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 1950's and ...

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22
votes
0answers
864 views

Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
20
votes
0answers
2k views

Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
6
votes
0answers
265 views

Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...
6
votes
0answers
396 views

Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
5
votes
0answers
140 views

If $X,Y$ are regular and of finite type over $S$, can $X\times _S Y$ be embedded into a regular $S$-scheme?

It seems to be well-known (see Is there an example of a variety over the complex numbers with no embedding into a smooth variety?) that a general finite type $S$-scheme does not embedd into a regular ...
5
votes
0answers
67 views

Category of the smooth formal p-groups over a local ring

Fontaine showed in Asterisque 47-48 that the category of finite dimensional smooth formal $p$-groups over the ring $A=W(k)$ of the Witt vectors over a finite field $k$ is anti-equivalent to the ...
5
votes
0answers
274 views

What is known about “singularity types” in the Murphy's Law sense?

In his "Murphy's Law" paper, Vakil gives a definition equivalent to the following: The singularity type of a pointed scheme $(X,p)$ its equivalence class, under the following equivalence relation: ...
5
votes
0answers
493 views

Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases? Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
4
votes
0answers
196 views

Differential Geometry of (Non-Abelian) Gerbes in the language of Brylinski

Context In an effort to have a definition for connection on a non-abelian gerbe, in the style of Brylinski, I am reading Breen and Messing's Differential Geometry of Gerbes [BM]. It seems that there ...
4
votes
0answers
110 views

formal smooth morphism with a formal smooth source

Let $f:X\rightarrow Y$ a morpism between $k$-schemes ( $k$ a field). We suppose that X is formally smooth and f is formally smooth and surjective. Do we have that $Y$ is formally smooth? Or if it's ...
4
votes
0answers
215 views

Is the pushout of smooth varieties along a smooth closed subvariety again a variety?

The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms. Suppose k is an algebraically closed field of ...
4
votes
0answers
534 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 $k$. ...
4
votes
0answers
491 views

Zariski tangent space of a scheme as the vector space of derivations

A standard lemma says that for a scheme $X$ of finite type over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent ...
4
votes
0answers
293 views

Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$. Does $R_\mathbb{R}$ send an ...
3
votes
0answers
102 views

Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles. ...
3
votes
0answers
216 views

Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces. Under these ...
3
votes
0answers
133 views

Does the following first order approximation of the Kapranov-Vasserot infinitesimal loops still do any job?

Let $X$ be a scheme over, say, a field $k$. Let us denote $\mathrm{Spec}(k[\varepsilon])$ by $T$ and its (unique) $k$-point by $0\in T$. Call the first order infinitesimal cone $C_{T,0}(X)$ over $X$ ...
3
votes
0answers
165 views

Does a section of a morphism of schemes give a subscheme?

Let $f:X\rightarrow Y$ be a morphism of schemes (or algebraic spaces), and $s:Y\rightarrow X$ is a section to $f$, i.e. $fs=1_{Y}$. Question: Is $s$ an (closed) immersion?
3
votes
0answers
55 views

on Neron defect of smoothness for groups schemes

Let $G$ a semisimple simply connected group over $\mathbb{C}$. Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$. We consider $I_{\gamma}$ the group ...
3
votes
0answers
262 views

A presentation of a scheme as a limit of smooth ones over finitely generated bases

Suppose that a scheme $S$ is separated, excellent, and has finite Krull dimension. Which of the following statements are true: If $S$ is regular, then it can be presented as a projective limit of ...
2
votes
0answers
63 views

Quasi-finite morphisms of stacks

Let $f:X\to Y$ be a morphism of ``nice" stacks over $\mathbf C$ such that the induced morphism on coarse moduli spaces is quasi-finite. Is $f$ quasi-finite? By a "nice" stack I mean a smooth finite ...
2
votes
0answers
107 views

Examples of non-projective morphisms with projective fibres

Let $X\to S$ be a morphism of noetherian schemes such that, for all $s$ in $S$, the morphism $X_s\to $ Spec $k(s)$ is projective. Then it doesn't follow that $X\to S$ is projective in general. In ...
2
votes
0answers
63 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 ...
2
votes
0answers
109 views

scheme of sections over complete local ring

Let $f:X\rightarrow S= Spec(k[[\pi]])$ a finite type faithfully flat morphism. Let $U\subset X$ be an open subset such that $U$ is smooth and surjective on $S$. We consider the $k$-scheme ...
2
votes
0answers
135 views

fpqc, formal smoothness

Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field), let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
2
votes
0answers
110 views

descent for formally smooth maps

Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?
2
votes
0answers
122 views

Regular subscheme of a projective limit of schemes

Let $S\cong \varprojlim S_i$, where $S$ and all $S_i$ are separated regular excellent of finite Krull dimension. Let $Z$ be a closed regular subscheme of $S$. As Theorem 8.8.2 of EGA4 shows, $Z$ comes ...
2
votes
0answers
111 views

Dualizing sheaf in mixed characteristic for regular schemes.

I've been looking many places, but everything I find seems to either talk about (a) varieties or (b) extremely general situations with dualizing complexes. As I am not in the situation of (a) (i.e. ...
2
votes
0answers
174 views

Segre class of cones and Base change of projective cones

I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme). ...
2
votes
0answers
321 views

cohomology of projective limit of schemes

Hello, Suppose that $X_i$ is a projective system of schemes and $F_i$ is a compatible projective system of abelian sheaves on the $X_i$ (i.e. if $p_{ij} : X_i \to X_j$ is the transition map, then we ...
2
votes
0answers
231 views

Can any radiciel morphism be presented as the composition of a universal homeomorphism with an immersion?

