All Questions
36 questions with no upvoted or accepted answers
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
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
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
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
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
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}\,...
user138661
2
votes
0
answers
212
views
Closed points and the absolute Galois group
Suppose $\chi$ is a scheme of finite type over the field $k$; define $\overline{\chi} := \chi \otimes_{\mathrm{Spec}(k)} \mathrm{Spec}(\overline{k})$, with $\overline{k}$ an algebraic closure of $k$. ...
- 4,547
2
votes
0
answers
212
views
Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$
This is a crosspost of this MSE question.
I have asked several questions in an attmept to get a general version of the Chinese remainder theorem without conditions on the ideals which will trivially ...
- 10.5k
2
votes
0
answers
186
views
completion of non-finitely generated ideal
Let consider $A=k[x_{1},x_{2}...]$, the polynomial ring with countably many indeterminates.
Then we can consider the completion $\hat{A}=\varprojlim_{r,l}k[x_{1},x_{2},..]/(x_{1}^{r},..x_{l}^{r},x_{l+...
- 3,472
2
votes
0
answers
102
views
semicontinuity of the conductor defined by Temkin
We say a principal pair $(X,\mathcal{I})$ where $X=Spec(A)$ is affine scheme and $\mathcal{I}=\tilde{I}$ where $I\subset A$ is a principal ideal generated by $\pi$ wich is a non zero divisor.
For a ...
- 3,472
1
vote
0
answers
69
views
Descent of $G$-invariant formal system of parameters using GAGF
Let $R=(R,\mathfrak{m})$ be a comm local regular ring of char $\neq 2$ (ie $2 \neq 0$ in $R$) with maximal ideal $\mathfrak{m}$ of (Krull) dimension $2$, ie $R$ admits system of parameters $x,y \in \...
- 6,018
1
vote
0
answers
52
views
Symmetric 0-dimensional schemes with generic Hilbert function and Grassmannians
I've came across this problem while thinking about some properties of fat schemes.
Let me give you an explicit (motivating) example:
We have $S=\mathbb{C}[x,y,z]$, the coordinate ring of $\mathbb{P}^2$...
- 1,343
1
vote
0
answers
330
views
Meaning of "cut out (scheme-theoretically)"
Let $V$ be a projectively normal closed subvariety of some projective space over an algebraically closed field $\mathbb{K}$. Let $R$ be the local ring at the vertex of the affine cone over $V$ ($R$ is ...
- 839
1
vote
0
answers
189
views
Existence of integral extension of DVR satisfying some conditions
Let $X^{\prime}$ and $X$ be integral noether schemes over $\mathbb{C}$, and $p:X^{\prime}\rightarrow X$ be a surjective morphism.
Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its ...
- 297
1
vote
0
answers
96
views
depth and extension of sections
Let $S$ be an affine scheme, $X$ smooth affine over $S$ and $U$ an open subset of $X$, fiberwise of codimension at least two.
Suppose that we have a function on $U$, can we extend it to $X$?
- 3,472
1
vote
0
answers
81
views
Geometry of componentially locally strongly separable algebras
Janelidze's categorical Galois theory yields, for nice adjunctions, a good notion of covering morphism.
The category of finitely affine schemes admits such an adjunction into the category of ...
- 10.5k
0
votes
0
answers
329
views
Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 149
0
votes
0
answers
267
views
completion and tensor product
Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue ...
- 3,472
0
votes
0
answers
130
views
Is a closed subsecheme contained in a Cartier divisor?
Let $X$ be a variety over a field $k$. For a closed subscheme $Z\hookrightarrow X$ and a closed point $x\in X$ such that $\text{codim}_XZ \geq 1$ and $x\notin |Z|$, is there an effective Cartier ...
- 349
0
votes
0
answers
163
views
Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford
In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
- 3,625
0
votes
0
answers
153
views
Locus of trivialization of an extension of a vector bundle
Let $X$ be a normal noetherian affine scheme. Let $j:U\rightarrow X$ be a codimension two open of $X$ and $\mathcal{E}$ a vector bundle on $U$.
We assume that $j_*\mathcal{E}$ is a vector bundle. In ...
- 3,472
0
votes
0
answers
301
views
Irreducible component of a scheme over a dvr
Let $\mathcal M$ be a (reduced) quasi-projective scheme over a dvr (of mixed caracteristics), $R$. Suppose that the generic fiber $\mathcal M_{\eta_R}$ is (nonempty) smooth and irreducible of ...
- 155