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 the 1960s.

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0answers
102 views

Quasi-compactness of irreducible separated scheme locally of finite type

Is an irreducible separated scheme locally of finite type necessarily quasi-compact?
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1answer
322 views

Finiteness of surjective etale morphisms

Is every surjective etale morphism from a connected separated scheme to $A^n_{\mathbb{C}}$ of finite type? Is it finite? We use Stacks project's definitions. EDIT: From Jason Starr's answer, we ...
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130 views

Invariants of etale topological type that are not homotopy invariants

Artin--Mazur theory attaches etale homotopy type to reasonable schemes. Associated to this homotopy type are certain invariants of the scheme, such as etale fundamental group and higher homotopy ...
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171 views

Cohomology groups on small fppf site and small etale site are not the same

Let $F$ be a quasi-coherent sheaf on a scheme $X$. Is there an example where cohomology groups of $F$ on small fppf site of $X$ and small etale site of $X$ are not isomorphic?
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192 views

Basic questions about crystals and Grothendieck connections

I have a few basic questions about Grothendieck connections and crystals. I know it's bad practice to ask a bunch of different questions at once, however I feel they naturally come bundled together. ...
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2answers
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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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2answers
456 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 ...
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1answer
319 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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297 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 ...
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1answer
150 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 ...
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2answers
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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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84 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 ...
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2answers
540 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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119 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_*(\...
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1answer
269 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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1answer
243 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{...
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1answer
197 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 ...
7
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1answer
547 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 ...
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1answer
275 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 $...
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2answers
349 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 ...
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1answer
196 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 ...
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1answer
158 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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1answer
438 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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152 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 : ...
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1answer
200 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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1answer
336 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'...
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5answers
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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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0answers
201 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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125 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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0answers
117 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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1answer
103 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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1answer
649 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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1answer
458 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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1answer
232 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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2answers
291 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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1answer
352 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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0answers
258 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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1answer
199 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
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1answer
134 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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86 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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109 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 ...
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151 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 ...
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0answers
146 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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2answers
728 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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1answer
171 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
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1answer
337 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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2answers
303 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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0answers
135 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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0answers
140 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
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0answers
277 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 $\...