Let $f:X\to Y$ be a radiciel (=universally injective) morphism, where $X$ is a regular connected scheme. Can it be presented as the composition of a universal homeomorphism with an immersion? This ...
2
votes
0answers
185 views

Induced groupoid schemes

This is a more direct version of this question, which was perhaps a bit obtuse. This is a more elementary formulation. Recall that for a groupoid scheme (or indeed any internal groupoid) $X = (X_1 ...
2
votes
0answers
221 views

Colimit of an etale diagram of schemes

It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More ...
2
votes
0answers
521 views

Quotient morphisms in the category of schemes

Which morphisms of schemes (or varieties, if you prefer) $\pi: X \rightarrow Y$ are quotient morphisms, i.e. satisfy the following universal property (*)? (*) For any morphism $f:X \rightarrow Z$, ...
1
vote
0answers
94 views

complex automorphisms acting on projective varieties

Consider a complex projective variety $X=\operatorname{Proj}\frac{\mathbb C[T_1,\ldots,T_n]}{(f_1,\ldots,f_n)}$ with $f_1,\ldots,f_n$ homogeneous polynomials. If $\sigma\in\operatorname{Aut}(\mathbb ...
1
vote
0answers
71 views

Holomorphic convergence conditions on $\mathbb C((z))$-valued points of a group $G$

Let $G$ be a complex, connected, simply connected, semisimple group. I'm trying to compare the following two spaces: The free loop space $LG$ of $G$, and the $\mathbb C((z))$-valued points of $G$, ...
1
vote
0answers
101 views

smooth morphism from a finite type source

Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
1
vote
0answers
74 views

ind scheme and Jacobson property

Let $G$ a semisimple group over $k$ and $k$ algebraically closed. Let $G(k((t)))$ the corresponding ind-scheme, does it satisfies the Jacobson property, say closed points are dense in it?
1
vote
0answers
67 views

closed subscheme of ind scheme

Let $X$ a ind-scheme of ind-finite type and ind-affine. (e.g, take a k- smooth, affine scheme of finte type $T$, $C$ a smooth projective curve over $k$ and $x$ a closed point, then $X=T(C-x)$ verifies ...
1
vote
0answers
265 views

“reduction” of a module

Let $X$ be a scheme over a field $k$. There is a well-known antiequivalence between locally free sheaves of $\mathcal{O}_X$-modules and vector bundles over $X$. Given a module $\mathcal F$ and a ...
1
vote
0answers
139 views

When inverse image is conservative; a reference or a generalization?

I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
0
votes
0answers
56 views

Group schemes decomposition

Given an abelian group scheme of finite type $(G,+)$ over $\mathbb{F}$ connected, and given two connected closed subgroup schemes of finite type $G$ over $\mathbb{F}$ connected $H$, $N$ of $G$. ...
0
votes
0answers
93 views

Normal cones and the geometry of closed subschemes

Let $S$ be a closed subscheme of a smooth variety $M$ and suppose its ideal sheaf factors as $\mathscr{I}_S=\mathscr{I}_{S_1}\cdot \mathscr{I}_{S_2}$ for closed subschemes $S_1$ and $S_2$. Then what ...
0
votes
0answers
46 views

open subset in constructible set of divisors

Let a smooth projective curve $X$ over $\mathbb{C}$. Let a pair $(x, D)$ a pair xith a closed point $x$ and $D$ an effective divisor on $X$, such that $d_{x}:=m_{x}(D)\neq 0$. Let $N=\deg (D)$ and ...
0
votes
0answers
59 views

change of topologies and functoriality

Let X be a scheme and $\epsilon_X:X_{FL}\to X_{et}$ be the morphism of topoi from the big flat topos to the small etale topos. Let $f:X\to Y$ be a morphism of schemes. I denote $f_{top}$ the induced ...
0
votes
0answers
115 views

Reducibility of fibers over closed points implies reducibility of the generic geometric fiber?

Suppose that $f\colon X\to Y$ is a proper (or even projective) morphism of (reduced) algebraic varieties over an algebraically closed field $k$. If fibers of $f$ over all closed points of $Y$ are ...
0
votes
0answers
70 views

On the intersection complex

Let $j:U\subset X$ an open immersion between $k$ schemes integral of finite type. Let $K\in D_{c}^{b}(X,\bar{\mathbb{Q}}_{l})$ a complex of $l$-adic sheaves, such that we have that $IC_{U}$ is a ...
0
votes
0answers
94 views

on the fibers over closed points

Let $X$ and $S$ $k$-schemes of finite type . ($k$ a field) and $U$ an open subset of $X$ Let $f:X\rightarrow S$ a $k$-morphism of finite type. We assume that for any closed point $s\in S(\bar{k})$, ...
0
votes
0answers
160 views

on flat morphisms

Let $j:U\rightarrow X$ an open immersion between k-schemes of finite type and $f:X\rightarrow S$ a surjective k-morphism of finite type. We suppose that $f\circ j:U\rightarrow S$ is faithfully flat, ...
0
votes
0answers
138 views

How would you call a subscheme of a smooth $S$-scheme?

In my preprint I propose to call $X/S$ quasi-smooth if $X$ can be embedded into a smooth $X'/S$. Does this sound fine? Upd. So, smoothly embeddable is better? Is it ok to call a morphism smoothly ...