The first purpose of schemes theory is the geometrical study of solutions of algebraic systems of equations, not only over the real/complex numbers, but also over integer numbers (and more generally over any commutative ring with 1).It was finalized by Alexandre GROTHENDIECK, during the 1950's and ...

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11
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1answer
1k views

What are the monomorphisms in the category of schemes?

Someone recently asked what the epimorphisms in the category of schemes are; the other day I had been wondering about the similar question: what are the monomorphisms in the category of schemes? I am ...
15
votes
1answer
1k views

What are the epimorphisms in the category of schemes?

Is there a known characterization of epimorphisms in the category of schemes? It is easy to see that a morphism $f : X \to Y$ such that the underlying map $|f|$ is surjective and the homomorphism ...
17
votes
2answers
870 views

Does Zariski's Main Theorem come with a canonical factorization?

Zariski's Main Theorem (EGA IV, Thm 8.12.6): Suppose $Y$ is a quasi-compact and quasi-separated scheme, and $f:X\to Y$ is quasi-finite, separated, and finitely presented. Then $f$ factors as ...
18
votes
1answer
900 views

Why and how did preschemes become schemes?

Originally (e.g., in the first edition of EGA and in Mumford's Red Book), what are now called "schemes" were referred to as "preschemes." The word "scheme" was reserved for what are now called ...
3
votes
0answers
271 views

A presentation of a scheme as a limit of smooth ones over finitely generated bases

Suppose that a scheme $S$ is separated, excellent, and has finite Krull dimension. Which of the following statements are true: If $S$ is regular, then it can be presented as a projective limit of ...
7
votes
2answers
804 views

Diagonal map and “infinitesimal points”

Let $f:X\to Y$ be a morphism between schemes. To construct the relative sheaf of differentials on $X$ (relative to $Y$), we first consider the diagonal map $\Delta: X \to X\times_Y X$ and then define ...
3
votes
1answer
485 views

How to compute cohomology groups of a closed subscheme Z of projective space, defined by a homogeneous polynomial of degree d?

Let $Z = \mathrm{Proj}\,k[x_{0},x_{1},\ldots,x_{r}]/f$ be a closed subscheme of degree $d$, i.e., $f$ is a homogeneous polynomial of degree $d$, and ...
2
votes
2answers
428 views

Do affine schemes form a Mal'cev category?

This may be a silly question, but I have no intuition in this direction. Every category internal to a Mal'cev category is a groupoid (this is why categories internal to $Grp$ are groupoids). If this ...
21
votes
4answers
2k views

The Frobenius morphism

I found the following list on the "Frobenius Page" by David Ben-Zvi, described by the author as "an outdated collection of intuitive ways to think about raising to the p-th power". Generates a ...
6
votes
0answers
271 views

Does one need l to be invertible in S in order to consider the l-adic cohomology of S-schemes and motives?

When Ivorra defines the $l$-adic realization of $S$-motives (i.e. of Voevodsky's motives over a scheme $S$) he demands $l$ to be invertible in $S$. Is this condition really necessary? What happens ...
0
votes
0answers
143 views

An (almost) terminological question: could one shorten the phrase 'the spectrum of the residue field of a point'?

For a scheme S I want to consider the spectra of the residue fields of points of S. Is there any way to make this phrase shorter? Is there a term for the morphism that connects such a spectrum with S? ...
22
votes
0answers
2k views

Why “open immersion” rather than “open embedding”?

When topologists speak of an "immersion", they are quite deliberately describing something that is not necessarily an "embedding." But I cannot think of any use of the word "embedding" in algebraic ...
2
votes
0answers
189 views

Induced groupoid schemes

This is a more direct version of this question, which was perhaps a bit obtuse. This is a more elementary formulation. Recall that for a groupoid scheme (or indeed any internal groupoid) $X = (X_1 ...
14
votes
1answer
919 views

Universal homeomorphisms and the étale topology

Let $f:X\to S$ be a universal homeomorphism of schemes. Assume $X(S')\neq\emptyset$ for some étale surjective $S'\to S$. Does $f$ have a section? The answer is yes if $S$ is reduced, by descent. ...
5
votes
3answers
728 views

Intuition for rational functions

I asked this on mathematics stack exchange and did not receive answer . I hope it is good manners to ask here. Thank you very much. Let $X$ be integral scheme and $\mathcal K$ sheaf of rationnal ...
6
votes
1answer
649 views

Scheme theoretic closure of a locallly closed subscheme

In the book "The Geometry of Schemes" of Eisenbud and Harris, page 26, it is said that the scheme theoretic closure of a closed subscheme Z of an open subscheme U is the closed subscheme of X defined ...
8
votes
2answers
641 views

Is this finite surjective flat morphism of 2 dimensional schemes a local complete intersection

Let $X$ be a regular integral noetherian scheme of dimension 2 and let $D$ be a simple normal crossings divisor in $X$. EDIT: Let $U = X-D$. Consider a finite etale morphism $V\longrightarrow U$ ...
22
votes
6answers
3k views

What is the inverse image sheaf necessary for in algebraic geometry?

