The concept of scheme mimics the concept of manifold obtained by glueing pieces isomorphic to open balls, but with different "basic" glueing pieces. Properly, a scheme is a locally ringed space locally isomorphic (in the category of locally ringed spaces) to an affine scheme, spectrum of a ...

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46 views

Being a regular embedding is an open condition for locally Noetherian schemes

Exercise $8.4.G$ of Vakil's algebraic geometry notes asks us to prove: If a locally closed embedding $\pi:X\rightarrow Y$ of locally Noetherian schemes is a regular embedding at $p$, then it is a ...
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2answers
105 views

Intersecting two pencils of plane curves

In $\Bbb{P}^2$, let $D_1,D_2$ be two curves of degree $d_1,d_2$ respectively. Choose two pencils $|D_1(t)|\subset|D_1|$ and $|D_2(t)|\subset|D_2|$ (free of fixed components) parametrized by the same ...
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56 views

Scheme $\{x\in \mathbb{R}^{d}:|x^{2}|\leq \varepsilon\}$ glued by which affine schemes?

I am trying to see why the $\{x\in \mathbb{R}^{d}:|x^{2}|\leq \varepsilon\}$ (equipped with a ring, say sheaf of continuous functions on that set) is a scheme i.e.to which affine schemes is it locally ...
3
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60 views

Differential forms on a projective curve: two constructions

Let $X$ be a projective curve on a perfect field $k$ ($k$-scheme integral, separated, of finite type) and let $K$ be the function field of $X$. Let's compare the following two constructions: 1. ...
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67 views

if the Hilbert polynomial of coherent sheaf is constant number,then locally free?

I'm reading "The Geometry of Moduli Spaces of Sheaves" (Huybrechts Lehn). I want to prove [$Quot_{X/S}(H,l)=Grass_S(H,l)$]where X=S and l is a number. Let T be a S-scheme and [ρ:$H_T \rightarrow F$]$...
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32 views

Checking a global property over the closed point of a local ring

Let $k$ be a field and let $T=\textrm{Spec }R$ be the spectrum of a local $k$-algebra $(R,\mathfrak m)$. Let $X\to T$ be a proper flat map from a $k$-scheme $X$, and suppose given a morphism of ...
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1answer
26 views

Details about the definition of: “deformation of a family”

Let $f:X\to Y$ be a flat, surjective morphism of $k$-schemes with connected fibres i.e. $f$ is a family. Definition: Let $T$ be a $k$-scheme. A deformation of $f$ (over $T$) is a family $g:\mathfrak ...
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2answers
75 views

Commutation of pushforward and pullback along cartesian squares

I am sure this is well-known, but I cannot find a reference. Consider a cartesian square $$\require{AMScd} \begin{CD} P @>{v}>> X\\ @V{g}VV @VV{f}V \\ Z @>{u}>> Y \end{CD}$$ where ...
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1answer
41 views

Intersection of locally closed subschemes

Let $S$ be a Noetherian scheme and $Y,Z$ two locally closed subschemes. What is the scheme theoretic intersection of $Y$ and $Z$. I am asking because in Mumford's "Lectures on curves on an algebraic ...
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1answer
43 views

sheaf defined on the support of a sheaf

Let $X$ be a topological space and $\mathcal{F}$ a sheaf on $X$. Let $Y$ be the support of $\mathcal{F}$ (Hartshorne, exercise 1.14), i.e., $Y = \left\{ P \in X: \, F_P \neq 0\right\}$. Is it true ...
3
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41 views

on the necessity of gluing conditions

Suppose we are given a family of schemes $\left\{X_i\right\}_i$, with $U_{ij}$ open in $X_i$ such that there exists isomoprhism $\phi_{ij}: U_{ij} \rightarrow U_{ji}$. Why do we need the condition $\...
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1answer
50 views

Maps to $\mathbb{P}^1$ induced by rational functions.

I am reading from Hartshorne, Corollary II.6.10 page 138. Given a nonsingular (is this necessary?) curve $X$ over a field $k$, let $f\in K(X)^*\setminus k$. Then the inclusion of fields $k(f)\...
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69 views

Visualize the affine to projective map $\mathbb{A}^{n+1}_k-\{O\}$ to $\mathbb{P}^n_k$

Classically, the natural map $\mathbb{A}_\mathbb{C}^{n+1}-\{O\} \rightarrow \mathbb{P}^n_\mathbb{C}$ maps every point in $\mathbb{A}_\mathbb{C}^{n+1}-\{O\}$ to the affine line that contains it. For ...
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1answer
43 views

Total space of a finite rank locally free sheaf, Vakil's 17.1.4 & 17.1.G

If $X$ is a scheme and $\mathcal{F}$ is a locally free sheaf of rank $n$, then in Vakil's book the total space of $\mathcal{F}$ is defined to be $Spec(\text{Sym}^\bullet \mathcal{F}^\vee)$, the ...
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1answer
37 views

Is the tangent bundle of a smooth $\Bbbk$-scheme an infinitesimal extension of it?

