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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4
votes
1answer
280 views

Flat family of normal schemes over a normal base

Let $f \colon X \to Y$ be a flat morphism of schemes over $\mathbb{C}$. Suppose that $Y$ is normal and that the fibers over the closed points of $Y$ are all normal. Can I say something about the ...
3
votes
3answers
448 views

Are schemes pushouts of neighbourhoods and formal neighbourhoods?

Hello, I have two questions, the first less important. Let $X$ be a scheme, $x \in X$ a schematic point. What is an elegant way of defining/characterizing the map $\operatorname{Spec}(O_{X,x}) ...
9
votes
1answer
732 views

Affine scheme on spec(A) of a ring A as the sheafification of a pre-sheave on spec(A)?

It is obvious that there is a parallel between the definition of structure sheaf of $\operatorname{Spec}(A)$ versus the sheafification of a pre-sheaf. The definition of the sheaf $\mathscr F^+$ ...
0
votes
1answer
327 views

Codimension of points in fibered products

This is a question about a proof in Hartshorne, but let me try to formulate it without reference to Hartshorne. Let $X$ be a noetherian scheme (which is also integral, separated and regular in ...
5
votes
2answers
1k views

Jacobian criterion for smoothness of schemes

An affine scheme $X = Spec(A)$ is said to be smooth if for any closed embedding $X\subset\mathbf A^n$, of ideal $I$, it is true that, locally on $x\in X$, the ideal $I$ can be generated by a sequence ...
3
votes
1answer
678 views

Is the degree of a finite morphism stable by base change

Let $f:X\longrightarrow Y$ be a finite morphism of schemes of degree $n$. Let $S\to Y$ be a morphism of schemes. Is the degree of the finite morphism $X\times_Y S \longrightarrow S$ equal to $n$? If ...
7
votes
5answers
1k views

pushforward of locally free sheaf is locally free?

Hi, Is there an example of a proper smooth map of schemes $f:X\to Y$ and a vector bundle $E$ on $X$ such that $f_*E$ is not locally free on $Y$? Thanks
1
vote
1answer
227 views

What kind of conditions we need to make morphisms of schemes quasi-projective?

What kind of conditions we need to make morphisms of schemes quasi-projective? I am really interested in the following case: If $f : X \to Y$ is an etale, of finite type and separated morphism of ...
3
votes
1answer
445 views

Vector space structure on the tangent bundle of a scheme and relation to the tangent sheaf

First a word of warning: I am not a trained algebraic geometer, so it is possible (likely) that these questions are inappropriate for MO, if so: my apologies. Said this: As far as I understand the ...
3
votes
1answer
262 views

The restriction of a global section which is not a zero divisor is still an non-zero divisor?

Let X be a scheme. U is an open subscheme of X. Assume f is a global section on X which is not a zero divisor, then the restriction of f to U is still an non-zero divisor? If X is affine, the answer ...
2
votes
2answers
388 views

Components of an exceptional divisor

Let $X$ be a projective variety and let $\tilde{X}$ be the blow-up of $X$ at a subscheme $Z$. Let $F$ be the exceptional divisor of $\tilde{X}$. I wonder: What is the number of irreducible ...
22
votes
0answers
879 views

Mikhalkin's tropical schemes versus Durov's tropical schemes

In Mikhalkin's unfinished draft book on tropical geometry, (available here) (page 26) he defines a notion of tropical schemes. It seems to me that this definition is not just a wholesale adaptation of ...
0
votes
1answer
310 views

Is the following morphism etale

Let $Y$ be a reduced noetherian $1$-dimensional scheme such that the normalization morphism $f:X \longrightarrow Y$ is finite. Let $g:Y\longrightarrow Z$ be a finite flat morphism, where $Z$ is a ...
5
votes
1answer
359 views

Limits of reduced schemes question from Eisenbud and Harris

My question pertains to exercise II-16 in Eisenbud and Harris' "The geometry of Schemes". For an algebraically closed field $K$ the question is as follows: Consider zero-dimensional subschemes ...
4
votes
1answer
322 views

When does Zariski closure commute with base change?

This should be an elementary question for anyone who knows SGA by heart (alas, not for me). It smells a lot like a descent problem. All schemes are supposed to be noetherian, and all morphisms to be ...
3
votes
2answers
768 views

Hartshorne's associated scheme for a variety

This question comes from Proposition 2.6 in Chapter 2 of Hartshorne's Algebraic Geometry. In my edition, that's on page 78. For a variety $V$, Hartshorne defines the topological space $t(V)$ to ...
3
votes
2answers
718 views

morphisms of affine schemes question

So, in chapter 2, section 2 of Hartshorne, (prop 2.3), he describes how if $\varphi : A\rightarrow B$ is a homomorphism of rings, then you get a morphism of (affine schemes): ...
2
votes
1answer
528 views

k rational points and base change

This could be a tricky question but could help me to better understand these very interesting things. Let $X$ be an algebraic variety over a field $k$ (in the sense of a k-scheme like in Qing Liu), ...
7
votes
1answer
676 views

