The concept of scheme is from algebraic geometry: A locally ringed space that is glued together (much like a manifold) from affine schemes, that is spectra of rings with Zariski topology and structure sheaf.

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Eisenbud/Haris, Exercice I-53: morphism between global spectra

Let $X=\operatorname{Spec}(\mathscr{F})$ and $Y=\operatorname{Spec}(\mathscr{G})$ two global spectra over a scheme $S$ and let $f:X\to Y$ be a morphism. I want to show that for all prime ideal sheaf ...
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higher order schemes or curvature to be smoothed

I want to calculate the radius of curvature of a meandering river. I have done a first estimated using a tool in arcgis. These are the results. The values of -99999 are recorded for points of no ...
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Restriction maps in an integral scheme are injective

Suppose $X$ is an integral scheme. I would like to show that the restriction maps $res_{U,V} : O_X(U) \rightarrow O_X(V)$ is an inclusion so long as $V$ is not empty. I was wondering if someone could ...
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64 views

Describing $Spec(\mathcal{O}_K[X])$

Let $K$ be an algebraic number field and $\mathcal{O}_K$ its ring of integers. I am trying to describe $Spec(\mathcal{O}_K[X])$ in terms of fibers of the map $g: Spec(\mathcal{O}_K[X]) \rightarrow ...
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Definition of $\mathbb{A}^n_S$ by glueing

In Eisenbud and Haris (Geometry of schemes I.2.4), if $S=\cup_\alpha U_\alpha$ with the $U_\alpha$ affines, to define $\mathbb{A}^n_S$ one take $X=\cup_\alpha \mathbb{A}^n_{U_\alpha}$ with ...
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Is $Spec(R_p)$ to $Spec(R)$ is an open immersion?

Let $R$ be a ring. $p$ be a prime ideal of $R$ . Let $R\rightarrow R_p$ be the canonical. Consider the map of schemes $Spec(R_p) \rightarrow Spec(R)$. Is it an open immersion. $Spec(R_p)$ is an open ...
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21 views

Closed subscheme of a projective scheme determined by homogeneous ideals

So in Ravi Vakil's notes Ex 8.2C, I have to prove that if $\pi:X\hookrightarrow\text{Proj}\ S_{\cdot}$ is a closed subscheme (here $S_{\cdot}$ is a graded ring finitely generated by elements of degree ...
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59 views

Why is the functor $S\mapsto\prod_{1\leq j\leq k}\mathcal{O}_{S}^{n}\left(S\right)$ representable by this scheme?

Let $S$ be a scheme and $\mathcal {O}_S$ the structure sheaf of rings over $S$. Question: Why is the functor $S\mapsto\prod_{1\leq j\leq k}\mathcal{O}_{S}^{n}\left(S\right)$ representable by a ...
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42 views

How to prove that a morphism of schemes preserves the surjectivity of ring endomorphisms?

Let $\left(T,\mathcal{O}_{T}\right)$ and $\left(S,\mathcal{O}_{S}\right)$ be schemes and $f:T\rightarrow S$. $U\subset S$ and $V\subset T$ are some open sets that correspond to the definition of ...
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75 views

Why can schemes of finite type over $\mathrm{Spec}\left(k\right)$ be considered to be affine?

Let $k$ be a field (not necessarily algebraically closed). We call $k$-variety a scheme of finite type over $\mathrm{Spec}\left(k\right)$. Let $X$ be a geometrically reduced $k$-variety and $Y$ a ...
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examples of interpreting schemes (Eisenbud)

I am having trouble understanding the role primary decomposition plays in ``interpreting'' the geometric picture of a scheme. Here are the examples I am struggling with from Eisenbud's Commutative ...
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34 views

Chow lemma + resolution of singularities

I am trying to understand the following simple reduction in Deligne's Theorie de Hodge II. He works in the category of schemes of finite type over $\mathrm{Spec}(\mathbf{C})$. Let $f : X \to S$ be a ...
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44 views

Does it hold $\mathcal{O}_X (U) =\bigcap_{x \in U} \mathcal{O}_{X, x} \in K (X)$?

maybe this is a stupid question, but I'm not seeing if this is true for some spaces (affines at least). Let $K (X) = \varinjlim\limits_{\emptyset \neq U \in \text{Open}(X)} \mathcal{O}_X (U)$ be the ...
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39 views

Alternative proof of Noether Normalization Lemma

On Mumford's Red Book (end of section 7 of chapter 2, pg 126, 127), there is an alternative proof of Noether's Normalization Lemma that goes like this: For an affine variety $X$ over an algebraically ...
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58 views

What is the canonical morphism of $\mathbb{P}^n_A$ to $\text{Spec }A$?

