All Questions
71 questions
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
26
votes
1
answer
4k
views
Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?
It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$
versus the sheafification of a pre-sheaf.
The definition of the sheaf $\mathscr F^+$ ...
- 363
12
votes
1
answer
480
views
Extending properties of commutative rings to schemes
I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\...
- 2,814
11
votes
2
answers
2k
views
Is a scheme Noetherian if its topological space and its stalks are?
Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?
- 564
11
votes
2
answers
2k
views
Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
- 10.5k
10
votes
1
answer
265
views
Finite coverings by closed subschemes
Let $X$ be a scheme. Assume we have two closed subschemes $Y_1$, $Y_2$ that cover $X$ set-theoretically. Are there closed subschemes $Y'_1$, $Y'_2$ with the same underlying sets, such that the ...
- 19.6k
10
votes
1
answer
599
views
Isbell Duality and Dualizing Scheme Objects
I'm not sure if this question is too elementary for MO; nevertheless, I have seen many helpful discussions surrounding this topic here. I'm interested in studying the adjunction $\operatorname{Spec}\...
- 223
9
votes
1
answer
1k
views
Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
- 3,472
9
votes
0
answers
400
views
Weierstrass division theorem for henselian rings
Let $A$ be an henselian local noetherian ring. There is an old result of Lafon ("Anneaux henséliens et théorème de préparation" (1967)), which says that if $A$ is analytically normal and of ...
- 3,472
8
votes
1
answer
472
views
Nonnegative additive functions on coherent sheaves
Let $(X,\mathcal{O}_X)$ be a Noetherian integral scheme and let $g$ be a (numerical) additive nonnegative function from coherent $\mathcal{O}_X$-modules to $[0,\infty)$. This question may be well ...
- 3,101
8
votes
0
answers
127
views
universally open and connected fibers
Let $A$ be a coherent ring, and consider the map:
$Spec(A[[t]])\rightarrow Spec(A)$,
in particular, we know that it's flat. Is it universally open? Does it have connected fibers?
N.B: Easy if A is ...
- 3,472
8
votes
0
answers
1k
views
Two definitions of smoothness?
This is confusing, there appear to be possibly two definitions of smoothness in algebraic geometry for a morphism $f: X \rightarrow Y$ of schemes of finite type over an arbitrary field $k$.
...
- 3,555
7
votes
2
answers
1k
views
Beauville-Laszlo for schemes
For a commutative ring $A$ and $f\in A$ a non-zero divisor, the Beauville-Laszlo theorem gives a gluing statement for vector bundles on $A$ in terms of a vector bundle on $A\big[\frac{1}{f}\big]$, a ...
- 3,472
7
votes
2
answers
327
views
How to understand the "boundary" of subscheme, as defined in "An elementary characterisation of Krull dimension"
In An elementary characterisation of Krull dimension and A short proof for the Krull dimension of a polynomial ring, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, ...
- 1,142
7
votes
1
answer
452
views
Recovering the Zariski topology from the Zariski topology over an extension
Suppose $A$ is a $k$-algebra, with $k$ a field, and let $\ell$ be a field extension of $k$. Is there an easy way to see/recover $\mathrm{Spec}(A)$ in/from $\mathrm{Spec}(A \otimes_k \ell)$, using the ...
- 4,547
6
votes
1
answer
892
views
A naive algebraic geometry question
Suppose $X$ is a scheme over a ring $A$, $B$ is an $A$-algebra, and $X\times_AB$ is affine. I am looking for conditions on $A$ and $B$ (and perhaps the structure morphism of $X$ over $A$) that will ...
- 61
6
votes
1
answer
434
views
Regular morphisms and formal power series
Let $A$ be a local noetherian ring. When (besides when $A$ is excellent) do we have that $\operatorname{Spec}(A[[t]])\rightarrow \operatorname{Spec}(A[t])$ is regular?
- 3,472
6
votes
0
answers
190
views
Computing the automorphism scheme of projective space
$\newcommand{\Spec}{\operatorname{Spec}}$I'm trying to understand why $PGL_{n}$ is the automorphism scheme of $\mathbb{P}^{n-1}_{\mathbb{Z}}$.
In Conrad's Reductive Group Schemes, the following ...