Closed points of valuation scheme

In the excellent book "Algebraic Geometry 1" of Görtz & Wedhorn, in exercise 3.14, one is asked to show that in the spectrum of a valuation ring with infinitely many primes, the complement of the ...
2
votes
0answers
324 views

cohomology of projective limit of schemes

Hello, Suppose that $X_i$ is a projective system of schemes and $F_i$ is a compatible projective system of abelian sheaves on the $X_i$ (i.e. if $p_{ij} : X_i \to X_j$ is the transition map, then we ...
7
votes
1answer
419 views

Complex analytic space with no (positive dim.) subscheme ?

Is there an example of a complex analytic space $X$ that doesn't have any (not necessarily open or closed) positive dimensional subspace $Y$ which is analytically isomorphic to (the complex ...
16
votes
2answers
656 views

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?

What is an explicit example of a variety X which is finite over Spec F_p but which does not lift to a scheme Y which is finite and flat over Spec Z_p?
1
vote
1answer
441 views

Is every regular (excellent) scheme separated?

Sorry for one more stupid AG question. I need schemes that are regular, excellent and separated. Are these three conditions independent?
4
votes
0answers
293 views

Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$. Does $R_\mathbb{R}$ send an ...
5
votes
1answer
295 views

Topological space associated to a real or complex scheme

Hi, consider a scheme $X$ of finite type over $\mathbb{R}$ (or $\mathbb{C}$). In Hartshorne's appendix B on 'transcendental methods' it is shortly mentioned how to assign a reasonable topological ...
2
votes
0answers
232 views

Can any radiciel morphism be presented as the composition of a universal homeomorphism with an immersion?

Let $f:X\to Y$ be a radiciel (=universally injective) morphism, where $X$ is a regular connected scheme. Can it be presented as the composition of a universal homeomorphism with an immersion? This ...
128
votes
14answers
8k views

What elementary problems can you solve with schemes?

I'm a graduate student who's been learning about schemes this year from the usual sources (e.g. Hartshorne, Eisenbud-Harris, Ravi Vakil's notes). I'm looking for some examples of elementary ...
2
votes
3answers
394 views

Sections of morphisms of schemes up to a finite morphism

Let $f:X\longrightarrow S$ be a flat projective morphism of regular integral noetherian schemes such that that the generic fibre $X_\eta\longrightarrow K(S)$ is a smooth projective connected curve ...
2
votes
1answer
366 views

More on universal homeomorphisms

I would like to understand this notion better; where could I find some examples? In particular, I am interested in the following questions (and references for the answers). Is a universal ...
4
votes
2answers
237 views

Is the pre-image of a regular subscheme with respect to a universal homeomorphism of regular schemes regular?

Let $f:X\to Y$ be a universal homeomorphism of regular (excellent finite-dimensional) schemes, $Z\subset Y$ be a regular subscheme. Is $f^{-1}(Z)$ necessarily regular?
1
vote
0answers
139 views

When inverse image is conservative; a reference or a generalization?

I am interested in the following question: for $f$ being a morphism of schemes, which conditions ensure that $Rf^*_{et}$ is conservative? This is true if $f$ has a section or if $f$ is an \'etale ...
3
votes
1answer
273 views

Classification of fat projective lines?

In section III.3.4 of Eisenbud & Harris's "The Geometry of Schemes," we/they construct an infinite family of double structures on $\mathbb{P}^1 \subset \mathbb{P}^3$ that are distinguished from ...
5
votes
0answers
500 views

Do all the main properties of constructible and perverse sheaves (in an 'arithmetic' situation) follow from results of Gabber?

This question is a continuation of Bad behaviour of perverse sheaves over 'general' bases? Let $S$ (for example) be a finite type separated scheme over $\mathbb{Z}$. I would like: (1) to ...
1
vote
2answers
621 views

Coproducts of schemes (“gluing construction”) ?

In this MO question it was raised the topic of "gluing constructions" in the category of schemes. I understand the phrase "gluing two schemes along maps to them" as "there exists a coproduct of the ...
1
vote
1answer
241 views

Group scheme of infinite dimensional linear groups ?

Hi there, I know there are fairly straightforward ways to write down the schemes of infinite dimensional projective spaces (not restricting myself to only countable dimensions), but what happens with ...
7
votes
2answers
529 views

Scheme-theoretic account of why every variety embeds in a complete variety

The standard reference for the statement that "any abstract variety is an open subscheme of a complete variety" is Nagata's 1962 paper Imbedding of an abstract variety in a complete variety. ...
11
votes
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
854 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
883 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
267 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
776 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
466 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
425 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 ...
20
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
265 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
140 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
186 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
901 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. ...