Given a continuous map $f \colon X \to Y$ of topological spaces, and a sheaf $\mathcal{F}$ on $Y$, the inverse image sheaf $f^{-1}\mathcal{F}$ on $X$ is the sheafification of the presheaf $$U \mapsto ...
6
votes
0answers
398 views

Ever seen a ringed group?

A locally ringed space is a common generalization of schemes and various manifolds. I am wondering about a locally ringed group which should be a common generalization of group schemes and various Lie ...
1
vote
1answer
791 views

Ringed and locally ringed spaces

A pair $(X,O_X)$ is a ringed space if $X$ is a topological space and $O_X$ is a sheaf of rings. If every stalk $O_{X,x}$ is a local ring, then we say that $(X,O_X)$ is a locally ringed space. In the ...
17
votes
4answers
2k views

When is an irreducible scheme quasi-compact?

The standard examples of schemes that are not quasi-compact are either non-noetherian or have an infinite number of irreducible components. It is also easy to find non-separated irreducible examples. ...
0
votes
1answer
288 views

Are morphisms of schemes generically affine

Let $f \colon X \to Y$ be a morphism of schemes, where $X$ and $Y$ are separated integral Noetherian schemes. Does there necessarily exist a nonempty open affine $U \subset Y$ such that $f^{-1}(U)$ is ...
9
votes
4answers
896 views

Replacing Spectrum with Valuations of a Field - An Alternative to Schemes?

A scheme is defined to be a sheaf which is locally isomorphic to the spectrum of a ring. The idea behind this is that given an affine coordinate ring of a variety over an algebraically closed field, ...
5
votes
1answer
736 views

Model of a scheme regular over the generic point

Let all schemes below be excellent. Let $X_0$ be a regular (not necessarily smooth, projective) non-empty scheme of finite type over the generic point $\eta$ of a regular connected scheme $S$. As the ...
3
votes
1answer
276 views

If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ a nil-immersion?

If $f:X\to S$ is a universal homeomorphism, is $f':X\times_S X\to X$ always a nil-immersion? This seems to be easy, yet possibly I miss something. Should I give references to this fact in a paper?
5
votes
1answer
560 views

For a morphism f from a regular scheme, should there exist an open subscheme U of the target such that fibre of f at each point of U is regular

For a finite type morphism $f:X\to S$, $X$ is a regular scheme, should there always exist an open (dense) subscheme $U\subset S$ such that the fibre of $f$ at each Zariski point of $U$ is regular? All ...
9
votes
1answer
802 views

Can a scheme be defined by gluing open affines such that the intersections are affine?

One way to think of a manifold is as a family of of open subsets $U_i \subset \mathbb{R}^n$, together with distinguished subsets $V_{ij} \subset U_i$ and isomorphisms $\psi_{ij}: V_{ij} \to V_{ji}$ ...
2
votes
0answers
226 views

Colimit of an etale diagram of schemes

It is known that the category of schemes is not cocomplete (e.g. see this question: Colimits of schemes). However, do diagrams of schemes for which every morphism is etale have colimits? More ...
2
votes
1answer
932 views

on the genus of a function field

Let $K$ be an algebraic function field of one variable. Then we can define its genus. On the other hand, it can also be seen as a scheme, so we can define the arithmetic and geometric genus. Could ...
17
votes
6answers
1k views

Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
9
votes
1answer
378 views

Extending properties of commutative rings to schemes

I'm trying to pin down the various ways we can extend a property of commutative rings to a corresponding property for schemes. Let $P$ be a property of commutative rings. We could define a scheme ...
5
votes
5answers
2k views

When is the push-forward of the structure sheaf locally free

Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module? Example 1. Suppose that $f$ is affine. Then ...
0
votes
1answer
216 views

Restriction of Proj S to D(f) is isomorphic to Spec S_{(f)}

$S$ is a graded ring (over non-negative integers), $f \in S_{+}$ is a homogeneous element of positive degree, $D(f)$ the elements of Proj $S$ not containing $f$. I don't see the bijection between ...
5
votes
2answers
657 views

