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.

learn more… | top users | synonyms

2
votes
1answer
57 views

Question on geometrically reduced, geometrically connected.

I have a question from a book which I am trying to attempt. Let $k$ be a field not of characteristic 2 and let $a\in k$ be not a square (i.e. for all $b\in k$, $b^{2}\neq a$). I want to show that ...
2
votes
1answer
45 views

Dimension of schemes over fields

In my reading I recently encountered the following problem: Let $A$ be a DVR and let $Y=\mathrm{Spec}\,A$. Let $K$ be the fraction field of $A$ and let $k$ be the residue field. The canonical ...
2
votes
1answer
62 views

Interpreting results concerning the global sections ring being finitely generated

Let $A$ be a ring and $X$ be an $A$-scheme. (Hartshorne exercise II.2.17) Suppose there exist $f_0,\dots,f_n\in \mathcal{O}_X(X)$ such that a) $X_{f_i}:=\{x\in X: f_i(x)\not=0\}\subset X$ is ...
0
votes
1answer
54 views

Properties of Quasi-Coherent Modules

Let $X=\mathrm{Spec}\,A$ be an affine scheme and $M$ an $A$-module. Show that the following two conditions are equivalent: (a) $\tilde{M}$ is a locally free $\mathcal{O}_{X}$-module of finite type. ...
2
votes
1answer
63 views

Scheme-Theoretic Nakayama's Lemma

Let $X$ be a noetherian scheme and $\mathscr{F}$ a coherent $\mathscr{O}_{X}$-module. For a point $x \in X$, let $k(x)=\mathscr{O}_{X,x}/\mathfrak{m}_{x}$ be the residue field at $x$. (a) Suppose $x ...
1
vote
1answer
61 views

Open immersion from a proper scheme to a separated, irreducible scheme.

Fix a scheme $S$ and let $X$ and $Y$ be $S$-schemes. Assume that $X$ is proper over $S$ and $Y$ is separated over $S$. Let $f: X \rightarrow Y$ be an open immersion of $S$-schemes. If $Y$ is ...
2
votes
2answers
195 views

Morphism from a proper irreducible scheme into an affine scheme of finite type

Let $K$ be a field and $X$ be a proper irreducible $K$-scheme. Show that the image of any $K$-morphism $X \rightarrow Y$ into an affine $K$-scheme $Y$ of finite type consists of a single point. I ...
7
votes
1answer
102 views

How to tell whether a scheme is reduced from its functor of points?

Suppose I have a scheme $X$ and I want to know if $X$ is reduced, but all I have access to is the functor $$ R\mapsto X(R)=Mor(\operatorname{Spec}(R),X) $$ from commutative rings to sets (rather ...
3
votes
1answer
168 views

Questions on Gluing Schemes

I am trying to do an exercise in a book and here is the question. I have attempted it but I am not sure if my answer is correct. I would appreciate if someone corrects my attempt. Note here that the ...
9
votes
2answers
307 views

Some basic examples of étale fundamental groups

I'm trying to get a better understanding of étale fundamental groups, and I think that the overall idea -- the big picture -- is beginning to become clear, but my computational ability seems to be ...
5
votes
1answer
158 views

Explicit description of discrete valuations corresponding to prime divisors

I'm reading Hartshorne p. 130 on Weil divisors. Let $X$ be a noetherian integral separated scheme regular in codimension 1, $Y$ a prime divisor on $X$, and $\eta \in Y$ its generic point. ...
4
votes
1answer
134 views

function field of an integral scheme

Suppose $X$ is an integral scheme, and let $\eta \in X$ be its generic point. Then the local ring $\mathcal{O}_{X,\eta}$ is a field, called the function field of $X$ and denoted $K(X)$. Why is $K(X)$ ...
2
votes
1answer
105 views

Graph morphism for a separated morphism of schemes

I want to prove the following: Let $f: X \rightarrow S$ be a separated morphism of schemes. Show that any section $g: S \rightarrow X$ of $f$ i.e. a morphism such that $f \circ g=\textrm{id}_{S}$ is ...
2
votes
1answer
130 views

Separated Morphisms of Schemes

(a) Let $f:X \rightarrow S$ be a separated morphism of schemes. Show that for any subscheme $U \subset X$, the restriction $f\mid_{U}:U \rightarrow S$ is separated. (b) Let $R$ be a commutative ring ...
3
votes
2answers
137 views

Why are non-separated schemes schemes?

