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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1answer
110 views

Base change of normalization map and scheme-theoretic surjectivity

Let $C$ be an affine, integral curve and $f: \widetilde{C} \to C$ be its normalization. Let $g:D \to C$ be a finite, affine, surjective morphism (note $D$ need not be reduced, but can assume ...
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121 views

Sections of non-reduced schemes

Let $X$ be an affine, irreducible (complex), generically reduced, scheme containing an embedded point, say at $x \in X$. Suppose further dimension of $X$ is strictly positive (can assume to be one ...
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85 views

Dimension of a linear system of divisors on singular curve

Consider an singular irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d>1$ over algebraically closed field $k$ which is given as vanishing locus $C=V(f(x,y,z))$ of a $f \in k[x,y,z]$ ...
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Subschemes of the affine line over a domain

Let $R$ be a domain with affine spectrum $S$ and consider the scheme $X=\mathbb A^1_R=\operatorname {Spec}R[T] $ over $S$. Let $P\subset R[T]$ be an ideal with $P\cap R=0$ and let $Y\subset X$ be the ...
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82 views

Reference for limits of schemes with non-affine transitions?

Inverse systems of projective schemes appear in several contexts, for example: in constructing the Zariski-Riemann space of a projective variety, in studying subvarieties of a projective variety ...
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90 views

Discriminant ideal in a member of Barsotti-Tate Group

Let $S = \operatorname{Spec} R$ an affine scheme (in our case latter a complete dvr) and $p$ a prime. Then Barsotti-Tate group or $p$-divisible group $G$ of height $h$ over $S$ is an inductive system ...
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1answer
131 views

Projection formula for flat morphisms

Let $f\colon X\to Y$ be a flat morphism between two smooth projective varieties. Let $L$ be a locally free sheaf on $X$ and $\mathcal{F}$ a coherent sheaf on $Y$. How to prove $f_*(L\otimes f^*\...
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160 views

Necessary condition to extend a morphism of schemes

Consider two schemes $X,Y$ over a locally noetherian scheme $S$. Let $p \in X$ and assume that $X$ is irreducible and not affine spectrum of a semilocal ring. We assume moreover we have a morphism $...
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135 views

What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?

The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...
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1answer
61 views

Does a morphism of etale sheaves restricting to a closed subscheme $Z$ induce a morphism of their subsheaves of sections supported on $Z$?

Let $X$ be a locally Noetherian scheme and $i:Z\to X$ be an immersion of closed subschemes. Let $\mathcal{F},\mathcal{G}$ be two etale abelian sheaves over $X_{et}$. We can define the subsheaf $\...
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1answer
301 views

The Serre duality theorem intuition

It is a well known fact that proper scheme $X$ over $k$ has a up to isomorphism unique dualizing sheaf (EGA I, Hartshorne). This dualizing sheaf $\omega_X$ comes with two striking properties: (i) ...
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37 views

local acyclicity when restricting to an hypersurface

Let $X$ be a smooth scheme over $\mathbb{C}$ and a constructible sheaf $K$ of complex vector spaces on $X\times\mathbb{A}^1$ and a function $g:X\rightarrow \mathbb{A}^1$. Suppose that $K$ is locally ...
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1answer
132 views

Injectivity of the cohomology map associated to the pullback of line bundles

Let $f:X\to Y$ be a flat, surjective, smooth morphism between smooth algebraic varieties (over $\mathbb C$). We assume that $f$ has relative dimension $n$ and we assume also that $\dim Y\ge 2$ (just ...
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113 views

Separable morphism of curves

A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows: Theorem 1.58 (M. Noether, 1871). Let $k$ be an algebraically closed field and $C \subset \...
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114 views

Surjective morphisms dominated by some fpqc covering

Now I'm reading Cornell and Silverman's "Arithmetic Geometry" and I have trouble with a statement in this book. On page 39 of this book, the author says "Now $G\to Q$ is a surjection, consequently it ...
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1answer
101 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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58 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. ...
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1answer
192 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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97 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 ...
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83 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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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 ...
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1answer
214 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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1answer
177 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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143 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" ...
2
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1answer
180 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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1answer
279 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
109 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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170 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,...
3
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1answer
205 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(...
5
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1answer
366 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
374 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 ...
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1answer
748 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
160 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 (=...
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1answer
112 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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120 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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162 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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229 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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235 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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184 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 ...
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1answer
301 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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1answer
180 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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93 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
186 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 ...
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143 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 $...
6
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1answer
223 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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1answer
222 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 ...
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0answers
108 views

When do the spectra of overrings glue to a proper morphism?

This question is motivated by the construction of blowups. Let $A \subset K$ be a commutative domain and its fraction field, and let $\{A_i\}$ be some finite collection of overrings in between. Let $...
3
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1answer
226 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 $\mathbb{P}_{k}^{1}\times\mathbb{P}_{k}^{1}$?
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90 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
108 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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