- 605
6
votes
0
answers
111
views
Relation between $\mathrm{Proj}(A[tI])$ and $\mathrm{Proj}(A[tJ])$ for ideals $I$ and $J$ of $A$ with $I^2 \subset J \subset I$
Let $k$ be a field and $A$ a noetherian local $k$-algebra.
Let $I$ and $J$ be two ideals of $A$ with $I^2 \subset J \subset I$.
Let $A[tI] = \bigoplus\limits_{i\geq 0} t^i I^i$ and $A[tJ] = \...
- 523
5
votes
1
answer
690
views
Gluing two points in an affine algebraic variety
Let $k$ be an algebraically closed field, $A$ a finitely generated $k$-algebra. Let $x,y$ be two distinct closed points of $\mathrm{Spec}(A)$. Is there an affine $k$-scheme of finite type obtained ...
- 796
4
votes
1
answer
2k
views
On Q-Cartier Divisors
I have my question on Q-Cartier Weil divisor.
People say $D$ is Q-Cartier divisor if $nD$ is Cartier for some $n \geq 1$. Especially for $n > 1$, I have never seen the `rigorous' definition of $...
- 781
4
votes
1
answer
904
views
Flat family of normal schemes over a normal base
Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal.
Can I say something about the ...
- 43
4
votes
1
answer
259
views
Embedding a finite morphism into a finite morphism of smooth varieties
Let $f\colon X\to Y$ be a finite morphism of quasiprojective varieties (I’m most interested in the case that $f$ is normalization and the varieties are over $\Bbb C$). Is the following statement true (...
- 3,079
4
votes
1
answer
385
views
Vector bundles on complete rings
Given a ring $A$ and an ideal $I$, consider the completion $\hat{A}$. What does usually mean by a vector bundle on $\hat{A}$? One way is to consider projective $\hat{A}$-modules. Another one is a ...
- 5,901
4
votes
0
answers
396
views
Non-Noetherian (classical) algebraic geometry
My starting point for this question is that, in a very classical sense, algebraic geometry is the study of solution spaces of systems of polynomial equations over an algebraically closed field. It is ...
- 365
4
votes
0
answers
833
views
When spreading out a scheme, does the choice of max ideal matter?
I'm looking at Serre's paper How to use Finite Fields for Problems Concerning Infinite Fields. Specifically I'm trying to use the techniques in the proof of Theorem 1.2 to write out the details of the ...
- 1,161
3
votes
2
answers
243
views
Minimal fields of isomorphism for varieties
Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...
- 11.3k
3
votes
1
answer
190
views
Irreducibility under etale ring map
Let $A\rightarrow B$ be a etale ring map between finite type algebra over algebraically closed field $k$.
If $A$ is one dimensional integral domain, is $B$ direct product of finite type integral ...
- 328
3
votes
1
answer
522
views
Flatness of Normalization of regular schemes
I have a followup to the following question: Flatness of normalization.
Suppose that $X$ is a regular scheme (of finite type over a $\mathbb{C}$ if one wants) and $X'$ is the normalization of $X$ in ...
- 2,824
3
votes
1
answer
514
views
characterisation of regular morphisms
Recall that a morphism of schemes $X\rightarrow Y$ is regular if it's flat with geometric fibres that are regular schemes.
Fix a field $k$ and consider a morphism $f:X\rightarrow Y$ of noetherian $k$-...
- 3,472
3
votes
1
answer
282
views
Separable extensions & topology vs inseparable extensions and algebra
In the note Properties of fibers and applications, Osserman writes above Definition 1.5:
Intuitively, the point is that phenomena relating to topology
can only change under separable extensions, ...
- 10.5k
3
votes
0
answers
347
views
Sections of non-reduced schemes
Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one ...
- 2,022
3
votes
0
answers
155
views
Bass theorem on non-affine scheme
A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free.
Now, consider a connected noetherian scheme $...
- 3,472
3
votes
0
answers
240
views
Smallest non-vanishing index of Local-Cohomology and Ext functor, as in the definition of depth, for not necessarily affine schemes
Let $(Z,\mathcal O_Z)$ be a closed subscheme of a Noetherian scheme $(X,\mathcal O_X)$. Then there is an ideal sheaf $\mathcal J$ on $X$ such that $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
- 642
3
votes
0
answers
205
views
Formally unramified morphisms and open diagonals
Let $R\to S$ be a commutative unital $R$-algebra with dual arrow $X\to Y$. If I understand correctly, an open diagonal $\Delta_{X/Y}:X\to X\times _YX$ always implies $X\to Y$ is formally unramified, ...