In "the old days", e.g. in the famous texts by Grothendieck and Mumford, a scheme was defined as what we now call a separated scheme. (i.e. a scheme where the image of the morphism $\Delta:X \to X ...
3
votes
1answer
78 views

Morphisms from spectra to schemes

Let $X$ be a scheme. Show that for any $x \in X$ there exists a canonical morphism $\textrm{Spec}\, \mathcal{O}_{X,x} \rightarrow X$. If $k(x)=\mathcal{O}_{X,x}/\frak{m}_{x}$ is the residue field at ...
1
vote
1answer
37 views

Mappings for an integral scheme with generic point

Let $X$ be an integral scheme with generic point $\eta$. Show that for any $x \in X$, there is a canonical ring homomorphism $\mathcal{O}_{X,x}\rightarrow \mathcal{O}_{X,\eta}$. For any open ...
2
votes
1answer
58 views

Frobenius twist commutes with fiber product

Let $k$ be a field, and let $X$ be a scheme over $k$. Let $f:k\to k$ be the embedding of fields defined by $f(\lambda)=\lambda^p$. Define the first Frobenius twist of $X$ to be the scheme $X\times_f ...
1
vote
1answer
91 views

The Frobenius twist of a scheme

Let $k$ be a field of characteristic $p$, and define an embedding of fields $f:k\to k$ by $f(\lambda)=\lambda^p$. Then, given any scheme over $k$, $X$, define the Frobenius twist of $X$ as ...
3
votes
1answer
73 views

map of graded rings and projective schemes

Let $\phi$ a morphism between $S$ and $R$, two graded rings (there is a $d>0$ such that $S_{n}$ maps to $R_{dn}$ for all $n$). How could I show that this induces a morphism of schemes ...
1
vote
0answers
96 views

Projective scheme

How can I prove that the two different construction of $\mathbb{P}_k^n$ (as $Proj K[x_0,x_1,...,x_n]$ and by gluing copies of $\mathbb{A}_k^n$) agree? And how can I prove that if $A$ is reduced also ...
3
votes
0answers
75 views

Grothendieck group of a variety and Grothendieck group of its derived category

Let $X$ be a smooth projective variety over a field $K$. Let $K_0(X)$ denote the Grothendieck group of the abelian category of coherent sheaves on $X$. Let $\newcommand{\Pf}{\mathbf{Pf}}K_0(\Pf(X))$ ...
1
vote
0answers
74 views

Closed sets of $Spec(R)$.

I have to dounbs: Let $R$ be the ring $\mathbb{C}[x]_{(x)}$. So we have that $Spec(R)=\{(0),(x)\}$. I have to descrive closed sets of $Spec(R)$. I think that they are $\{\emptyset, Spec(R), (x) \}$ ...
3
votes
1answer
48 views

Relationship between maps and maps of rings

I take this exercise from Ravi Vakil's book. Consider the map of complex manifolds sending $\mathbb{C} \rightarrow \mathbb{C}$ via $x \mapsto y=x^2$. We interpret $\mathbb{C}$ as the $x$-line and ...
0
votes
2answers
67 views

$Spec(k[x]_{(x)})$ as a shred of curve.

Let $k$ be a field and consider the ring $k[x]$. x is a prime ideal so we can take the localization $k[x]_{(x)}$ in which there are all inverses except for $x$. So we have that ...
1
vote
0answers
66 views

How can I calculate $\mathrm{Spec}(\mathbb{Z}_{(3)})$? And $\mathrm{Spec}(\mathbb{Z}_3)$?

Let $\mathbb{Z}_{(3)}$ be the localization (in $\mathbb{Z}$) of the ideal generated by $3$. So I have to put in $\mathbb{Z}$ all the inverses of the complement of $(3)$. How can I calculate ...
2
votes
1answer
204 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
votes
1answer
101 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
votes
1answer
86 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 ...
3
votes
1answer
146 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
vote
1answer
135 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 ...
4
votes
1answer
57 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
47 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
111 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
votes
0answers
109 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 ...
6
votes
2answers
334 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
votes
2answers
135 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
votes
1answer
113 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 ...
0
votes
1answer
56 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
82 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
147 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 $g:=f\times_S \text{id}_Y:X'\times_SY\to X\times_SY$. ...
4
votes
1answer
133 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
73 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
75 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
138 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 ...
4
votes
1answer
257 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
75 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
248 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
126 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 ...
8
votes
0answers
218 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 ...