All Questions
Tagged with ac.commutative-algebra ag.algebraic-geometry
2,098 questions
0
votes
2
answers
617
views
An element in the product of schemes
Let $ f : X \to S, g : Y \to S$ be the scheme morphisms, and $ X \times_S Y$ be the product of shemes. Let $ z \in X \times_S Y$ , and $ x=p(z) , y=q(z) ,$ where $ p: X \times_S Y \to X, q: X \times_S ...
- 95
1
vote
1
answer
1k
views
resolution of singular points on curve
After reading Fulton's book "Algebraic Curves", I know how to do resolution of singular points on curves. Given an affine equation, I can get it's non-singular affine model, i.e the normalization of ...
- 1,109
4
votes
1
answer
552
views
Factorization of schemes
Let $k$ be a ring (perhaps a field). Let $M$ be the "set" of isomorphism classes of $k$-algebras and regard it as a commutative monoid with multiplication $\otimes_k $ and unit element $k$. There is ...
- 63.1k
3
votes
1
answer
538
views
Comparation of dimensions of rings
Let $k$ be a field, $A$ be an integral domain, $B \subset A$, and $A, B$ are both finitely generated $k$ algebra. Let $p \subset B$ be a prime ideal. Suppose there exists prime ideals $q \subset A$, ...
- 95
18
votes
4
answers
4k
views
Flatness of normalization
Let $X$ be a noetherian integral scheme and let $f \colon X' \to X$ be the normalization morphism. It is known that, if non trivial, $f$ is never flat (see Liu, example 4.3.5).
What happens if we ...
- 3,704
7
votes
3
answers
2k
views
Products of Ideal Sheaves and Union of irreducible Subvarieties
Assume I have a nonsingular, irreducible, algebraic variety $X$ and irreducible, nonsingular subvarieties $Z_1,\ldots,Z_k\subseteq X$. Let $\mathcal{I}_i$ be the ideal sheaf of $Z_i$ and $\mathcal{I}:=...
- 4,902
37
votes
3
answers
3k
views
What does it mean geometrically that an element in a domain is irreducible?
Consider a domain $A$ and a non-zero element $f\in A$. That element $f$ is prime if and only if the subscheme $V(f)\subset \operatorname{Spec}(A)$ is integral and this is a completely satisfactory ...
- 47.5k
3
votes
1
answer
2k
views
Multiplicity of a singular point
Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is ...
- 2,444
12
votes
2
answers
3k
views
Vector bundles on affine scheme
I have already asked similar questions before, but now I realized that there a nice
general way to ask what I want. Namely let $X$ be a normal affine variety over
a field $k$. Assume first that $k$ is ...
- 7,037
11
votes
3
answers
2k
views
When is a blow-up Cohen-Macaulay?
Let $X$ be a smooth projective variety and $Z$ a closed subscheme. Let $X'$ be the blow-up of $X$ with center $Z$.
Under what conditions on $Z$ is $X'$
Cohen-Macaulay?
In the case $Z$ is non-...
- 11.6k
2
votes
2
answers
492
views
Model Theoretic Localization
This is a re-post on a previous question I asked. My first question was too vague to warrant detailed responses. Really, I have two specific questions to ask.
1) Let $\sigma = (A; \{0,1\}; +, \times)...
- 761
2
votes
1
answer
521
views
Kahler differentials of a hypersurface over a non-algebraically closed field
The following was recently on my algebraic geometry homework:
Let $k$ be an algebraically closed field, $f\in B=k[x_1,\ldots,x_n]$, and $A=B/(f)$. Show that $\Omega_{A/k}$ is locally free of rank $...
- 6,792
9
votes
3
answers
1k
views
Quasi-compact maps in Number Theory
Can someone give me an example of a non-quasi-compact morphism of schemes which arises naturally in the field of Algebraic Number Theory?
- 761
6
votes
1
answer
970
views
Reflexive sheaves on singular surfaces
Let $S$ be a normal surface over an algebraically closed field $k$ and let
$s$ be a point of $S$. Let ${\mathcal F}$ be a reflexive sheaf on $S$ of generic rank $n$ . Consider the (derived) fiber of $...
- 7,037
6
votes
2
answers
2k
views
When does the conormal bundle sequence split?
Let $X\subset \mathbb{P}^n$ be a smooth projective variety with ideal sheaf $I_X$. The conormal sequence is given by
$$
0\to I_X/I_X^2\to \Omega_{\mathbb{P}^n}|_X\to \Omega_{X}\to 0.
$$
For which ...
- 11.6k
7
votes
2
answers
512
views
Is the reduction of a flat, finite, surjective scheme over an integral base still flat?