On Liu's book Algebraic Geometry and Arithmetic Curves, he gives a general definition of the projective scheme $\mathbb{P}^n_A$, for $A$ a ring, as $Proj (B)$, for $B=A[x_0,\dots,x_n]$. Later, he ...
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Some questions about Qing Liu's proof of Valuative Criterion of Properness

I'm studying algebraic geometry by following Mumford's Red Book, but recently I found Qing Liu's book "Algebraic Geometry and Arithmetic Curves", which seems to be more careful and detailed in its ...
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30 views

Closed subscheme vs open subscheme in ring of dual numbers

If we take the ring of dual numbers $R=k[x]/(x^2)$ for algebraically closed field $k$, we note that by the ideal correspondence theorem, the only prime ideal in $R$ is $(x)$. Thus the scheme $spec ...
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36 views

Question about composition of morphisms of schemes (Mumford's)

On section 7 of Chapter 2 of Mumford's Red Book, there is the following statement: Suppose $X\stackrel{f}{\rightarrow}Y\stackrel{g}{\rightarrow}Z$ are morphisms of schemes, such that $g$ is of finite ...
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34 views

Simple questions about morphisms of finite type and proper morphisms

I'm studying algebraic geometry (it is my first course), by following Mumford's Red Book, and now I'm stacked in some (probably silly) questions about schemes, more precisely in the section 7 of ...
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Composition of morphisms and critical values.

Definition: Let $\varphi:X\longrightarrow Y$ be a morphism between varieties over $k$. We say that $\varphi$ is smooth at $x\in X$ if the following properties holds: $\varphi $ is flat at $x$ . ...
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The Theorem on Formal Funtions - Harthshorne Theorem 11.1

Let $f:X \rightarrow Y$ a projective morphism of noetherian schemes, let $\mathcal{F}$ be a coherent sheaf on $X$, and let $y\in Y$ be a point. For each $n\geq 1$ we define $X_n=X \times_Y Spec ...
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Hilbert series computation for Hilbert scheme of $n$ points on $\mathbb C^2$

How can we show that $$\sum_{n = 0}^\infty q^n \operatorname{character}_T S^n(\mathbb C[x,y])= \prod_{p_1,p_2\geq 0}\frac{1}{1-t_1^{p_1}t_2^{p_2}q}$$ where $T$ acts on $x,y$ as $(t_1,t_2)$? ...
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69 views

Hartshorne's Exercise II. 2.15 (fully faithful functor)

I'm struggling with the last part of the exercise. Namely, let $V,W$ be any two varieties over a field $k$. We build the functor $t$, which induces a natural map $$ ...
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46 views

integral ring homomorphism

Consider a homomorphism $f: A\to B$ of commutative rings and let $b\in B$. Let $g\colon A[X]\to B[X]$ be defined by $g(X) = X$. Put $I = g^{-1}((bX-1))$ (contraction of the ideal $(bX-1)\subseteq ...
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53 views

Point at infinity of the scheme $\mathbb{P}^1$

Let $\mathbb{P}^1_k$ be the scheme defined as either $\text{Proj }k[t]$ or obtained by gluing two affine lines appropriately. What is the point at infinity which in topology usually is given by ...
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44 views

Questions about tangent and cotangent bundle on schemes

In differential geometry, for a smooth manifold $M$ we have the definition of the tangent bundle and the cotangent bundle and then $k$-forms are defined to be (smooth) sections of the $k$-exterior ...
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50 views

When the fibers of a flat morphism are varieties.

Notations: As in Hatshorne's book. Suppose that $f:X\longrightarrow Y$ is a flat morphism between two non-singular projective varieties over an algebraically closed field. Are the fibers of $f$ ...
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55 views

A particular example of a non-reduced scheme (with a reduced ring of global sections)?

I am searching for a scheme $X$ which can be obtained by gluing two affine schemes (along open subsets) such that: 1) X is non-reduced; 2) $\Gamma(X,\mathcal O_X)$ is a reduced ring. Any ideas? ...
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Is Spec ($k[x_1,x_2,\ldots])$ a smooth $k$-scheme?

Let $k$ be a field and let $A = k[x_1,x_2,\ldots]$. Note that $A$ is not of finite type over $k$. Is $\operatorname{Spec} A\to \operatorname{Spec} k$ a smooth morphism of schemes? I think it is ...
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41 views

Confused about short exact sequence involving $\mathcal{O}_{\mathbb{P}^n}$ and $\mathcal{O}_Z$ for $\pi:Z\rightarrow \mathbb{P}^n$ closed embedding

If $\pi: Z\rightarrow \mathbb{P}^n$ is a closed embedding where $Z$ is the zero set of some degree homogeneous polynomial, we have: $$0\rightarrow \mathcal{O}_{\mathbb{P}^n}(-d)\rightarrow ...
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61 views

Layman's Question on Schemes

I am reading Jordan Ellenberg's article on Arithmetic Geometry in the Princeton Companion to Mathematics. I have forgotten most of the algebra I learned since passing my qualifying exams more than 30 ...
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43 views

$(-1)$-curves and base change along a field automorphism.

