasheff polytopes, associahedra for the Coxeter $A_n$ root system?
Richard Stanley keeps a famous running compilation of different guises of the celebrated Catalan numbers. The number of vertices of the associahedron is one instantiation among the multitude, and the ...
- 10.5k
17
votes
1
answer
6k
views
Bijection implies isomorphism for algebraic varieties
Let $f:X\to Y$ be a morphism of algebraic varieties over $\mathbb C$. Assume that
a) $f$ is bijective on $\mathbb C$-points
b) $X$ is connected
c) $Y$ is normal.
Does it imply that $f$ is an ...
- 7,037
78
votes
5
answers
14k
views
Is there a "geometric" intuition underlying the notion of normal varieties?
I first got concious of the notion of normal varieties around 3 years ago and despite the fact that by now I can manipulate with it a bit, this notion still puzzles me a lot.
One thing that strikes me ...
- 14.3k
15
votes
3
answers
3k
views
which homogeneous polynomials split into linear factors?
Let $R$ be the set of homogeneous polynomials of degree $n$ in $d$ variables over $\mathbb{C}$. When $n>2$, the set of elements of $R$ that split into a product of linear factors forms a proper ...
- 323
22
votes
6
answers
8k
views
A finitely generated $\mathbb{Z}$-algebra that is a field has to be finite
I was trying to understand completely the post of Terrence Tao on Ax-Grothendieck theorem. This is very cute. Using finite fields you prove that every injective polynomial map $\mathbb C^n\to \mathbb ...
- 14.3k
65
votes
4
answers
22k
views
When is the product of two ideals equal to their intersection?
Consider a ring $A$ and an affine scheme $X=\operatorname{Spec}A$ . Given two ideals $I$ and $J$ and their associated subschemes $V(I)$ and $V(J)$, we know that the intersection $I\cap J$ corresponds ...
- 914
52
votes
2
answers
4k
views
a categorical Nakayama lemma?
There are the following Nakayama style lemmata:
(the classical Nakayama lemma) Let $R$ be a commutative ring with $1$ and $M$ a finitely generated $R$-module. If $m_1, \ldots, m_n$ generate $M$ ...
user19475
52
votes
2
answers
7k
views
Ring-theoretic characterization of open affines?
Background
Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
- 5,407
42
votes
2
answers
3k
views
Is every Noetherian Commutative Ring a quotient of a Noetherian Domain?
This was an interesting question posed to me by a friend who is very interested in commutative algebra. It also has some nice geometric motivation.
The question is in two parts. The first, as stated ...
32
votes
7
answers
5k
views
Invariant polynomials under a group action (hidden GIT)
Let's say I start with the polynomial ring in $n$ variables $R = \mathbb{Z}[x_1,...,x_n]$ (in the case at hand I had $\mathbb{C}$ in place of $\mathbb{Z}$).
Now the symmetric group $\mathfrak{S}_n$ ...
- 1,993
30
votes
6
answers
8k
views
Algebraic stacks from scratch [closed]
I have a pretty good understanding of stacks, sheaves, descent, Grothendieck topologies, and I have a decent understanding of commutative algebra (I know enough about smooth, unramified, étale, and ...
18
votes
1
answer
1k
views
Smith Normal Form of powers of a matrix
What invariants of a matrix determine the Smith Normal Form (SNF) of all the powers of a matrix?
The question makes sense over any PID $R$. If we let $M = M_n(R)$ and $G=Gl_n(R)$, then SNF is a ...
- 3,526
17
votes
4
answers
4k
views
Completion of a local ring of a curve
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ ...
- 7,722
14
votes
0
answers
957
views
What is the state of art in Groebner bases
How big polynomial systems can we deal with? How do you know when you don't even have to try?
Motivation:
Recently I tried to solve a problem posed in another MO question and ultimately I got stuck ...
- 8,597
12
votes
5
answers
5k
views
reduced ⊗ reduced = reduced; what about connected?
Several questions actually.
All rings and algebras are supposed to be commutative and with $1$ here.
(1) Let $k$ be a field, and let $A$ and $B$ be two $k$-algebras. I need a proof that if $A$ and $...
- 33.8k
11
votes
0
answers
629
views
Inversion, Koszul duality, combinatorics and geometry
According to this MO answer Koszul duality is related to operations on generating series;
1) multiplicative inversion for quadratic algebras,
2) compositional inversion for quadratic operads,
3) ...
0
votes
1
answer
429
views
Separability of $\mathbb{C}[x]$ over its $\mathbb{C}$-subalgebras
For commutative rings $R \subseteq S$,
recall that $S$ is separable over $R$, if $S$ is a projective $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$.
