**2**

votes

**1**answer

510 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

306 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

259 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 ...

**2**

votes

**3**answers

422 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

431 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

286 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

568 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

937 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

256 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

629 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 ...

**18**

votes

**2**answers

920 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 ...

**19**

votes

**1**answer

993 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

317 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

909 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

532 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

**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

280 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?
...

**32**

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

217 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

768 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

761 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

680 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 ...

**18**

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

328 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

791 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

294 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

588 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

990 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

**0**answers

236 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

**1**answer

1k 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 ...

**19**

votes

**6**answers

2k 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" ...

**10**

votes

**1**answer

401 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 ...

**6**

votes

**5**answers

3k 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

**1**answer

222 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

**2**answers

681 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 ...

**3**

votes

**2**answers

1k 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

**0**answers

584 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 $f:X \rightarrow Z$, ...