Philosophy : seeking examples illuminating deeper geometric ideas behind base change of schemes

So for certain base changes, it's clear that 'base change' really means base change from high school : for example, if a curve is defined over a field, it will be of course defined over any extension ...
1
vote
2answers
870 views

Closed subschemes and pulling back the structure sheaf via the inclusion map

I would just like a clarification related to closed subschemes. If $(X,{\cal O}_X)$ is a locally ringed space and $A\subset X$ is any subset with the subspace topology then $i^{-1}{\cal O}_X$ will be ...
2
votes
0answers
532 views

Quotient morphisms in the category of schemes

Which morphisms of schemes (or varieties, if you prefer) $\pi: X \rightarrow Y$ are quotient morphisms, i.e. satisfy the following universal property (*)? (*) For any morphism $f:X \rightarrow Z$, ...
10
votes
3answers
827 views

What is the difference between Grothendieck groups K_0(X) vs K^0(X) on schemes?

More specifically, I was wondering if there are well-known conditions to put on $X$ in order to make $K_0(X)\simeq K^0(X)$. Wikipedia says they are the same if $X$ is smooth. It seems to me that you ...
11
votes
4answers
2k views

Extending vector bundles on a given open subscheme

Many people seem to know the following. Personally, I don't quite understand it though and maybe I'm wrong. It's the fact that "a vector bundle on an open subscheme extends in only one way to a vector ...
4
votes
1answer
475 views

When singular points of a reduced scheme are not dense in it?

A stupid AG question: could singular (Zarisky) points be dense in a reduced (Noetherian) scheme $S$? If yes, which 'standard' restrictions on $S$ could ensure that this does not happen? For example, ...
4
votes
2answers
405 views

Given a morphism from X to Y, when is the morphism from O_Y to the pushforward of O_X injective

I would like to know under what condition the morphism $\mathcal{O}_Y\longrightarrow f_\ast \mathcal{O}_X$ induced by a morphism $f:X\longrightarrow Y$ of schemes is injective. Let me give an example ...
3
votes
1answer
556 views

Quasi-coherent sheaves of O_X-algebras

Let $X$ be a quasi-compact scheme and let $\mathcal{A}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-algebras on $X$. $X$ being quasi-compact, we can write $X = U_1 \cup \dots \cup U_n$ with each ...
24
votes
2answers
2k views

Ring-theoretic characterization of open affines?

Background Recall that, given two commutative rings $A$ and $B$, the set of morphisms of rings $A\to B$ is in bijection with the set of morphisms of schemes $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...
1
vote
3answers
1k views

Properties stable under base change in algebraic geometry

I remember to have seen a big list in the EGA of properties $(P)$ such that: if $f : X \to Y$ has $(P)$ then, $f_{(S')} : X_{(S')}\to Y_{(S')}$ has $(P)$, where $f_{(S')}$ is the morphism $f$ after a ...
10
votes
1answer
695 views

Construction of the petit Zariski topos out of the gros topos of a scheme

Let S be a scheme. Let (Sch/S) be a small category of schemes over S (including essentially all finitely presented schemes affine over S). Let E = (Sch/S)zar denote the gros Zariski topos with its ...
1
vote
3answers
594 views

Functoriality of Hironaka's resolution of singularities

Is Hironaka's resolution of singularities functorial? I know that the resolution is not unique, there are flips etc. But if we have a rational map f:X---> Y, can we chose resolutions X'->X and Y'->Y ...
12
votes
1answer
952 views

Images and Monomorphisms of Schemes

If $X$ is an object in an arbitrary category, there is a natural definition of a subobject of $X$ as a monomorphism into $X$ (or really an equivalence class of monomorphisms). If $X$ is a scheme, ...
9
votes
3answers
1k views

Infinite projective space

Is infinite (say complex) projective space a scheme? More generally, can schemes have infinite cardinal dimension? It seems that infinite dimensional projective space is not a manifold, since it is ...
1
vote
2answers
332 views

morphism of schemes that is closed at topological space level

Suppose $X\stackrel f\to Y$ be a morphism of finite type $k$-schemes, where $k$ is a field; for the time being let me say that $k$ is algebraically closed. Then one knows that $f$ takes $k$-valued ...
6
votes
2answers
675 views

Functorial characterization of morphisms of schemes

This question is akin in spirit to this one: Functorial characterization of open subschemes? In the above MO question, a "functorial" characterization is given for closed immersions and open ...
4
votes
2answers
427 views

Relationship between Line Bundles with isomorphic ring of sections

Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where ...