# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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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 ...
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 $... 1answer 236 views ### The Idea of Kroneckerian geometry Let$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$. Let's ... 1answer 237 views ### vector bundles over projective line over an affine line Let$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}$? 33answers 26k views ### What should be learned in a first serious schemes course? I've just finished teaching a year-long "foundations of algebraic geometry" class. It was my third time teaching it, and my notes are gradually converging. I've enjoyed it for a number of reasons (... 0answers 101 views ### Pullback of ideal sheaf under base change by completion of base ring Assume$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$be a flat locally ... 0answers 115 views ### Formula for fibre square (from Fulton's Intersection Theory) I have a question about argumentation used in the proof of Proposition 1.7 in Fulton's book on Intersection Theory on page 18: Proposition 1.7 Let$\require{AMScd}$\begin{CD} X' @>{g'}>> ... 0answers 170 views ### Finite Flat Group Scheme over a field$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$(a,b)$... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 : ...
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 ...