Suppose that $E\subseteq S$ is a $(-1)$-curve inside a non-singular complex projective surface. By a $(-1)$-curve $E$, I mean that $E\cong\mathbb P^1_\mathbb C$ and $E^2=-1$. Now consider a field ...
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54 views

Varieties over a field $K$ are also varieties over any subfield of $K$.

Suppose that $f:X\longrightarrow\text{Spec} K$ is a variety over $K$, namely $X$ is an integral, separated $K$-scheme of finite type. Now if $L$ is a subfield of $K$, it is clear that there exists a ...
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94 views

Prove that two definitions are equivalent

Preliminary notion: Suppose that $X$ is an algebraic variety over a field $K$ ($K$-scheme, integral separated, of finite type). If $L$ is a subfield of $K$, we say that $X$ is defined over $L$ if ...
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80 views

What are the points of some schemes?

Let $X=\operatorname{Spec}\mathbb{C}[x,y,t]/(xy-t)$, $Y=\operatorname{Spec}K[x,y]/(xy-t)\rightarrow \operatorname{Spec}K$ and $Z=\operatorname{Spec}R[x,y]/(xy-t)\rightarrow \operatorname{Spec}R$, ...
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QR decomposition, borel groups and generalizations

Then every matrix $M$ in $M_{m\times m} (\mathbb{C})$ can be written in the form: $QR=M$, where $Q$ is unitary and $R$ is upper-triangular. My question is simple, does this generalize in the ...
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33 views

Existence of a rational function on a nonsingular curve with a simple zero at P and order 0 at Q

Let $X$ be a nonsingular curve over an algebraically closed field $k$ (by curve over $k$, I mean an integral separated scheme of finite type over $k$ which has dimension 1). Let $P$ and $Q$ be ...
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Eisenbud & Harris's The Geometry of Schemes proof of Prop I-18

There seems to be an error in the proof of Proposition I-18. The first inset equation (line 7) on page 20 only holds in $X_{f_a f_b}$. But it is used on line 15, where it needs to hold in $X_{f_b}$. ...
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Split extension of linear algebraic groups given by unipotent radicals and reductive groups

I've read that there is a split extension of any linear algebraic group $G$ over a perfect field given by $$ 1 \to R_u(G) \to G \to H \to 1$$ where $R_u(G)$ is the radical unipotent subgroup of $G$ ...
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contracted products of torsors

I have a question about contracted products of torsors: Is $(A \times^B C) \times^D E \cong A \times^B (C \times^D E) $?
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47 views

Fibred product of schemes

I'm learning algebraic geometry, and I'm having some difficulties in developing intuition for the fibred product of schemes. I can take schemes to be over a field (but not necessarily separably ...
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64 views

short exact sequences of linear algebraic groups and $K$-forms

This is probably a stupid question, but I can't figure it out. Let $G, G', G''$ be linear algebraic groups over a characteristic $0$ field $K$. Say that $G' \in A$ and $G'' \in B$, where $A,B$ are ...
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53 views

degree of morphism of schemes

Let $\phi: Y \to X$ be a finite etale morphism of proper smooth connected schemes over a field $K$ and suppose that the induced morphism $\phi: \overline{Y} \to \overline{X}$ has degree $n$, where ...
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trivialising cover for etale morphisms

Let $f:Y \to X$ be a finite etale morphism of smooth and proper schemes over a field $k$ (not necessarily sep closed). Is there a geometrically connected etale cover $\{U_i\}$ of $X$ which ...
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1answer
86 views

Fibres of the map $Spec\mathbb{Z}[x] \rightarrow Spec\mathbb{Z}$

I am trying to understand what is a spectrum of the ring $\mathbb{Z}[x]$. I have read Spectrum of $\mathbb{Z}[x]$ but because of my very restricted knowledge of schemes I do not understand the ...
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74 views

etale neighborhoods

I've read that quasi-compact etale morphisms of schemes over a not necessarily algebraiclly closed field $F$ (I'm happy to take $F$ a field of char $0$) are the algebraic analogs of local ...
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37 views

Notation in the Semicontinuity Theorem

In Hartshorne's book Algebraic Geometry in Chapter III Section 12 we have the following situation: $f\colon X\to Y$ is a projective morphism of schemes, $Y$ is the affine spectrume of a ring $A$ , ...
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When is the map “attaching irreducible components” an effective isomorphism?

Let $\mathcal C$ be a category with fiber products. We say that a morphism $X \to Y$ in $\mathcal C$ is an effective epimorphism if the sequence of sets $$ \text{Hom}(Y,S) \to \text{Hom}(X,S) ...
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51 views

Weil restriction for schemes

I'm trying to understand the Weil restriction of a scheme (since I'm reading a paper which uses it). I'm even having troubles trying understanding the following "toy" example. Toy example. Let $X$ be ...
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1answer
49 views

Morphism schemes after base extension

I have been reading Ravi Vakil notes on algebraic geometry, and one exercise asks if $phi:X\rightarrow Y$ and $\pi:X\rightarrow Y$ are morphisms of $k$-schemes and $\ell/k$ a field extension, if ...