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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600 views

Examples of affine schemes

I have to introduce affine schemes as topological spaces in a small seminar. Could you suggest me three examples of affine schemes in order to put in evidence the correlation among traditional ...
2
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1answer
201 views

Why do the Zariski distinguished open subsets form a base?

Let $R$ be commutative ring with unit. I have to prove that the distinguished open sets form a base for the Zariski topology i.e. any non-empty open set is a union of distinguished ones. We have that ...
3
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1answer
216 views

Generic point of a curve in affine plane

Consider the affine plane over $k$, i.e. Spec $k[x,y]$. There are three kinds of prime ideals: $(0)$, $(x-a,y-b)$, and $(f(x,y))$, for $f$ irreducible. Let the ideal $(f(x,y))$ correspond to the point ...
4
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1answer
307 views

What is Galois theory for schemes?

I have heard about "Galois theory for schemes" in this note. I haven't read it yet and I know Galois theory and a litle bit about schemes (as what it is, some properties like seperated, irreducible, ...
1
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1answer
211 views

Composition of Cartesian product is Cartesian

Let $X_1,X_2, X_3$ and $Y$ be schemes over $\mathbb{C}$. Let $f_1:X_1 \to Y, f_2:X_2 \to Y$ and $f_3:X_3 \to X_2$ be three morphism of schemes. Under the composition map $f_2 \circ f_3$ from $X_3$ to $...
5
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1answer
91 views

Tensoring a flasque resolution by a line bundle give s a flasque resolution

I had an argument explained to me the other day and I didn't quite understand one of the steps. Here's my best reconstruction: Let $\mathscr{F}$ be a quasi-coherent sheaf, and $\mathscr{L}$ a ...
5
votes
1answer
55 views

Concentrated schemes are closed under finite gluings

Let $X$ be a scheme. Assume that $X = \cup_i X_i$ is finite open covering, such that the $X_i$ as well as their intersections $X_i \cap X_j$ are concentrated, i.e. quasi-compact and quasi-separated (...
5
votes
1answer
193 views

Can I check smoothness after a base-change

Let $X\to S$ be a flat morphism of noetherian schemes. I know that I can check smoothness on the geometric fibers to see whether $X\to S$ is smooth. Let $T\to S$ be a surjective morphism. Under what ...
4
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0answers
155 views

Orbits of affine group scheme actions

Motivation Let $F$ and $K$ be distinct algebraically closed fields. Then $GL_n(F)$ acts on $M_n(F)$ by conjugation, and the nilpotent orbits of this action can be characterized by the Jordan normal ...
8
votes
2answers
846 views

Why is every Noetherian zero-dimensional scheme finite discrete?

In the book The geometry of schemes by Eisenbud and Harris, at page 27 we find the exercise asserting that Exercise I.XXXVI. The underlying space of a zero-dimensional scheme is discrete; if ...
5
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2answers
211 views

understanding schemes finite over Spec $K$

I am following Vakil's FOAG, exercise 7.3.H: Let $X\to $Spec $K$ be a finite morphism, prove that $X$ is a finite union of points with the discrete topology. I am following the guidance there. If we ...
4
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1answer
173 views

Is this the smallest non-affine scheme?

Exercise I.XXV. of the book Geometry of Schemes by Eisenbud and Harris claims that the smallest non-affine scheme has three elements with a constructed topology and sheaf. But I am wondering if this ...
1
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1answer
113 views

morphism of schemes factors through diagonal

Suppose that $X$ is a scheme over $Y$. Let $f:U \to X \times_{Y}X$ be a morphism such that $p_1 \circ f = p_2 \circ f$ where $p_1, p_2$ are projections maps from $X \times_{Y} X \to X$. Show that $f$ ...
0
votes
1answer
95 views

Are the hom sets in the category of varieties abelian groups?

This is supposedly (though I know of the proof bud haven't read it) for the Hom sets of noetherian schemes. Since every variety can be thought of as a noetherian scheme then it seems right... when ...
4
votes
1answer
260 views

Fiber product in scheme category

Let $X'$, $X$, and $Y$ be $S$-schemes. Let $f:X'\to X$ be an $S$-morphism such that $f_{x'}^{\#}$ is an isomorphism for all $x'\in X'$. Denote $g:=f\times_S \text{id}_Y:X'\times_SY\to X\times_SY$. ...
5
votes
1answer
225 views

Sheaf of meromorphic functions on an integral scheme

It is a theorem that the sheaf of meromorphic functions on an integral scheme is equal to the constant sheaf where each open set is assigned the function field of the scheme. See, for example, the ...
3
votes
1answer
79 views

Where can I find this result?

A Noetherian scheme $X$ over an algebraically closed field $k$ the set of derivations $\mathcal{O}_{X,x} \to \kappa(x)=k$, is isomorphic to the Zariski tangent space $(\mathfrak{m}/\mathfrak{m}^2)^*$ ...
2
votes
1answer
137 views

Is a prevariety the same as a notherian scheme?

By a pre-variety I mean a quasi-compact locally ringed space which can be covered by (a finite # of) affine varieties. I was wondering, this seems to be the same in scheme language as a Noetherian ...
6
votes
2answers
162 views

How does Liu intend his readers to compute $H^0(X,\Omega^1_{X/k})$ of this scheme?