- 10.5k
3
votes
0
answers
294
views
Formal smoothness implies local freeness of the sheaf of relative differentials
What is the least restrictive finiteness assumptions guaranteeing that for a formally smooth morphism of schemes $f:X\rightarrow Y$, the sheaf of relative differentials $\Omega _{X/Y}$ would be ...
- 953
3
votes
0
answers
565
views
going down theorem
Typical maps that satisfy "going down theorem" are flat morphisms and integral extensions of normal rings that are integral.
Let $Spec(B)\rightarrow Spec(A)$ be a finite type morphism of k-noetherian ...
- 3,472
2
votes
1
answer
177
views
Constructibility of the locus of points where the fiber is an isomorphism modulo nilpotents
Let $f: X \rightarrow S$ be a finitely presented morphism of schemes and let $$E = \{s \in S \mid \text{ $X_s$ is a point with residue field $\kappa(s)$ } \}$$
Is $E$ a constructible set?
The basic ...
- 938
2
votes
1
answer
437
views
Extending functors between K-algebras to schemes
Assume we have $K$ and $L$ (comm.) rings, and we have a functor $F$ from the category of $K$-Algebras to the category of $L$-Algebras (I work only with commutative rings). What conditions need to ...
- 608
2
votes
1
answer
656
views
Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$
Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
- 3,625
2
votes
1
answer
185
views
Finite étale cover of factorial ring
Let $A$ be a regular factorial ring.
Consider $B=A[X]/(P)$ such that $B$ is finite étale over $A$. When do we have that $B$ is also factorial?
- 3,472
2
votes
1
answer
350
views
Base change of normalization map and scheme-theoretic surjectivity
Let $C$ be an affine, integral curve and $f: \widetilde{C} \to C$ be its normalization. Let $g:D \to C$ be a finite, affine, surjective morphism (note $D$ need not be reduced, but can assume ...
- 2,022
2
votes
1
answer
1k
views
Affine communication lemma and finite limits in the category of rings
Let $X$ be a scheme and $\mathrm{Spec}(B) = V \subseteq X$ be an open affine subset. When using the affine communication lemma (c.f. Theorem 6.3.2, Vakil's notes, Foundations of Algebraic Geometry), ...
- 201
2
votes
0
answers
169
views
Principle of degeneration as precursor of Zariski's connectedness theorem (geometric intuition)
I have following question about so-called "principle of degeneration"
in algebraic geometry (which in modern terms is an immediate consequence
of Zariski's main theorem and goes in it's ...
- 6,018
2
votes
0
answers
730
views
What algebraic condition corresponds to injectivity of a morphism of varieties?
$\DeclareMathOperator{\Spec}{Spec}$
Let $X = \Spec A$, $Y = \Spec B$ be affine complex varieties, that is reduced $\mathbb{C}$-schemes of finite type. Equivalently we can say that $A$ and $B$ are ...
- 1,141
2
votes
0
answers
148
views
etale locally infinitesimal lifting property
For a morphism $X\rightarrow Y$ of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair $(R,I)$ with $I^2=0$, if there is a point $y\in Y(R)$ such that $y_{\vert R/I}$...
- 3,472
2
votes
0
answers
231
views
Necessary condition to extend a morphism of schemes
Consider two schemes $X,Y$ over a locally noetherian scheme $S$. Let $p \in X$ and assume that $X$ is irreducible and not affine spectrum of a semilocal ring.
We assume moreover we have a morphism $...
- 6,018
2
votes
0
answers
216
views
On an application of the going-down theorem of Cohen-Seidenberg in Mumford
There is a following result in Mumford's red book of schemes (Chapter II Section 8). Here $R$ is a valuation ring with algebraically closed fraction field $k$.
Let $Z \subset \mathbb{P}^n_k$ be an ...
- 3,625
2
votes
0
answers
416
views
Henselization and completions of local rings & schemes
That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...
- 6,018
2
votes
0
answers
114
views
Noetherian affine schemes for which localization computes the values of the structure sheaf
Is there a non-tautological characterization of the class of commutative unital Noetherian rings satisfying the following property: for any open subset $U$ of the topological space of $\mathrm{Spec}\,...
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