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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2
votes
1answer
206 views

Formal neighbourhood of a closed subscheme

Let $X$ be a variety and $Y \subset X$ a closed subvariety. Edit: Assume they are both smooth. Denote $N_{Y / X}$ the normal bundle of $Y$ in $X$. The formal neighbourhood of $Y$ in $X$ is the ...
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1answer
106 views

Connections on vector bundles over elliptic curves - concrete computations

This is linked to my question on math.Stackexchange for which I had no answer. I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me ...
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0answers
62 views

Existence of integral open subscheme with some nice properties

Let $R$ be a discrete valuation ring and $Spec(R):=S= \{\sigma, \eta\}$ it's affine scheme with closed point $\sigma$ and generic $\eta$. Let $f:Y \to S$ a dominant morphism of schemes of finite type. ...
2
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0answers
105 views

Morphism between jet spaces smooth

In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets: Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...
2
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1answer
235 views

Closed points of a closed subscheme of $\mathbb{P}^n$ over the residue field and the fraction field of a valuation ring $R$

Let $(R, M)$ be a valuation ring with algebraically closed fraction field $k$. Let $L = R/M$ be the residue field of $R$; it follows that $L$ is algebraically closed. I would like to understand the ...
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0answers
86 views

On an application of the going-down theorem of Cohen-Seidenberg in Mumford

There is a following result in Mumford's red book of schemes (Chapter II Section 8). Here $R$ is a valuation ring with algebraically closed fraction field $k$. Let $Z \subset \mathbb{P}^n_k$ be an ...
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0answers
126 views

Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford

In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows. Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...
5
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1answer
391 views

Schemes over (locally) ringed spaces: working over complex-analytic spaces, rigid-analytic spaces, formal schemes, etc

Monique Hakim developed in her doctoral thesis [1] the theory of relative schemes. These comprise, as a special case, the theory of schemes over (locally) ringed spaces. What makes the study of these ...
2
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1answer
204 views

Arc space & formal loops in motivic integration

One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points ...
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0answers
156 views

Preserved invariants by a flat family

Let $X, C$ be schemes and $f: X \to C$ be a "flat family". That is $f$ is flat morphism. For sake of simplicity we can say that $f$ is surjective and $C$ is an irreducible curve that "parametrizes" ...
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1answer
261 views

Pullback map on global sections surjective

Let $X,Y$ be two irreducible, projective $k$-schemes. $k$ is assumed to be algeraically closed. Consider a dominant morphism $f: X \to Y$ between them which is not an isomorphism! Let $\mathcal{L}$ ...
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0answers
178 views

Henselization and completions of local rings & schemes

That's the second part of my coarse becoming acquainted with Henselizations of fields and local rings. (in this question we focus on local rings as it is more algebro geometric motivated). So let $(R,...
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1answer
294 views

Field extensions in Grothendieck rings

Let $k$ be a field, and consider the Grothendieck ring of $k$-varieties, $K_0(V_k)$. Let $K/k$ and $K'/k$ be field extensions of $k$. We view $\mathrm{Spec}(K)$ and $\mathrm{Spec}(K')$ as $k$-schemes ...
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1answer
218 views

Double points in the Grothendieck ring

Let $K_0(V_k)$ be the Grothendieck ring of $k$-varieties, and consider the scheme $X = \mathrm{Spec}(k[x]/(x^2))$. I understand that this scheme has one point, but I am missing the fact that in $K_0(...
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1answer
128 views

Galois action on $\overline{k}$-valued points extends to action on $k$-scheme $X$

Let $X$ be $k$ variety or more genrally a $k$-scheme. Denote the algebraic closure of $k$ by $\overline{k}$. it's a fact that $X(\overline{k}):=Hom(\operatorname{Spec} \ \overline{k}, X)$ as set is ...
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1answer
787 views

Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...
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1answer
376 views

schemes having same reduced underlying space and same cotangent sheaf are isomorphic?

Let $X,Y$ be two closed subschemes of $\mathbb{A}^n_{\mathbb{C}}$ for some fixed $n$. Assume that I have an isomorphism of the underlying reduced spaces: $$f : X_\text{red} \stackrel{\sim}\...
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1answer
164 views

property of rational functions on projective curves

I have a couple of question about the proof of Lemma 1.20.5 from Janos Kolloar's Lecture on Resolution of Singularities (page 19): Lemma 1.20.5 Let $C$ be a reduced, irreducible projective curve (=1D ...
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1answer
492 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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1answer
130 views

Galois action on morphism between $\overline{k}$ schemes

I have a question on a certain property of morphisms between schemes endowed with Galois action. The motivation arises from a comment by Phil Tosteson on this question. Phil wrote: "If the map ...
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1answer
191 views

When do generizations (“generalizations”) lift uniquely?

If $f : X \to Y$ is proper, then specializations lift along $f$, and uniquely. (This means, if $R$ is a discrete valuation ring with fraction field $K$ and I choose a factorization $\text{Spec}K \to ...
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0answers
129 views

Application of Galois descent

I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it: Question: Why the assumption $k= \...
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0answers
326 views

Road map for moduli space/moduli problem/moduli stack

I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic ...
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0answers
169 views

Finiteness results in the category of schemes up to $\mathbb{A}^1$-homotopy

In algebraic geometry, we know that there exist geometrical conditions on a scheme $X/k$ for having finitely many rational points when $k$ is a number field. Namely for curves there is the Mordell ...
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1answer
230 views

Regular morphisms and formal power series

Let $A$ be a local noetherian ring. When (besides when $A$ is excellent) do we have that $\operatorname{Spec}(A[[t]])\rightarrow \operatorname{Spec}(A[t])$ is regular?
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0answers
263 views

Artin's “On Isolated Rational Singularities of Surfaces”

My question refers to M. Artin's paper "On Isolated Rational Singularities of Surfaces"; more precisely the proof of Thm 4 on page 133. Here the relevant excerpt: The Setting: Let $\bar{V}=Spec(A)$ ...
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0answers
272 views

What is the left adjoint to base change of schemes?