...
- 2,837
54
votes
8
answers
58k
views
Modern algebraic geometry vs. classical algebraic geometry
Can anyone offer advice on roughly how much commutative algebra, homological algebra etc. one needs to know to do research in (or to learn) modern algebraic geometry. Would you need to be familiar ...
4
votes
2
answers
2k
views
Cohen-Macaulay sheaves which are not locally free
A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth_{\mathcal{O}_x}(\mathcal{F}_x) = \dim\mathcal{O}_x$ for any $x\in X$, where $\mathcal{O}_x$ is ...
- 2,444
74
votes
1
answer
6k
views
$R$ is isomorphic to $R[X,Y]$, but not to $R[X]$
Is there a commutative ring $R$ with $R \cong R[X,Y]$ and $R \not\cong R[X]$?
This is a ring-theoretic analog of my previous question about abelian groups: In fact, in any algebraic category we may ...
- 63.1k
36
votes
3
answers
2k
views
Are large powers of polynomials linearly independent?
Let $P_1,\dots,P_k$ be polynomials over $\mathbf{C}$, no two of them being proportional.
Does there exist an integer $N$ such that $P_1^N,\dots,P_k^N$ are linearly independent?
- 4,653
32
votes
6
answers
9k
views
What is the universal property of normalization?
What is the universal property of normalization? I'm looking for an answer something like
If X is a scheme and Y→X is its
normalization, then the morphism
Y→X has property P and any ...
29
votes
5
answers
9k
views
Local complete intersections which are not complete intersections
The following definitions are standard:
An affine variety $V$ in $A^n$ is a complete intersection (c.i.) if its vanishing ideal can be generated by ($n - \dim V$) polynomials in $k[X_1,\ldots, X_n]$. ...
- 303
23
votes
2
answers
3k
views
Criteria for irreducibility of polynomial
If $f, g\in \mathbb C[a,b]$ are polynomials in two variables, are there easy criteria that allow to see if $f(x,y)-g(t,z)\in \mathbb C[x,y,t,z]$ is irreducible?
Thank you very much,
best
- 669
19
votes
4
answers
2k
views
What is the geometric object corresponding to a subalgebra in a polynomial ring
Many introductory texts on algebraic geometry set up some sort of algebra-geometry dictionary in which radical ideals correspond to varieties, and so on. I am wondering if there is a geometric way to ...
- 1,961
19
votes
1
answer
825
views
Is the regularity of finitely generated rings decidable?
Q: Is there an algorithm to decide whether a given finitely generated (over $\mathbb{Z}$) commutative ring is regular?
I mean by regular that the localization at every prime ideal is a regular local ...
- 343
18
votes
3
answers
702
views
Existence of a ring with specified residue fields
Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$?
To prevent things from being too easy, I require ...
- 706
14
votes
1
answer
2k
views
Some questions about the ring Z((x))
$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...
- 10.7k
13
votes
2
answers
1k
views
When does a quasicoherent sheaf vanish?
Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the ...
- 4,949
12
votes
1
answer
419
views
Is height preserved in a normalization?
Let $R$ be a domain and $\tilde R$ its integral closure in its fraction field: $R\subset \tilde R\subset Frac(R)$.
Is it true that a prime ideal $ \tilde {\mathfrak p} \subset \tilde R$ and its ...
- 47.5k
11
votes
2
answers
1k
views
Is there a relation between Gelfand duality and the spectrum of a ring (with its Zariski topology)?
Compare the following two results:
Thm A) Let $A$ be a commutative $C^*$-algebra and let $X$ be its Gelfand spectrum. Gelfand duality says that there's a natural isometric $*$-isomorphism from $A$ to ...
- 711
9
votes
2
answers
2k
views
Is an elementary symmetric polynomial an irreducible element in the polynomial ring?
Let $S=\mathbb{C}[x_1,x_2,\dots,x_n]$ be a polynomial ring. Let $e_a$ denotes the elementary symmetric polynomials of degree $a$ in $S$.
For $n=2$:
$e_1=x_1+x_2$;
$e_2=x_1x_2$.
For $n=3$:
$e_1=x_1+...
- 446
8
votes
1
answer
1k
views
Is the sheaf of smooth functions flat?
Let $X$ be a smooth algebraic variety over $\mathbb{C}$. Is the sheaf of smooth functions on $X$ flat as an $\mathcal{O}_X$ module?
- 338
6
votes
2
answers
798
views
When does glueing affine schemes produce affine/separated schemes?