In the chapter on duality theory in Liu's Algebraic Geometry and Arithmetic Curves, there is the following exercise. I'm having trouble seeing how one can apply the duality theory developed in Liu to ...
5
votes
1answer
593 views

When are complete intersections also local complete intersections?

First, let us recall some definitions. Let $P=\mathbb P^d_k$ be a projective space over a field $k$. Let $X$ be a closed subvariety of $P$ of dimension $r$. We say that $X$ is a complete ...
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vote
1answer
77 views

On an openess property

Let $S=Spec(A)$ where $A$ is a noetherian integral domain. Let $f:X\rightarrow S$ be a flat, proper morphism of schemes. Let $U\subset X$ be an open and $V=f(U)$ (in particular $V$ is open by flatness)...
8
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1answer
293 views

Computing Kähler differentials on $\mathbb P^1_A$ (What did Liu intend?)

Let $A$ be a ring and $X=\mathbb P^1_A$. On $D_+(T_0)\cap D_+(T_1)$, we have $$T^2_0d(T_1/T_0)= -T^2_1d(T_0/T_1).$$ How does one formally compute this transition? Of course this is obvious ...
4
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1answer
162 views

What is the stalk at a point of the quotient of a scheme?

This came up when I was talking to @Benjalim on the chat. Consider the space $X=\operatorname{Spec} \mathbb C[x]$. With the usual structure sheaf, this is a scheme. Let $Y$ be $X$ with the points $(...
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0answers
292 views

Why do these two constructions of a sheaf associated to a module on a projective scheme agree?

Let $B$ be a graded ring an $M$ be a $\mathbb Z$-graded $B$-module. We can associate to $M$ a sheaf of modules on $\operatorname{Proj} B$ by defining the sheaf on the principal open sets $D_+(f)$ to ...
3
votes
1answer
497 views

Some questions on the basics of invertible sheaves

Let $X$ be a scheme. A $\mathcal O_X$-module $\mathcal L$ is called invertible if, for every point $x\in X$, there is an open neighborhood $U$ of $x$ and an isomorphism of $U$-modules $\mathcal O_X|_U ...
4
votes
1answer
310 views

Elementary questions on ample invertible sheaves

Let $X$ be a quasi-compact scheme. Let $\mathcal L$ be an invertible sheaf on $X$. We say $\mathcal L$ is ample if for any finitely generated quasi-coherent sheaf $\mathcal F$ on $X$, there exists ...
2
votes
1answer
162 views

$f(Y)$ is closed iff stable under specialization

Let $f\colon Y \to X$ be a quasicompact morphism of schemes and suppose that $f(Y)$ is stable under specialization. Then $f(Y)$ is closed. I'm trying to follow the proof given here http://math.ucsd....
2
votes
2answers
162 views

A canonical homomorphism of sheaves of modules

Let $\mathcal F$ be a sheaf of $\mathcal O_X$ modules on a scheme $X$. Fix an affine open subset $U$. If $M$ is a module over the coordinate ring of $U$, we let $\tilde{M}$ denote the associated sheaf ...
4
votes
2answers
98 views

Problem 3.1.2 in Liu — Omission in problem statement?

Exercise 3.1.2 in Liu's Algebraic Geometry and Arithmetic Curves is as follows. Let $f:X\rightarrow Y$ be a morphism of schemes. For any scheme $T$, let $f(T):X(T)\rightarrow Y(T)$ denote the ...
4
votes
1answer
133 views

Finite fiber of scheme morphism is zero-dimensional?

Let $X$ and $Y$ be locally Noetherian schemes and $f:X\rightarrow Y$ be an étale morphism of finite type. Let $x\in X$ and $y=f(x)$. I would like to know why the fiber $X_y$ is a zero-dimensional ...
3
votes
1answer
139 views

Does this morphism necessarily give rise to a finite extension of residue fields?

Let $f:X\rightarrow Y$ be a morphism of finite type of locally Notherian schemes. Let $x\in X$ and $y=f(x)$. Recall that $f$ is said to be unramified if the map of stalks $g:\mathcal O_{Y,y} \...
4
votes
1answer
119 views

Flat closed immersion into a Noetherian scheme is open

Let $X$ be an irreducible Noetherian scheme. Consider some flat closed immersion into it. I want to show that it is also open, so that the morphism is surjective. I have a few thoughts, but I can't ...
4
votes
1answer
191 views

Stalk map on fiber products of schemes

Let $X$ and $Y$ be schemes and $x$ a point on $X$. Let $f:X\rightarrow Y$ be a morphism. Recall this induces a map $f_x:\mathcal O_{Y,f(x)}\rightarrow \mathcal O_{X,x}$ on the level of sheaves. ...
10
votes
2answers
624 views

Inverse of open affine subscheme is affine

This seems ridiculously simple, but it's eluding me. Suppose $f:X\rightarrow Y$ is a morphism of affine schemes. Let $V$ be an open affine subscheme of $Y$. Why is $f^{-1}(V)$ affine? I noted that $...
4
votes
2answers
109 views

Why is this composition of scheme morphisms proper?

