**5**

votes

**1**answer

390 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

**1**answer

365 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 ...

**4**

votes

**2**answers

946 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 ...

**4**

votes

**2**answers

929 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):
$\newcommand{\Spec}{\...

**3**

votes

**1**answer

701 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

**1**answer

733 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

**0**answers

375 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

**1**answer

462 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 analytic ...

**16**

votes

**2**answers

706 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?

**2**

votes

**1**answer

519 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

**0**answers

307 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

**1**answer

317 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

**0**answers

262 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 ...

**155**

votes

**14**answers

11k 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 self-...

**2**

votes

**3**answers

425 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

**1**answer

444 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

**2**answers

246 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?

**2**

votes

**0**answers

149 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

**1**answer

289 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

**0**answers

577 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 ...

**2**

votes

**2**answers

972 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

**1**answer

258 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

**2**answers

655 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. ...

**15**

votes

**1**answer

2k 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 ...

**16**

votes

**1**answer

2k 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 $f^\...

**18**

votes

**2**answers

924 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 $X\...

**19**

votes

**1**answer

998 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

**0**answers

325 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

**2**answers

919 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

**1**answer

545 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 $\mathcal{O}_{Z}(1)=i^{*}\mathcal{O}_\mathbb{P}(1)$....

**2**

votes

**2**answers

446 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 ...

**22**

votes

**4**answers

3k 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

**0**answers

281 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

**0**answers

145 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?

**33**

votes

**1**answer

3k 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

**0**answers

219 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 \...

**15**

votes

**1**answer

1k 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

**3**answers

775 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

**1**answer

781 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

**2**answers

685 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 $V\longrightarrow U$ ...

**26**

votes

**6**answers

4k 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

**0**answers

400 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

**1**answer

1k 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 ...

**19**

votes

**4**answers

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

**1**answer

333 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 ...

**11**

votes

**4**answers

1k 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

**1**answer

797 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

**1**answer

299 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

**1**answer

592 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

**1**answer

1k 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 $\psi_{ij}: V_{ij} \to V_{ji}$ ...