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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Applications of schemes to mathematical physics
Could anyone cite some applications or developments in mathematical physics or string theory that use schemes?
I find curious the fact that while things like derived algebraic geometry and stacks ...
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Universal homeomorphisms and the étale topology
Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section?
The answer is yes if $S$ is reduced, by descent. ...
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Infinite projective space
Is infinite (say complex) projective space a scheme? More generally, can schemes have infinite cardinal dimension? It seems that infinite dimensional projective space is not a manifold, since it is ...
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Minimal number of generators for $A^n$
Let $A$ be a commutative ring and $n \in \mathbb{N}$. What is the minimal number $e_A(n)$ of generators of the $A$-algebra $A^n$? Here is what I already know (I can add proofs if necessary) from a ...
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Can an algebraic space fail to have a universal map to a scheme?
Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there ...
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Making the étale topos construction a fully faithful 2-functor from schemes to Grothendieck topoi
For nice topological spaces (say Haudorff spaces) $X$ and $Y$, there is a bijection between continuous maps $X\to Y$ and isomorphism classes of geometric morphisms $\mathrm{Sh}(X)\to \mathrm{Sh}(Y)$.
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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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Non-simply-connected smooth proper scheme over Z?
Source
This question came up in the discussion between Kevin Buzzard and Minhyong Kim in the comments to Smooth proper scheme over Z. It was 2 weeks ago, so I took the liberty of posting it as ...
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Constructing a degeneration (as a group scheme) of G_m to G_a
SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga.
What is such an example?
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explanation on a scheme which is not affine scheme
Hartshorne at the end of page 76 of his Algebraic Geometry book gives an example of a scheme which is not an affine scheme. The scheme is constructed by gluing two affine lines together along their ...
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Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?
A scheme is defined to be a sheaf which is locally isomorphic to the spectrum of a ring. The idea behind this is that given an affine coordinate ring of a variety over an algebraically closed field, ...
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Which local ringed spaces are schemes?
(This was originally asked on math.stackexchange, but didn't get any responses. I figured it might be worthwhile to move it here and try again.)
This paper gives a proof that the underlying ...
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Is there an analogue of projective spaces for proper schemes?
Does there exist a countable set of connected proper smooth $\mathbb{C}$-schemes such that any connected proper smooth $\mathbb{C}$-scheme admits a $\mathbb{C}$-immersion into one of them?
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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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Do disjoint unions and fiber products commute?
Do disjoint unions and fiber products commute?
In other words, is the following statement true?
Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...
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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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Is the degree of a finite morphism stable by base change
Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes.
Is the degree of the finite morphism $X\times_Y S \longrightarrow S$ equal to $n$?
If ...
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Jacobian criterion for smoothness of schemes
An affine scheme $X = Spec(A)$ is said to be smooth if for any closed embedding
$X\subset\mathbf A^n$, of ideal $I$, it is true that, locally on $x\in X$, the ideal $I$
can be generated by a sequence $...
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Obstructed automorphisms of schemes
Let $X$ be a smooth projective scheme over a field $\mathbf{k}$ of characteristic zero such that $\mathrm{H}^0(X, \mathrm{T}X)$ vanishes, and let $f$ be an automorphism of $X$. I would like to have an ...
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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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Motivation for Henselian rings in algebraic geometry
In Andrew Kobin's script on Algebraic Geometry
I found on page 355 a comment I would like better understand. It states
Another
way to view formal smoothness is as an abstraction of Hensel's Lemma.
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Extending properties of commutative rings to schemes
I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme $(X,\...
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When is the push-forward of the structure sheaf locally free
Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then $f_\...
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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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What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?
More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
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The category of finite locally-free commutative group schemes
I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian ...
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Nonalgebraic complex manifolds
I'm turning here (a variation of) a question asked by a friend of mine. For the purposes of this question I will say that a compact complex manifold is projective if it is isomorphic to a subvariety ...
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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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About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$
I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...
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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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Sheaf associated to presheaf Aut
Let $S$ be a scheme and let $C$ be the category of schemes flat and locally of finite presentation over $S$. Endow $C$ with the fppf topology (or perhaps any subcanonical topology). Let $\mathcal P$ ...
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pushforward of locally free sheaf is locally free?
Hi,
Is there an example of a proper smooth map of schemes $f:X\to Y$ and a vector bundle $E$ on $X$
such that $f_*E$ is not locally free on $Y$?
Thanks
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Isomorphic schemes over DVR
Let $S, S'$ be flat schemes over a DVR. Their generic fibers are isomorphic and their special fibers are isomorphic as well.
Does that imply $S$ and $S'$ isomorphic? If not, what can go wrong?
Thanks ...
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Parahoric Group Scheme
I am looking for the definition of a parahoric group scheme in the sense of Bruhat and Tits? I couldn't find a reference for that? at least a "clear" reference!
thanks
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How does descent theory imply a sheaf is a scheme?
I've noticed that often authors will comment that "descent theory" shows that some sheaf in the étale topology is actually a scheme. I was wondering what result in descent theory actually implies ...
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Finite coverings by closed subschemes
Let $X$ be a scheme. Assume we have two closed subschemes $Y_1$, $Y_2$ that cover $X$ set-theoretically. Are there closed subschemes $Y'_1$, $Y'_2$ with the same underlying sets, such that the ...
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Is the functor of points of a scheme cofinally small?
Background: In functorial algebraic geometry one would like to consider the category of all functors $\mathsf{CRing} \to \mathsf{Set}$ and define/characterize the category of schemes as a full ...
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Can a scheme be defined by gluing open affines such that the intersections are affine?
One way to think of a manifold is as a family of of open subsets $U_i \subset \mathbb{R}^n$, together with distinguished subsets $V_{ij} \subset U_i$ and isomorphisms $\psi_{ij}: V_{ij} \to V_{ji}$ ...
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Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
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Can the algebraic geometry of schemes be developed internally in topoi?
Using the internal logic of a topos it's often possible to derive newer theorems about sheaves from earlier ones about simpler objects, assuming that you can prove the earlier ones constructively. In ...
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Diagonal map and "infinitesimal points"
Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define $...
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Are higher etale homotopy groups topological groups in a natural way?
Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. ...
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Is the category of schemes wellpowered? regularly wellpowered?
Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...
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Simple examples of colimits of affine schemes (evaluated in the presheaf category) which are not affine schemes
Notation and Setting: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{...
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Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence?
Main Question: What Is the correpondence between flows and vector
fields in algebraic geometry?
Here is a more precise statement could be an answer If it was true (I have no idea it is):
"...
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Representable diagonal map $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$ for DM-Stacks/algebraic spaces
Following the standard definitions of a algebraic space or Deligne–Mumford stack one imposed condition is that the diagonal morphism $\Delta: \mathcal{X} \to \mathcal{X} \times \mathcal{X}$
has to be ...
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Tube of a mod p point on a smooth Z_(p)-scheme
Let $R$ be a smooth, integral, finite-type $\mathbb{Z}_{(p)}$-algebra of relative dimension $n$ and $\overline{f} \colon R \to \mathbb{F}_p$. Then Hensel's lemma tells us that this lifts to a map $R \...
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Picard group and reduced schemes
$\DeclareMathOperator\Pic{Pic}$If $A$ is a ring, then we know that $\Pic(A)=\Pic(A_\text{red})$, but for a scheme $X$ it is false in general.
On the other hand, we have that $\Pic(X)=H^{1}_{et}(X,\...
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Torsors trivializing over a fixed finite etale cover
Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.
Is the set of $S$-isomorphism classes of $G$-torsors ...
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on the local structure of schemes
Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:
$\...
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