Let $X$ be an affine scheme with an open affine subscheme $U\subset X$. Given an automorphism of $U$, we can glue $X$ with itself along $U$ to get a new scheme. Is there a description in terms of ...
user142965
5
votes
1
answer
1k
views
Ideal generated by two univariate, coprime, integer polynomials
Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\...
- 7,746
5
votes
0
answers
162
views
Nullstellensatz with nilpotents and $I=J(V(I))$
Let $R$ be the ring $$\mathbb{R}[t_1,t_2\ldots]/(t_1^2,t_2^2,\ldots)$$
Let $p_1,\ldots p_\ell$ be polynomials in $R[x_1,\ldots,x_n]$ whose constant terms are 0.
Let $f$ be a polynomial which is zero ...
user44143
5
votes
1
answer
453
views
an algebraic variety for a boolean circuit
There is a polynomial reduction from a $3-CNF$ $SAT$ problem to some system of polynomial equations over $\mathbb{F}_2$.
I mean there is polynomial reduction $F$ such that for every boolean ...
- 2,576
4
votes
5
answers
2k
views
What properties define open loci in families?
This question is somehow related to the question What properties define open loci in excellent schemes?.
Let $f:X\to S$ be a proper (or even projective) morphism between schemes (of finite type over ...
- 16.1k
2
votes
0
answers
238
views
A special type of ideals
I am looking for some references that contained a study of ideals with the following *-property:
Let $I $ be an ideal of a commutative ring with ideantity. The ideal $I $ has the *-property if $I\...
- 61
2
votes
1
answer
205
views
Deformation of isolated singularities and non zero divisors
Consider $f \in \mathbb{C}\{x_1,\dots,x_n\}$ such that $(V(f),0)$ has an isolated singularity.
Let $F \in \mathbb{C}\{x_1,\dots,x_n,t\}$ be a deformation of $f$ such that there exists some integer $m$...
1
vote
0
answers
274
views
Does analytic isomorphism imply local isomorphism?
If $ \mathfrak{p} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(A) $, and $ \mathfrak{q} $ is a (not necessarily closed) point of a variety $ \operatorname{Spec}(B) $ such ...
- 912
222
votes
8
answers
35k
views
How to memorise (understand) Nakayama's lemma and its corollaries?
Nakayama's lemma is mentioned in the majority of books on algebraic geometry that treat varieties. So I think Ihave read the formulation of this lemma at least 20 times (and read the proof maybe ...
- 14.3k
70
votes
2
answers
9k
views
What is the insight of Quillen's proof that all projective modules over a polynomial ring are free?
One of the more misleadingly difficult theorems in mathematics is that all finitely generated projective modules over a polynomial ring are free. It involves some of the most basic notions in ...
- 44.7k
59
votes
4
answers
12k
views
Geometric meaning of Cohen-Macaulay schemes
What is the geometric meaning of Cohen-Macaulay schemes?
Of course they are important in duality theory for coherent sheaves, behave in many ways like regular schemes, and are closed under various ...
- 63.1k
48
votes
4
answers
4k
views
Are there more Nullstellensätze?
Over which fields $k$ is there a reasonable analogue of Hilbert's Nullstellensatz?
Here is a more precise formulation: let $k$ be an arbitrary field, $n$ a positive integer, and $R = k[t_1,..,t_n]$. ...
- 65.4k
46
votes
4
answers
8k
views
What does "linearly disjoint" mean for abstract field extensions?
All definitions I've seen for the statement "$E,F$ are linearly disjoint extensions of $k$" are only meaningful when $E,F$ are given as subfields of a larger field, say $K$. I am happy with the ...
- 11.2k
44
votes
5
answers
6k
views
What is the cotangent complex good for?
The cotangent complex seems to be a pretty fundamental object in algebraic geometry, but if it's treated in Hartshorne then I missed it. It seems to be even more important in derived algebraic ...
- 63.9k
42
votes
4
answers
8k
views
Serre intersection formula and derived algebraic geometry?
Let $X$ be a regular scheme (all local rings are regular). Let $Y,Z$ be two closed subschemes defined by ideals sheaves $\mathcal I,\mathcal J$. Serre gave a beautiful formula to count the ...
- 30.5k
40
votes
1
answer
3k
views
Is every connected scheme path connected?
Every (?) algebraic geometer knows that concepts like homotopy groups or singular homology groups are irrelevant for schemes in their Zariski topology. Yet, I am curious about the following.
Let's ...
- 47.5k
39
votes
2
answers
6k
views
What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
- 30.5k