Restriction of Scalars and Functoriality of Presheaves. Let $\phi\colon R\longrightarrow S$ be a morphism of rings. There is associated to $\phi$ a natural functor from $\mathrm{Alg}_S$ to $\mathrm{...
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0answers
200 views

An application of Grothendieck's version of Hensel's Lemma

Assume $R$ is a Henselian local ring with $k=R/m_R$ ($m_R$ is the unique maximal ideal of $R$) and $G$ a finite flat group scheme over $R$. We denote by $G_k= G \otimes_R k$ the closed fiber. There ...
8
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1answer
345 views

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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0answers
95 views

Symmetric schemes/symmetric spaces in the category $\mathsf{Sch}_k$

From nlab, a symmetric space is a quandle object in the category of smooth manifolds. My questions are then the following: Is there a meaningful way to define symmetric spaces in the category of ...
8
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1answer
219 views

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 ...
3
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0answers
144 views

Bass theorem on non-affine scheme

A famous theorem of Bass tells that over a noetherian ring $A$, with $\operatorname{Spec}(A)$ connected, every projective module of infinite type is free. Now, consider a connected noetherian scheme $...
5
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1answer
236 views

The Idea of Kroneckerian geometry

Let $X$ be a complex, projective algebraic variety and assume that $X$ has a model $X_0$ over $\mathbb Z$ i.e. $X\cong X_0\times_{\operatorname{Spec }\mathbb Z}\operatorname{Spec }\mathbb C$. Let's ...
3
votes
1answer
237 views

vector bundles over projective line over an affine line

Let $k$ be a field and $E$ be a vector bundle over $\mathbb{P}_{k}^{1}\times\mathbb{A}_{k}^{1}$, does it extend to $\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
167
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33answers
26k views

What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (...
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0answers
101 views

Pullback of ideal sheaf under base change by completion of base ring

Assume $R$ is a discrete valuation ring with uniformizing parameter $t$, i.e. $\mathfrak{m}_R=(t)$. We denote $\widehat{R}$ the completion of $R$ with respect to $(t)$. Let $Y$ be a flat locally ...
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0answers
115 views

Formula for fibre square (from Fulton's Intersection Theory)

I have a question about argumentation used in the proof of Proposition 1.7 in Fulton's book on Intersection Theory on page 18: Proposition 1.7 Let $\require{AMScd}$ \begin{CD} X' @>{g'}>> ...
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0answers
170 views

Finite Flat Group Scheme over a field $k$ of characteristic $0$ is always Etale

I have a question about a claim about classification of finite flat group schemes: https://en.wikipedia.org/wiki/Group_scheme#Finite_flat_group_schemes A fin flat group scheme $G$ is of type $(a,b)$...
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1answer
232 views

Formal Schemes Methods: Applications

Possibly this question is bit too broad but up to now I was not able to find a satisfying answer. Let $X$ be a locally Noetherian scheme and $X' \subset X$ be a closed subscheme of $X$ which is ...
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0answers
71 views

Existence of an open dense of a connected scheme such that the preimage under a surjective finite etale map is connected

Let $K$ be a field, $G$ a smooth finite linear algebraic group over $K$, $X$ a proper reduced connected separated scheme of finite type over $K$, $g: Y \to X$ a connected etale $G$-torsor over $X$ (so ...
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2answers
2k views

Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...
16
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3answers
1k views

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 ...
4
votes
1answer
264 views

Gluing two points in an affine algebraic variety

Let $k$ be an algebraically closed field, $A$ a finitely generated $k$-algebra. Let $x,y$ be two distinct closed points of $\mathrm{Spec}(A)$. Is there an affine $k$-scheme of finite type obtained ...
3
votes
1answer
447 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 $f:C \to C’$ is a rational morphism from ...
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0answers
107 views

Application of Stein factorisation: rigidity lemma

Let $X,Y$ Noetherian schemes and $f:X \to Y$ proper map. The Stein factorisation factorizes $f$ as $X \xrightarrow{g} Spec \text{ } f_* \mathcal{O}_X \xrightarrow{h} Y$ with $h$ finite and $g$ has ...
3
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0answers
175 views

Abelian varieties by Moonen and van der Geer: proof of rigidity lemma

I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...
2
votes
1answer
301 views

Geometric interpretation of sections $H^0(\Theta_X, X)$ of the Tangent sheaf over curve

I'm reading Mumford's & Oda's Algebraic Geometry II and I'm confused about explanations on geometric intuition of sections $H^0(\Theta_X, X)$ of the tangent sheaf on page 287: Let $X$ a ...
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0answers
92 views

Formally smooth maps of schemes and factorization systems

I am thinking about how formally smooth maps of schemes relate to factorization systems. Let $C$ be the category of schemes. Let $E$ be the class of morphisms of schemes consisting of closed ...
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0answers
138 views

Spreading out a morphism of the generic fibers

Let $X$ and $Y$ be finite type schemes over $\mathrm{Spec} \mathbb{Z}$ and let $f_\xi : X_\xi \rightarrow Y_\xi$ be a morphism between the generic fibers. Then $f_\xi$ spreads out to a morphism $g_U : ...
2
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0answers
104 views

Zero section of quasi-coherent bundle

Let $S$ be a scheme and let $\mathcal{E}$ be a quasi-coherent $\mathcal{O}_S$-module. Then we can construct a graded quasi-coherent $\mathcal{O}_S$-algebra $\mathscr{A}:= Sym(\mathcal{E})$ and define ...

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