An infinitesimal extension of an affine scheme $\operatorname{Spec}R$ is a surjection $\hat R\twoheadrightarrow R$ with nilpotent ideal. The scheme case is defined by globalizing. I read somewhere on ...
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1answer
106 views

Question about proof that the Grassmannian is a parameter space

Edit: If it is easier to give a reference where this is written down in detail, I would gladly accept that as an answer. Fix a base scheme $B$, and fix $n$ and $k$ with $k<n$. In section 28.3 ...
4
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1answer
49 views

closed immersion onto an affine scheme - showing affineness

Let $A$ be a ring, $X=\operatorname{Spec}A$ and $f: Z \rightarrow X$ a morphism of schemes such that i) $f$ is a homeomorphism of topological spaces and ii) $f^{\#}:\mathcal{O}_X \rightarrow f_* \...
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1answer
35 views

local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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58 views

Cohomology of structure sheaf of abelian variety

Let $X$ be an abelian variety over $\mathbb{C}$ of dimension $n$. Consider the structure sheaf $O_X$. It's Euler characteristic is zero, because $\chi(O_X)= (O_X^n)/n!$. And the self intersection of $...
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0answers
45 views

The hypothetical fibre over an infinite prime in $\mathbb{A}^1_{\mathbb{Z}}$

This is just a speculative question I've been wondering about; I hope that others may find it interesting too, but if it's too vague please let me know! We can picture $\mathbb{A}^1_{\mathbb{Z}} = \...
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1answer
49 views

Isotrivial family: different definitions

Let $f:X\to B$ a flat morphism of varieties over an algebraically closed field $k$. If $f$ is flat and with connected fibres we say that $f:X\to B$ is a family over $B$. In literature you can find ...
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38 views

Factorization through diagonal morphism

The following question came up when reading Hartshorne's proof of the valuative criterion. Let $f\colon X\to Y$ be a $Y$-scheme and let $K$ be a field. Let $g\colon \operatorname{Spec}K\to X \times_Y ...
2
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1answer
104 views

Support and stalks at generic points

Let $X$ be an noetherian scheme, $Y$ an irreducible closed subscheme of $X$ with generic point $y$ and $\mathscr G$ a coherent sheaf of $\mathscr O_X$-modules. Consider the following statement: If $\...
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1answer
31 views

Trace of a linear system on a smooth projective variety

In Hartshorne's Algebraic Geometry, Chapter 2, section 7, the trace of a linear system is defined as follows. Let $i:Y\hookrightarrow X$ be a closed immersion of nonsingular projective varieties over $...
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1answer
39 views

Stalks of ringed space

Let $X$ be a locall ringed space (more narrowly a scheme, if you like) and $A=\Gamma(X,\mathcal{O}_X)$ its ring of global sections. Given a point $x\in X$, is there a prime ideal $p$ of $A$ such that $...
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1answer
77 views

Geometrical meaning of $\operatorname{Spec}\widehat{\mathcal O}_{X,x}$

Let $X$ be a complete smooth irreducible variety over a field $K$ and consider a closed point $x \in X$. Moreover let $\widehat{\mathcal O}_{X,x}$ be the $\mathfrak m_x$-adic completion of the local ...
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1answer
72 views

Pullback of an invertible sheaf through an isomorphism

Consider an isomorphism of schemes $(f,f^{\#})(X,\mathcal{O}_X)\to(Y,\mathcal{O}_Y)$. Moreover let $\mathcal F$ be an invetible sheaf on $Y$ and let $f^{*}\mathcal{F}$ be its pullback. Is it true ...
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1answer
62 views

Preimage of diagonal subscheme is a closed subscheme

Let $\alpha: X\to S$ and $\beta:Y\to S$ be $S$-schemes and let $\Delta\subseteq Y\times_S Y$ be the diagonal subscheme defined as follows (following Eisenbud-Harris): for each affine open subscheme $\...
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1answer
67 views

Synthetic differential geometry and formally étale morphisms?

Upon looking throug Kostcki's synthetic differential geometry notes, I stumbled upon the following definition. (Here $R$ is the geometric line, $W$ is a Weil algebra, and $\operatorname{Spec}_RW$ is ...
4
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1answer
180 views

Usefulness of the notion of Hilbert scheme in algebraic geometry.

Could someone tell me why and how Hilbert schemes and relative Hilbert schemes are important and useful in algebraic geometry? Could anyone give me some applications of this notion in concrete terms? ...
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1answer
58 views

Conormal bundle of Cartier divisors

Given any closed immersion of schemes $i:Z\to X$ defined by a sheaf of ideals $\mathcal{I}$ on $X$, apparently the conormal bundle is $\mathcal{C}_{Z/X}:= {\mathcal{I}}/{\mathcal{I}^2}$ "seen as a ...
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0answers
73 views

Confusion with arithmetically Cohen-Macaulay varieties

I'm a bit struck about this fact; I think it's really a silly question, but I'm not completely sure about it. Let $X\subseteq \mathbf{P}^m$ be a projective variety; choose the best hypotheses ...
2
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1answer
55 views