Is the reduction $X_{red}$ of a flat, finite, surjective scheme $X$ over an integral base $S$ still flat?
I could possibly add that I am already aware we can assume the base $S$ to be local and ...
- 1,347
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
3
votes
2
answers
667
views
Normality and rational singularities via Hilbert series
Let $A$ be a finitely generated ${\mathbb Z}_{\geq 0}$-graded algebra over a field without zero divisors;
assume that all graded components are finite-dimensional and that $Spec(A)$ is smooth
outside ...
- 7,037
2
votes
2
answers
406
views
Extending a polynomial function from an open subset
I am a bit embarrassed to ask this question, but still: assume that I have
a finite morphism $\pi:X\to Y$ of affine algebraic varieties over a field (probably
finiteness is too strong an assumption, ...
- 7,037
8
votes
1
answer
289
views
Top degree local cohomology under action by a non-zerodivisor
Let $R$ be a noetherian commutative ring of dimension $n$, and let $M$ be a faithful finite $R$-module. Let $I$ be a proper ideal of $R$, and let $x\in I$ be a non-zerodivisor on $M$.
When does ...
- 19.6k
7
votes
2
answers
2k
views
An effective way to tell if the saturation of a homogeneous ideal is the irrelevant ideal
Let $\Bbbk$ be an algebraically closed field, let $R$ denote the graded ring $\Bbbk[x_0, \dotsc, x_N]$, and let $f_1, \dotsc, f_n \in R_m$ be nonconstant homogeneous polynomials. Then the common ...
- 7,328
7
votes
0
answers
518
views
An elementary question in singularities
The following problem came up in something I am working on. It has a really elementary statement but I couldn't crack it in a couple of hours of thinking about it. It isn't clear to me if I am being ...
- 3,758
2
votes
1
answer
186
views
Behaviour of Primes under Regular Coefficient Extensions
Let $K\hookrightarrow K'$ be a regular extension of fields, and $K[x_{1},\cdots,x_{n}]\hookrightarrow K'[x_{1},\cdots,x_{n}]$ the corresponding ring extension. Does every prime ideal of the first ring ...
- 125
3
votes
3
answers
549
views
Castelnuovo-Mumford Regularity of Ideals of Maximal Minors
I have an $m \times 2m$ matrix of linear forms over $\mathbb{C}[x,y,z,w]$. It is of the form $$M = ( x I - A z -B w \mid y I - C z - D w).$$ Here $A,B,C$ and $D$ are $m \times m$ scalar matrices. Let $...
- 431
3
votes
1
answer
901
views
Behaviour of Hilbert functions
Let $G$ be a complex simple reductive group. Then the set of isomorphy classes $Irr G$ is isomorphic to the set of dominant weights $\Lambda_+$ in the weight lattice of the maximal torus of a Borel ...
- 31
5
votes
3
answers
411
views
CM for primary ideal
Let $R$ be a regular local ring, $I$ a prime ideal and $J$ an $I$-primary ideal in $R$. Is it true that if $R/I$ is CM then also $R/J$ is CM?
This question is in some way the inverse of this one.
- 205
4
votes
2
answers
468
views
Maximal separable extensions of residue fields
Assume that $(A,m)$ is a Noetherian normal local domain, $K = Quot(A) \subset E, F$ Galois extensions of $K$. If $B=\overline{A}^{E}$, $C=\overline{A}^F$, and $D=\overline{A}^{EF}$ and we choose ...
- 43
1
vote
1
answer
307
views
A problem on Moebius transformations
We have the following result:
Let $R=\mathbb{C}[t]_f$, with $f=(t-a_1)(t-a_2)\cdots (t-a_n)$. Then the automorphism group of $R$ is isomorphic to the group of all Moebius transformations which fix (...
- 73
13
votes
1
answer
3k
views
When are complex polynomial maps almost surjective?
Consider a complex polynomial map $f: \mathbb{C}^n \rightarrow \mathbb{C}^n$.
For $n = 1$, the fundamental theorem of algebra says that, for any $y \in \mathbb{C}$ there exists $x \in \mathbb{C}$ ...
- 133
28
votes
2
answers
3k
views
Maximal Ideals in Formal Laurent Series Rings?
Setup: Let $k$ be a field, let $n$ be a positive integer, and let $R := k[[x_1,\ldots,x_n]]$ denote the commutative ring of formal power series over $k$ in $x_1,\ldots,x_n$. We know that there is ...
- 283
7
votes
1
answer
757
views
Characterizing intersection of zero sets of elementary symmetric polynomials on R^n
Stated simply, the question is:
Consider two elementary symmetric polynomials $\sigma_{k}$ and $\sigma_{k+1}$ on $\mathbb{R}^{n}$ with zero sets $U_{k}$ and $U_{k+1}$. Let $V_{i_{1}i_{2}\dotsb i_{j}...