I am learning about proper morphisms from Liu's book. I have a question about the proof of the Lemma 3.17 on page 104. Let $A$ and $B$ be rings and suppose $\operatorname{Spec} B$ is proper over $A$....
8
votes
1answer
373 views

Why is the category of coherent sheaves not grothendieck?

Let $(X, \mathscr{O}_X)$ be a ringed space. It is well-known that the category $\mathbf{Mod}(\mathscr{O}_X)$ of $\mathscr{O}_X$-modules is a grothendieck abelian category (see e.g. Grothendieck's ...
7
votes
2answers
353 views

Scheme over S and morphisms

Quoting from Hartshorne Let $S$ be a fixed scheme. A scheme over $S$ is a scheme $X$, together with a morphism $X \to S$. If $X$ and $Y$ are schemes over $S$, a morphism of $X$ to $Y$ as schemes ...
10
votes
2answers
179 views

Is the product of non-separated schemes non-separated?

My question is the title, but let me be more specific: for schemes $X$ and $Y$ over $S$, with at least one non-separated over $S$, is it true that the fibered product $X\times_S Y$ is also not ...
0
votes
1answer
98 views

Why are projective schemes $\mathbb P_A^n$ over a ring not affine for $n>1$?

I recently posted a very similar question, but I hid the question I really wanted answered in it. I'm posting this to make that question explicit. Let $A$ be a nonzero commutative ring with unit. ...
3
votes
1answer
294 views

When is the image of a closed point closed under a morphism between schemes?

Let $f: X \rightarrow Y$ be a morphism between schemes. When is the image of a closed point closed? In another question , some remarks were already made. For example if $X$ and $Y$ are of finite type ...
7
votes
1answer
230 views

Why are projective spaces over a ring of different dimensions non-isomorphic?

Let $A$ be a nonzero commutative ring with unit. Define $\mathbb P_A^n$ to be the scheme $\operatorname {Proj} A[T_0,\dots,T_n]$, where the grading on the polynomial ring is by degree. Why is it true ...
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votes
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124 views

'$R$-rational points,' where $R$ is an arbitrary ring

On page 49 of Liu's Algebraic Geometry and Arithmetic Curves, we find Example 3.32. In it, he shows that if $k$ is a field and $X=k[T_1,\dots,T_n]/I$ is an affine scheme over $k$, the sections $X(k)$ (...
9
votes
1answer
286 views

Quasicoherent ideal sheaves on open subschemes

Let $X$ be an open subscheme of an affine scheme $\operatorname{Spec} A$, let $f : X \to \operatorname{Spec} A$ be the inclusion, and let $\mathscr{I}$ be a quasicoherent ideal sheaf on $X$. Since $\...
2
votes
2answers
114 views

A particular closed subscheme

Look at the following definition: Let $(X,\mathscr O_X)$ be a scheme. A closed subscheme of $(X, \mathscr O_X)$ is a scheme $(Z, \mathscr O_Z)$ such that: $Z$ is a closed subset of $X$ ...
8
votes
1answer
414 views

Toric Varieties: gluing of affine varieties (blow-up example)

Let $\Delta$ be a fan, consisting of cones $\sigma_0=conv(e_1,e_1+e_2)$ and $\sigma_2=conv(e_1+e_2,e_2)$ and $\tau=\sigma_0\cap\sigma_1=conv(e_1+e_2)$. The dual cones are $\sigma_0^\vee= conv(e_2,e_1\!...
3
votes
1answer
293 views

Morphism of finite type between affine schemes is quasi-projective

I want to prove that given $A \to B$ a ring homomorphism of finite type, then the induced morphism of schemes $X \to Y$ is quasi-projective. A morphism is quasi-projective if it factors into an open ...
4
votes
1answer
218 views

Morphisms between locally ringed spaces and affine schemes

I need some hints to understand the conclusion of the proof of the following lemma from the Stacks Project: Lemma $\mathbf{6.1.}\,$ Let $X$ be a locally ringed space. Let $Y$ be an affine scheme. ...
2
votes
1answer
46 views

Points contained in the diagonal of the product of schemes

Let $X,Y$ schemes over $S$, and $f,g$ two $S$-morphisms of schemes, $h:X \to Y\times_{S} Y$ the morphism obtained from $f$ and $g$ and $\Delta:Y \to Y \times_{S} Y$ the diagonal morphism. I tried to ...
1
vote
0answers
35 views

von Neumann Stability help

Using the forward time centered space scheme, I transformed the equation: $u_t-2u_{xx}-u_{yy}=0$ to $u_{i,j}^{n+1}=(1-2s_{x}-2s_{y})u_{i,j}^{n}+s_{x}(u_{i+1,j}^{n}+u_{i-1,j}^{n})+s_{y}(u_{i,j+1}^{...
6
votes
1answer
459 views

Dominant morphism, equal dimensions: always finite?

Let $f:X\to Y$ be a dominant morphism of varieties (integral separated schemes of finite type over an algebraically closed field) such that dim $X$ = dim $Y$. Question: must f be finite? It seems ...