Intuitive way to understand identity/gluing axioms of sheaf

Is there an intuitive way to understand the identity and gluing axiom of a sheaf, specifically in the setting where the source category is affine schemes? What is the motivation for such a definition? ...
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1answer
38 views

Affine scheme obtained from (commutative) group algebra

Let $G$ be a finite abelian group (written multiplicatively), $R$ a commutative ring and let $R [G]$ denote the set of all formal linear combinations of elements of $G$ with coefficients in $R$. Then $...
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1answer
73 views

“Lifting” fibres of morphism of arithmetic schemes to get rid of “nongeometric” ramification

This is a soft question and really a request for pointers towards a certain rigorous formulation of geometric intuition I've had for some "arithmetic schemes". I'm looking for ideas and key references ...
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0answers
57 views

Does affine open set equal to distinguished open subset in an affine scheme?

For $A$ a commutative ring, does it always hold that all affine open subschemes of $\text{Spec }A$ lie over a distinguished open subset of $\text{Spec} A$?
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1answer
77 views

Is surjectivity preserved in open neighborhoods?

Let $X,S$ be schemes of finite type over a field and let $f:X\times S\to S$ be the projection. Suppose we have a morphism of coherent sheaves $\phi:\mathscr E\to \mathscr F$ on $X\times S$. Is it ...
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1answer
64 views

Differential forms on a scheme: unclear equation

Disclaimer: In this question I assume that the reader is familiar with the construction of the module of differentials $\Omega^1_{B|A}$ where $B$ is an $A$-algebra. (If you need more details about ...
3
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113 views

Is the function field of a variety a function field?

Let $X$ be an integral Noetherian scheme of dimension $n$ over a field $k$ (arbitrary field). The function field of $X$ is defined as $K(X):=\mathcal O_{X,\eta}$ where $\eta$ is the generic point of ...
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1answer
37 views

Function field on a regular scheme of dimension $1$

Let $(X,\mathcal O_X)$ be a locally noetherian scheme of dimension $1$ and suppose that $X$ is regular, that is: $\mathcal O_{X,x}$ is a regular local ring. We have no other hypothesis on $X$. What ...
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1answer
31 views

Is the set of affine morphisms the smallest set in this specific sense?

One motivation for affine morphisms I have seen is that: $\operatorname{Spec} A \to \operatorname{Spec} \Bbb Z$ should be an affine morphism for any ring $A$. The set of affine morphisms should be ...
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1answer
54 views

What is the relative version of a reduced scheme?

I have often heard it said that it is important to think of properties of a scheme $X$ as really a special case of a property of morphisms applied to the morphism $X\to \operatorname{Spec} \Bbb Z$. ...
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53 views

Is the scheme-theoretic image stable under taking products?

Let $f:X\to Y$ be a morphism of schemes and let $Z\subset Y$ be its scheme-theoretic image. If $T$ is any other scheme, consider the induced morphism $g=f\times 1_T:X\times T\to Y\times T$. ...
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1answer
57 views

Stalks of the ideal sheaf of an irreducible subscheme

Suppose that $X$ is a noetherian scheme such that $Z\subseteq X$ is a closed subscheme. Clearly $Z$ define an ideal sheaf $\mathscr I\subset\mathscr O_X$. Now let $z\in Z$ be a point such that it is ...
6
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2answers
149 views

Vakil's definition of smoothness — what happens at non-closed points?

The following is definition 12.2.6 in Vakil's notes. A $k$-scheme is $k$-smooth of dimension $d$, or smooth of dimension $d$ over $k$, if it is pure dimension $d$, and there exists a cover by ...
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0answers
31 views

Calculating sheaf of differential operators for smooth scheme

I have heard that if $X$ is a smooth scheme over $k$, then we can calculate the sheaf of differential operators $\mathcal{D}_X$ by considering étale morphisms from an affine open set to $\mathbb{A}^...
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1answer
72 views

Hartshorne Exercise II.2.18(d)

The Exercise: Let $\phi: A \rightarrow B$ be a ring homomorphism and let $X = \operatorname{Spec} A, Y = \operatorname{Spec} B$. Let $f: Y \rightarrow X$ be the morphism of schemes induced by $\phi$. ...
3
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1answer
54 views

Local properties of morphisms of schemes

In Hartshorne Proposition II.5.8, he shows, given a morphism $f \colon X \to Y$ where X and Y are schemes and $\mathcal{G}$ a quasi-coherent sheaf of $\mathcal{O}_{Y}$- modules, that $f^{*}\mathcal{G}$...
2
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1answer
41 views

Quasicoherent sheaf on the functor of points is the same as on the scheme itself

I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor $$ X : CRing \to Sets $$ as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ ...
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1answer
25 views

Infinitely many non-isomorphic degree 8, dimension zero schemes in the plane

In Geometry of Schemes by Eisenbud and Harris, it is claimed in Exercise II-19 that: There are infinitely many isomorphism types of degree 7 subschemes supported at the origin in 3-space, and ...