- 83
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
5
votes
2
answers
1k
views
Unramified (finite) extensions of fields complete with respect to a discrete valuation
Hello,
I've been reading the excellent online book on Algebraic Number Theory by J.S.Milne. In the section described above there is a footnote maintaining that the separability of the residue field ...
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
5
votes
5
answers
4k
views
Unique factorisation and the fact that $\mathbb A^2-0$ is not an affine variety?
While learning commutative algebra and basic algebraic geometry and trying to understand the structure of results (i.e. what should be proven first and what next) I came to the following question:
...
- 14.3k
16
votes
2
answers
4k
views
A geometric reference for (affine) Gorenstein varieties and singularities
I would like to ask for a reference to some text that explains in relatively down to earth (if possible geometric) terms (for dummies) what is a Gorenstein singularity and Gorenstein variety (for a ...
- 14.3k
4
votes
0
answers
367
views
criteria for reduced fibres
I was wondering if it is foolish to ask if there is a criteria on a morphism $f: X \to Y$ between separated schemes of finite type over a perfect field which will assure that all the scheme theoretic ...
- 1,347
17
votes
4
answers
2k
views
Are units of rings of functions on algebraic varieties finitely generated (mod. constants)?
Hello,
Consider the following question. Let $A$ be a finitely generated reduced algebra over an algebraically closed field $k$. Consider the group of units of $A$, modulo the group $k^*$. Is this ...
- 5,562
16
votes
1
answer
2k
views
Deformation to the normal cone
Deformation to the normal cone appears in several places including Intersection theory and Verdier specialisation of construtible sheaves or D-modules. I'd like to understand what happens when we ...
- 7,527
8
votes
2
answers
425
views
Doing explicit computations with coordinate rings
Suppose that we are given an integral $k$-algebra $A$ of finite type explicitly, by which I mean that we are given the generators of the defining ideal $J$ where $A = k[x_1,...,x_n]/J$. What kinds of ...
- 3,918
3
votes
1
answer
497
views
Formally smooth maps between adic rings and regular immersions
Suppose $(A,\mathfrak{a})$ and $(B,\mathfrak{b})$ are two adically complete (commutative) noetherian rings. Let $f:A \to B$ be a continuous formally smooth formally of finite type map (that is, $B/\...
- 1,214
2
votes
3
answers
1k
views
General hyperplane sections and projection from a point
Let $k$ be an algebraically closed field, and consider some subscheme $X\subset \mathbb{P}_k^n$. Let $x$ be a closed point of $X$, and $H$ a general hyperplane containing $x$. There is a regular map $\...
- 45
6
votes
2
answers
456
views
Immerse an affine schemes into $A^n_S$
Suppose $f: X\rightarrow S$ is of finite type, S is Noetherian. Now X=Spec B is affine, but the morphism f is not an affine morphism. S is not affine (or really f does not factor through any affine ...
- 1,160
3
votes
3
answers
681
views
on the relative conductor of curve singularity and quotient of ideals
Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ ...
- 2,232
8
votes
2
answers
537
views
Prime avoidance in adjacent degrees
Let $\mathfrak{p}_1, \dotsc, \mathfrak{p}_k$ be relevant homogeneous primes ideals in the graded ring $R := \Bbbk[x_0, \dotsc, x_n]$, where $\Bbbk$ is a field. Prime avoidance (in Eisenbud's ...
- 7,328
3
votes
2
answers
534
views
An easy example of a (1/quasi-)Gorenstein ring with non-trival canonical divisor class.
Suppose that $R = S/I = k[x_1, \dots, x_n]/I$ is a (normal) domain of finite type over a field (or any semi-local ring $k$ with a dualizing complex). In this case, I can define $\omega_R = \textrm{...
- 20.5k
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
1
vote
1
answer
320
views
covers of complete regular local rings
It is well-known that if one assumes algebraic closedness and characteristic 0 of the residue field then finite covers of complete DVRs are all of the form $A[x]/(x^m-a)$ for some $a \in A$ (direct ...
- 4,070
0
votes
0
answers
198
views
why a reduced ring can be embedded into a sum of integral rings?
Hi,
the question is exactly
"why a reduced ring (commutative with 1) can be embedded into a sum of integral rings?"
Is this simply because in the normalization process we can have many irreducible ...
- 141
5
votes
0
answers
994
views
Maximal ideals in polynomial rings over algebraically closed fields - when Weak Nullstellensatz does not apply
Weak nullstellensatz describes maximal ideals in polynomial rings over algebraically closed fields at least when the cardinality number of variables is finite. Lang obtained the same conclusion also ...
- 17.6k