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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43
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2answers
4k 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 ...
34
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0answers
1k 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 ...
17
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0answers
1k views

History of the functor of points

Until now, I thought the functor of points approach was introduced by Grothendieck at the 1973 Buffalo seminar. However, in this note by Lawvere the author writes: "I myself had learned the ...
16
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0answers
933 views

Are all formal schemes *really* Ind-schemes?

I'm trying to understand whether there's a fully faithful functor $LRS \supset FormalSch \to IndSch$ and in what sense. Here's my progress so far: Let $\mathsf{A}$ be the category of adic rings. The ...
9
votes
0answers
255 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 ...
8
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0answers
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 ...
8
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0answers
217 views

A Hartogs-type criterion for flatness

Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two. Now, let $Y$ be a smooth quasi-affine connected variety ...
8
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0answers
346 views

Arithmetic zeta function and local zeta functions

For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product \begin{equation} \zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s), ...
8
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0answers
746 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 $k$. ...
7
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0answers
245 views

Algebraic geometry “over the function field” of the base

This is vaguely similar to, but quite different from, this question. In the above linked question the focus is on fields of large cardinality per se. Here we fix a base field $k$ (say algebraically ...
7
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0answers
217 views

Weil homotopy theory

In algebraic geometry, we have something called Weil cohomology theories, which formalize the notion of a "good" cohomology theory of smooth projective varieties. I believe that for $l$-adic ...
7
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0answers
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 ...
7
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0answers
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 ...
7
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0answers
318 views

What Spec-like functors are there?

The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there ...
7
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0answers
246 views

$\Bbb A^1$-Localisation of Schemes, and $\Bbb A^1$-Rigid Schemes

Question 1: Are there some publications or preprints that provide $\Bbb A^1$-fibrant replacements of certain classes of (smooth) schemes? Of course, smooth schemes that are $\Bbb A^1$-fibrant are $\...
6
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0answers
91 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] = \...
6
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0answers
298 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 ...
6
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0answers
179 views

Comparison of sheaves of modular forms

Let $\pi:E\to X$ the universal generalized elliptic curve over the compactified modular curve, with zero section $e: X\to E$. Now consider the following two sheaves on $X$: $e^*\Omega^1_{E/X}$ and $\...
6
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0answers
246 views

Is the stack of varieties with a big line bundle algebraic

In Starr's paper https://www.math.stonybrook.edu/~jstarr/papers/moduli4.pdf the folk result that the fibred category of pairs $(X\to S, L)$, where $S$ is an affine scheme, $X\to S$ is flat proper ...
6
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0answers
211 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 $...
6
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0answers
291 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
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0answers
413 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
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0answers
131 views

Henselian Schemes

I have a question about properties of Henselian ring/schemes exploited in M. Artin's article "On Isolated Rational Singularities of Surfaces": Here the relevant excerpt: In the excerpt we start with ...
5
votes
0answers
333 views

Schemes admitting a cover by isomorphic affine opens

Let $X$ be a Noetherian integral scheme. When does $X$ have a cover by affine open subschemes that are isomorphic as abstract schemes? Does there exist an example when we have a cover by $n$ ...
5
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0answers
230 views

Schemes monomorphing into affine scheme of dimension 1

Let $Y$ be an affine scheme of Krull dimension 1. Let $X\rightarrow Y$ be a monomorphism in the category of schemes. If $X$ is connected, is $X$ necessarily affine? What if we assume that $Y$ is a ...
5
votes
0answers
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?
5
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0answers
278 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 $\...
5
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0answers
815 views

A course on modern algebraic geometry from “The Stacks Project”

I hope this question is viable for this site. I'm sincerely sorry, if you think it isn't. For a lot of time, "EGA" by Alexander Grothendieck and Jean Dieudonne was "the" reference on the basics of ...
5
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0answers
514 views

Equivalent definitions of the Hasse invariant

As probably many others before me, I got stuck in verifying all the nice properties of the Hasse invariant. Let me start by recalling one definition: Let $E\to S$ be an elliptic curve in ...
5
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0answers
358 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 ...
5
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0answers
238 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 ...
5
votes
0answers
338 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 ...
5
votes
0answers
83 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
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0answers
393 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
1k 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 ...
5
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0answers
668 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
354 views

Why are algebraic schemes called algebraic?

In scheme theory, an algebraic scheme is the data of a scheme + a morphism of finite type to the spectrum of a field. Where does the term "algebraic scheme" come from? It does not seem intuitive to me ...
4
votes
0answers
113 views

Embed FPPF group scheme into smooth one

Let $A$ be a ring and $G$ be an affine commutative FPPF group scheme over $A$. Can we embed $G$ into a smooth group scheme over $A$?
4
votes
0answers
419 views

A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...
4
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0answers
281 views

Compact quasi-coherent sheaves

Let $X$ be a scheme. What are the compact objects in the category of quasi-coherent $\mathcal{O}_X$-modules? All references seem to discuss the derived category but I need the abelian category.
4
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0answers
211 views

Irreducible Smooth Proper one-dimensional Schemes isomorphic to $\mathbb{P}^1$

I have a curious question about an argument/hint given in following thread: https://math.stackexchange.com/questions/3062986/irreducible-smooth-proper-one-dimensional-mathbbz-schemes The OP asked if ...
4
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0answers
195 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. ...
4
votes
0answers
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 ...
4
votes
0answers
239 views

When is a coherent subsheaf determined by its global sections

I am reading an article in which a proof is based on defining a subsheaf by only giving its global sections. The exact setting is that, one has a surjective finite morphism $f:Y\to X$ between ...
4
votes
0answers
200 views

A suspected typo, and Deligne's image of the general fiber swallowing the special

In SGA 4.5 (Arcata V.1) Deligne writes: Let $X$ be a complex analytic variety and $f:X\rightarrow D$ map $X$ into the disk. Write $[0,t]$ for closed line segment with extremities 0 and $t$ in ...
4
votes
0answers
154 views

Can we classify reductive group schemes over curves

Let $C$ be a smooth quasi-projective connected curve over the complex numbers. Can one classify all reductive group schemes over $C$? Certainly, you have the trivial ones (coming from pulling-back ...
4
votes
0answers
277 views

Tannaka categories and reductive groups

The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple. Pro-reductive groups make sense over any scheme. Is there an extension of the theory ...
4
votes
0answers
317 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
150 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 $i_*(\mathcal O_Z) \cong \mathcal O_X/\mathcal J$ , ...
3
votes
0answers
166 views

Tangent Space of Picard Scheme

Let $X$ be a scheme and it's Picard scheme $\underline{Pic}(X)$ exist. Denote by $\underline{Pic}^0(X)$ the connected component of the origin of the Picard scheme. My question is what the geometric ...