# Questions tagged [schemes]

The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1). It was finalized by Alexandre Grothendieck, during the 1950s and ...

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772 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?

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votes

**2**answers

415 views

### Defining algebraic manifold without referring to schemes

Let $M$ be a complex manifold admitting an atlas with each chart biholomorphic to $\mathbb{C}^n$ and transition maps being rational functions.
Is it true that there exists a smooth integral ...

**3**

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**1**answer

237 views

### Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...

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votes

**0**answers

41 views

### Proper family with connected fibers over a connected scheme

I was reading Q. Liu's textbook and there is an exercise:
Let $f : X \rightarrow Y$ be a surjective morphism. We suppose that $Y$ is connected
and that all of the fibers $X_y$ are connected. Show ...

**6**

votes

**0**answers

272 views

### Where is the local structure theory of étale morphisms needed?

In Stacks 02GH it is mentioned étale morphisms are locally standard étale. This seems to be the analogue of the local form of local diffeomorphisms in the smooth category. In the latter, local ...

**4**

votes

**1**answer

148 views

### When can a scheme be recovered from its descent groupoid?

Suppose that $ Y $ is a scheme and $ f\colon X\to Y $ a covering of $ Y $ in some Grothendieck topology on the category of schemes (i.e. if $\{ U_i\to Y\}$ is a covering in the topological sense, then ...

**21**

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**2**answers

1k views

### Hodge theory (after Deligne)

In an interview with Deligne on the Simons Foundation website, I heard Robert MacPherson say that at the time Deligne's papers on Hodge theory were being published, the results seemed absolutely ...

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**0**answers

63 views

### Explicit description of the scheme obtained by relative gluing data over a base scheme

I have recently been trying to get a better understanding of the projective space bundle of a quasi-coherent sheaf of graded algebras over a scheme $X$. The key idea is the following construction ...

**12**

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515 views

### Defining abstract varieties and their morphisms over a finitely generated subfield of the base field

Let $k$ be an algebraically closed field.
By a finitely generated subfield of $k$ I mean a subfield $k_0\subset k$ that is finitely generated over the prime subfield of $k$ (that is, over $\mathbb Q$ ...

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vote

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109 views

### Exceptional Curves of a Fibration

Let $f:X \to Y$ be a dominant morphism between two integral proper surfaces (therefore $2$-dimensional, proper $k$-schemes).
Futhermore we assume
for the structure sheaf holds $\mathcal{O}_Y= f_*(\...

**6**

votes

**1**answer

259 views

### Breaking a morphism with generic fiber $\mathbb{F}_n$

Assume we are working over $\mathbb{C}$, and we have a projective morphism with connected fibers $f: X \rightarrow Z$ whose geometric generic fiber $X_\overline{\eta}$ is isomorphic to a Hirzebruch ...

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206 views

### On non-representability of certain hom schemes

Let $k$ be an algebraically closed field. It is well-known that the isom-sheaf Isom$(\mathbb{A}^1_k,\mathbb{A}^1_k)$ is not representable by an algebraic space. (To be clear, the functor Isom$(\mathbb{...

**8**

votes

**1**answer

179 views

### Presentation for a Finite Etale Cover of an (Affine) Elliptic Curve

I posted this question on MSE a few days ago, but I did not get much interest in it. So I thought I would try my luck here. If you are interested in answering the question, there is a bounty over on ...

**6**

votes

**1**answer

452 views

### Are relative curves $X \to S$ determined by their fibers?

Consider relative curves $X \to S$, defined to be flat, integral, projective schemes of relative dimension 1 over $S$. When are these objects determined by their fibers?
So if $X,Y$ are $S$-schemes ...

**5**

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**1**answer

243 views

### What is the spectral interpretation of the arithmetic zeta function?

I recently stumbled upon the slides of a talk given by Kedlaya, in which the following appears:
For $X$ of finite type over $F_q$, a Weil cohomology theory, mapping $X$ to
certain vector spaces $...

**3**

votes

**2**answers

325 views

### Finite maps and jacobian condition

Let $k$ be an algebraically closed field and take $f_{1}, ..., f_{n} \in k[X_{1},..., X_{n}]$ with the jacobian condition: $\det J_{f} = 1$. Let $A:= k[X_{1},...,X_{n}]/(f_{1},...,f_{n})$ and ...

**4**

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**1**answer

169 views

### Pairing on arithmetic surfaces

Let $f: X \to S$ be an arithmetic surface, where $S=\operatorname{Spec } O_K$ for a number field $K$. It is well known that if we want to introduce a reasonable intersection theory on $X$ we have to ...

**4**

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**1**answer

153 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 (...

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**1**answer

410 views

### The étale topos of a scheme is the classifying topos of which groupoid?

[Sent here from Math.StackExchange by suggestion of an user.]
By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. ...

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**0**answers

144 views

### Weaker version of smooth base change for étale sheaves

Consider the cartesian square of schemes
$$ \require{AMScd}
\begin{CD}
X' @>{g'}>> X \\
@V{f'}VV @VV{f}V \\
S' @>>{g}> S
\end{CD}
$$
and the base change map
$$ \eta : ...

**2**

votes

**1**answer

158 views

### Properties of log smooth schemes

Let $k$ be a field and $M$ be sharp monoid (with no invertible element) consider the log point $\eta_M=(\operatorname{Spec}(k), M)$. Let $X$ be a fine saturated scheme over $\eta_M$ such that the ...

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votes

**1**answer

305 views

### Picard group and reduced schemes

If $A$ is a ring, then we know that $Pic(A)=Pic(A_{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $Pic(X)=H^{1}_{et}(X,\mathbb{G}_m)$ and étale cohomology doesn'...

