# 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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### Natural correspondence between the set of morphisms and the set of global sections

I'm trying to prove the following claim. Let $X$ : smooth projective scheme over $\mathbb{C}$, $L$ : line bundle over $X$ and $A$ : $\mathbb{C}$-algebra. $X_A=X\times_{\mathbb{C}} \operatorname{Spec}A$...
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### What is the natural motivation for smooth/étale/unramified morphisms restricting from formally smooth/étale/unramified morphisms?

(I asked it first in MathStackExchange but I haven't get an answer yet) Smooth (resp. étale) morphisms are just locally finitely presented + formally smooth (resp. étale) morphisms. For unramified ...
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### “Real algebraic varieties” vs finite type separated reduced $\mathbb{R}$-schemes with dense $\mathbb{R}$-points

This question is partly motivated by a few comments here. Let me denote by $R$ the (real-closed) field of real numbers $\mathbb{R}$; everything is probably the same over an arbitrary real-closed field....
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### Closed map of schemes and Frobenius reciprocity

A map of locales $f : X \rightarrow Y$ is closed if it satisfies the reciprocity relation $f_*(f^*(X) \vee Y) \cong X \vee f_* (Y)$. How can we express a that a map of schemes $f : X \rightarrow Y$ ...
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### Use of Flattening Stratification part 2 (Nitsure's construction of Hilbert and Quot schemes)

I study Nitin Nitsure's paper Construction of Hilbert and Quot Schemes (arXiv:math/0504590) and have some problems with the content of imposed universal property (F) in the section "Use of ...
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### What are the sections of an ideal sheaf on a scheme?

Suppose $X$ is a scheme and $f_1,...,f_n\in \Gamma(X,\mathcal O)$ are global sections. One often reads about the ideal sheaf $\mathcal I=\mathcal (f_1,...,f_n)\subset \mathcal O$, but I have never ...
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### Algebraic spaces in the étale topology (proof from Stacks project)

I have a question about the proof of Lemma 78.12.1 from Stacks Project. The aim of the last paragraph of the proof is to verify that the map of sheaves in the étale topology $F \to U/R$ is an ...
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### Algebraic spaces as quotients of schemes (Definition from wikipedia)

I think that wikipedia article on Algebraic spaces contains a serious content error in the part on the definition of Algebraic spaces as quotients of schemes and I would like to discuss if it is ...
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### Projection from closure of locally closed subscheme is Etale

Let $S$ an arbitrary scheme and denote by $\Delta: S \to S \times S$ the diagonal immersion and $p_i: S \times S \to S$ the both projections to first resp second factor. (in following we will wlog ...
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### arithmetic del Pezzo surfaces in comparison with del Pezzo surfaces over a field

A del Pezzo surface is a smooth, 2-dimensional projective variety $X$ with ample anticanonical divisor, i.e. a 2-dimensional Fano variety. I am interested in the arithmetic analogue, a 2-dimensional ...
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### $\mathscr Coh_{X|S}$ is algebraic and of finite type

Let $S$ be a Noetherian scheme and $X$ a projective $S$-scheme. Reading Laumon-Moret--Bailly's "Champs Algebriques", Theorem 4.6.2.1: $\mathscr Coh_{X|S}$ and $\mathscr Fib_{X|S,r}$ are ...
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### Global regular functions and restriction to the fiber

Let $S$ be a local scheme (spectrum of a local ring) with closed point $s$ and $f:X \to S$ a morphism of schemes. Under which conditions on $f$ and $S$ is the natural map  H^0(X,\mathcal{O})\otimes ...
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### Upper semi-continuity of intersection numbers

Consider a smooth projective morphism of schemes $X \rightarrow S$ with relative dimension $n$ (the application I have in mind is with $S$ = an open subset of $\text{Spec } \mathbb{Z}$) and assume ...
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### Fixed point scheme definition

I'm sorry if this is a trivial question, but it seems I can't find a clear answer. I have a finitely generated Poisson algebra $A$, the Poisson scheme $X=Spec(A)$ and an automorphism $g$. What is ...
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### 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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### 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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### 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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### 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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### 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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### 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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### Separable extensions & topology vs inseparable extensions and algebra

In the note Properties of fibers and applications, Osserman writes above Definition 1.5: Intuitively, the point is that phenomena relating to topology can only change under separable extensions, ...
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 $\... 0answers 133 views ### Other interesting notions when we change topology on$\text{Sch}/S$Let$\text{Sch}$be the category of schemes. Let$S$be an object of$\text{Sch}$. Consider the category$\text{Sch}/S$. Some interesting topologies on$\text{Sch}/S$are Zariski, fpqc, étale, fppf..... 0answers 41 views ### local acyclicity when restricting to an hypersurface Let$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$is locally ... 1answer 1k views ### Why Use Hypercohomology When Defining the de Rham Cohomology of a Smooth Scheme over$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-$scheme. In these ... 1answer 154 views ### Injectivity of the cohomology map associated to the pullback of line bundles Let$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$(just ... 0answers 140 views ### Separable morphism of curves A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows: Theorem 1.58 (M. Noether, 1871). Let$k$be an algebraically closed field and$C \subset \...
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 ...