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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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
583 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 ...
1
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 ...
8
votes
1answer
292 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 ...
4
votes
1answer
160 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 ...
9
votes
0answers
290 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
479 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
304 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
161 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 ...
2
votes
2answers
161 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
130 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
138 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
190 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
607 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 ...
8
votes
1answer
371 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
344 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
96 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
285 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
229 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 ...
9
votes
2answers
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
405 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= ...
3
votes
1answer
291 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
215 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 ...
6
votes
1answer
449 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 ...
6
votes
1answer
165 views

Finishing the proof that $\textrm{Spec}K[X,Y]\setminus\{(X,Y)\}$ is not an affine scheme

If $K$ is a field, $\mathbb A^2_K=\textrm{Spec}K[X,Y]$ and $U=\mathbb A^2_K\setminus\{(X,Y)\}$, I want to prove that $(U,\mathscr O_{\mathbb A^2_K|U})$ is not an affine scheme. I know that this topic ...
6
votes
1answer
365 views

Degree of a Cartier Divisor under pullback

This is question 7.2.3 in Liu's book Algebraic Geometry and Arithmetic Curves and I have been trying with this for some time now. Let $f:X \rightarrow Y$ be a morphism of Noetherian schemes, and ...
6
votes
1answer
229 views

Hartshorne Proposition II.6.5

The statement of part (a) of this proposition is as follows: Let $X$ be a noetherian integral separated scheme which is regular in codimension 1. Let $Z$ be a proper closed subset of $X$, and let ...
4
votes
2answers
573 views

Intersection of open affines can be covered by open sets distinguished in *both*affines

Suppose $X$ is an arbitrary scheme and $U \cong \operatorname{Spec} A$ and $V \cong \operatorname{Spec} B$ are affine upon subsets of $X$. It's not true in general that $U \cap V$ is affine, so if we ...
5
votes
1answer
1k views

Finite morphisms of schemes are closed

I want to prove that finite morphisms of schemes are closed, but I cannot prove the affine case, namely: Given a finite morphism of rings $\varphi :B \to A$ prove that the induced morphism of ...
1
vote
0answers
71 views

Algebraic analogue of maximum modulus principle applied to Riemann surface.

Let $X$ be an abstract curve in the following sense: $X$ is a scheme, proper over $k$ which is noetherian, integral, dimension 1, and normal. The important thing to point out is that I am not assuming ...
7
votes
1answer
256 views

Definition of prime ideal sheaf

In the chapter I.3.3 of Eisenbud & Harris "The Geometry of Schemes" they give a definition of "prime ideal sheaf": Let $\mathcal{O}_S$ be the structure sheaf of a scheme $S$, and let ...
2
votes
1answer
249 views

criterion for geometrically integral scheme

I want to prove the remark 3.2.9 of the book Algebraic Geometry of Arithmetic Curves (of Quing Liu) that is: let $X$ be an integral scheme with function field $K(X)$, if $K(X)\otimes_k \overline{k}$ ...
3
votes
1answer
290 views

The image of the diagonal map in scheme

Let $X\rightarrow S$ be a separated morphism of schemes, that is, the diagonal map $\Delta:X\rightarrow X\times_S X$ is a closed inmersion. In general (see Closure of image of diagonal morphism of ...
4
votes
3answers
155 views

When are the sections of the structure sheaf just morphisms to affine space?

Let $X$ be a scheme over a field $K$ and $f\in\mathscr O_X(U)$ for some (say, affine) open $U\subseteq X$. For a $K$-rational point $P$, I can denote by $f(P)$ the image of $f$ under the map ...
11
votes
1answer
283 views

Reality check: $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(1)$ and $\mathcal{O}_{\mathbb{P}_\mathbb{R}^1}(-1)$ are both Möbius strips?

This question follows up this previous question, which has an accepted answer that I am having trouble believing. (Edit: the answer has now been fixed; thanks Georges!) I have been working on giving ...
3
votes
1answer
156 views

Does the direct image functor on sheaves reflect epimorphisms?

Let $f:X\to Y$ be a morphism of schemes, and let $f_*:\mathbf{Sh}(X)\to\mathbf{Sh}(Y)$ be the direct image functor. Does $f_*$ reflect epimorphisms? That is, suppose ...
4
votes
2answers
173 views

Irreducibility is preserved under base extension

I want to prove that if $A$ is a finitely generated $k$-algebra ($k$ is a field) with prime nilradical then for any field extension $k\rightarrow K$, the $K$-algebra $A\otimes_kK$ has also prime ...
6
votes
1answer
172 views

Glueing sheaves on Grothendieck sites

Let $X$ be a topological space and $\{V_i\}$ a cover of $X$. Let $F_i:\mathsf{Open}(V_i)^{\mathrm{op}}\to \mathsf{Sets}$ be a family of sheaves. One can glue this family to obtain a sheaf ...
3
votes
1answer
111 views

How to understand elements of the scheme $GL_n$ as linear isomorphisms?

Let as usual $GL_n$ be the scheme given by the equation $$det(\{x_{kl}\}_{1\leq k,l\leq n})y=1$$ in $\mathbb{A}^{{n^2}+1}$. I have seen that one considers elements of $GL_n$ likewise as linear ...
8
votes
1answer
189 views

Deducing that a locally ringed space is a scheme

I would like to know if the following is true: Let $\mathscr{F}$ and $\mathscr{G}$ be two sheaves on a topological space $X$ and let $\varphi:\mathscr{F}\rightarrow\mathscr{G}$ be a morphism such that ...
3
votes
1answer
91 views

A scheme from a quasi-compact morphism

Let $f: X \to Y$ be quasi-compact, set $\mathcal{J} = \ker f^{\#}$, and $Z = V(\ker f^{\#}) = \{ y \in Y \: | \: \mathcal{J}_y \neq \mathcal{O}_{Y,y} \}$. $Z$ is a locally ringed topological space ...
2
votes
1answer
370 views

Nice proof for étale of degree 1 implies isomorphism.

Let $f: X \rightarrow Y$ be a finite étale covering of degree 1 of varieties over some field $k$, so $$ [K(Y):K(X)] = 1. $$ If $k=\mathbb{C}$ and the varieties are smooth, one can apply complex ...