**17**

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**5**answers

938 views

### Research in applied algebraic geometry that essentially needs a background of modern algebraic geometry at Hartshorne's level

By applied algebraic geometry, I don't mean applications of algebraic geometry to pure mathematics or super-pure theoretical physics. Not number theory, representation theory, algebraic topology,...

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168 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 ...

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115 views

### Do residues commute with transverse base change?

Fix a number $n > 0$. Given a smooth $\mathbb{Z}[1/n]$-scheme $X$ (i.e., a smooth scheme such that $n$ is invertible in its ring of functions), we may consider the étale sheaf $\mu_n$ on $X$ which ...

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vote

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114 views

### Which object related to families of algebraic varieties over a scheme $ S $ corresponds to the tensor product of vector bundles?

I asked yesterday on math.stackexchange.com that if the fiber product of two vector bundles seen in general as the fiber product of two families of special algebraic varieties over a scheme $ S $ ...

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vote

**1**answer

102 views

### Functorial description of a certain subgroup scheme

We work with schemes over an arbitrary field $k$. Let $X$ be a scheme, and $G$ a group scheme acting on $X$. Let $Y\subseteq X$ be a locally closed subscheme. Consider the following functor $N$: for ...

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vote

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114 views

### A question on Hilbert polynomial and flat morphisms

Note: this question was previously asked at math.stackexchange (under the same title) to no avail.
I have been recently reading Dr. Kaledin's notes on algebraic geometry. There is a statement in ...

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**1**answer

432 views

### Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over $k$?

Hopefully this question is of an appropriate level for this site: I'm reading some notes by Claire Voisin titled Géométrie Algébrique et Géométrie Complexe. Let $X$ be a smooth $k-$scheme. In these ...

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**1**answer

322 views

### Basic questions about formal schemes

I have some questions related to formal schemes. Essentially I would like to understand how much formal schemes are different from usual schemes. First of all, let me specify the definition I prefere ...

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**1**answer

161 views

### Extension of morphism of quasiprojective varieties

I am trying to understand when a morphism defined in an open dense subset of a variety can be extended to the whole variety.
For curves, it is known that if f:C->C’ is a rational morphism from the ...

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vote

**2**answers

290 views

### Embeddings of fields and rational points

Let $S$ be a irreducible scheme over a field $k$ (for example a smooth projective curve over algebraically closed field). Denote by $k(S)$ its field of fractions. Let $K$ be a(n algebraically closed) ...

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331 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 ...

**3**

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163 views

### Base change of integral scheme of finite type over a field

Let $X$ be an integral scheme of finite type over a field $k$. If $k'\supseteq k$ is a field extension, then $X' = X\otimes_k k'$ is not necessarily integral.
Why does each irreducible component of ...

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votes

**1**answer

188 views

### Does cohomology with compact support contain “ample” elements

Let $X$ be a quasi-projective integral variety over $\mathbb{C}$. If $X$ is projective, then $\mathrm{H}^2(X,\mathbb{Z})$ contains "ample" classes. These "ample" classes are defined as being the image ...

**4**

votes

**1**answer

127 views

### Group algebraic spaces that are locally of finite type and have only finitely many points

Let $k$ be an algebraically closed field of characteristic zero. Let $G$ be a group algebraic space over $k$ such that $G\to $ Spec $k$ is locally of finite type.
Suppose that $G(k)$ is finite.
...

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79 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, ...

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93 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 ...

**4**

votes

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147 views

### On representable abelian sheaves vs abelian sheaves

Let $S$ be a scheme, $(\text{Sch}/S)_{\rm Ét}$ a big étale site, and $A$ a representable
(either in schemes or algebraic spaces over $S$) abelian sheaf on $(\text{Sch}/S)_{\rm Ét}$.
Suppose there is ...

**2**

votes

**0**answers

131 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 $k$. ...

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593 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 ...

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vote

**1**answer

146 views

### Automorphism group of fiber products of schemes

Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ?...

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votes

**1**answer

300 views

### Dualizing sheaf and determinant of cohomology

Let $\pi:X\to S=\operatorname{Spec } O_K$ be an arithmetic surface in the sense of Arakelov geometry. Here $K$ is a number field $\pi$ is a flat map and $X$ is a projective surface. For any coherent ...

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votes

**2**answers

292 views

### Base change of a finite morphism

Let $X$, $Y$, $Z$ be integral schemes of finite type over a field $K$ (i.e., locally affine opens are finitely-generated algebras over $K$). Suppose we have the following condition$\colon$
$f \colon ...

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vote

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123 views

### direct image and commutative diagram

Suppose we have following commutative diagram (not a square i.e not a base change) of schemes:
$X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$.
...

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131 views

### Inverse of reduction mod $p$ functor?

I have a very general, and possibly not very precisely stated question, which comes up quite often in my work, and I would be very happy to be able to address. To my dismay, I only have some very ...

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232 views

### Injective sheaf of $\mathcal{O}_X$ modules

I am trying to verify that:
A sheaf of $\mathcal{O}_X$ modules $\mathcal{F}$ is an injective object in the category of $\mathcal{O}_X$ modules iff their local rings $\mathcal{F}_x$ are injective $\...

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vote

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270 views

### Is finite union of locally closed subscheme, a scheme

Let $X$ be a projective, noetherian $k$-scheme for an algebraically closed field $k$ of characteristic zero. Let $Y_1,...,Y_r$ be locally closed subschemes (open subschemes of closed subschemes) of $X$...

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votes

**1**answer

293 views

### Pushouts of schemes along closed immersions

Is there a closed immersion $i:Z\to X$ and a finite morphism $f:Z\to Y$ of schemes such that the pushout of the span $Y\stackrel{f}{\leftarrow} Z\stackrel{i}{\rightarrow} X$ does not exist in the ...

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157 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 ...