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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 ...

11
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2answers
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?
0
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2answers
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
votes
1answer
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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0answers
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
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0answers
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
1answer
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
votes
2answers
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 ...
1
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0answers
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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2answers
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 $\mathbb Q$ ...
1
vote
0answers
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
1answer
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 ...
11
votes
1answer
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
1answer
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
1answer
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
votes
1answer
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
2answers
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
votes
1answer
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
votes
1answer
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 (...
10
votes
1answer
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. ...
2
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0answers
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
1answer
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 ...
8
votes
1answer
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
votes
5answers
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,...
4
votes
0answers
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 ...
7
votes
0answers
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 ...
1
vote
0answers
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 $ ...
1
vote
1answer
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 ...
1
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0answers
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 ...
6
votes
1answer
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 ...
4
votes
1answer
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 ...
3
votes
1answer
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 ...
1
vote
2answers
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) ...
7
votes
1answer
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
votes
0answers
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 ...
2
votes
1answer
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
1answer
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. ...
3
votes
0answers
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, ...
7
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0answers
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
0answers
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
0answers
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 $k$. ...
11
votes
2answers
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 ...
1
vote
1answer
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 $B$ ?...
4
votes
1answer
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 ...
0
votes
2answers
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 ...
1
vote
0answers
123 views

direct image and commutative diagram

Suppose we have following commutative diagram (not a square i.e not a base change) of schemes: $X\xrightarrow{p_1} Y$, $Y\xrightarrow{\pi_2} Z$, $X\xrightarrow{\pi_1} W$, $W\xrightarrow{p_2} Z$. ...
3
votes
0answers
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 ...
5
votes
0answers
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 $\...
1
vote
2answers
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 $X$...
4
votes
1answer
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 ...
3
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0answers
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 ...