# Questions tagged [schemes]

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 1950s and the 1960s.

54
questions

**43**

votes

**2**answers

4k 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 $\mathrm{Spec}(B)\to\mathrm{Spec}(A)$. ...

**23**

votes

**4**answers

4k views

### Extending vector bundles on a given open subscheme

Let $U$ be a dense open subscheme of an integral noetherian scheme $X$ and let $E$ be a vector bundle on $U$. Suppose that the complement $Y$ of $U$ has codimension $\textrm{codim}(Y,X) \geq 2$. Let $...

**13**

votes

**2**answers

2k views

### Is there an example of a formally smooth morphism which is not smooth?

A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth.
What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...

**11**

votes

**3**answers

1k 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 ...

**3**

votes

**1**answer

438 views

### Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that,
If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable?
I could not find the statement (...

**206**

votes

**16**answers

49k 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-...

**21**

votes

**4**answers

4k 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. ...

**17**

votes

**1**answer

4k 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

**2**answers

1k views

### Homotopy types of schemes

Let $X$ be a scheme over $\mathbb{C}$.
When does the topological space $X\left(\mathbb{C}\right)$ of $\mathbb{C}$-points have the homotopy type of a finite CW-complex?
When does the topological ...

**2**

votes

**1**answer

411 views

### Noetherian stalks imply locally Noetherian

Is there an example of a non-Noetherian integral affine scheme with Noetherian space and Noetherian stalks? What if we replace "integral" with "reduced"?

**167**

votes

**33**answers

26k views

### What should be learned in a first serious schemes course?

I've just finished teaching a year-long "foundations of algebraic
geometry" class. It
was my third time teaching it, and my notes are gradually converging.
I've enjoyed it for a number of reasons (...

**37**

votes

**2**answers

2k views

### Commutative rings to algebraic spaces in one jump?

Typically, in the functor of points approach, one constructs the category of algebraic spaces by first constructing the category of locally representable sheaves for the global Zariski topology (...

**40**

votes

**1**answer

2k views

### Useful, non-trivial general theorems about morphisms of schemes

I apologize in advance as this is not a research level question but rather one which could benefit from expert attention but is potentially useful mainly to novice mathematicians.
I'm trying to ...

**24**

votes

**2**answers

5k views

### Intuition behind generic points in a scheme

In a scheme, each point is a generic point of its closure. In particular each closed point is a generic point of itself (the set containing it only), but that's perhaps of little interest. A point ...

**9**

votes

**5**answers

6k 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 $f_\...

**24**

votes

**2**answers

2k views

### Reference for de Rham cohomology in positive characteristic

It is known in characteristic $0$ that (algebraic) de Rham cohomology is a Weil cohomology theory. However, in characteristic $p > 0$ it isn't, if only because it has mod $p$ coefficients, whereas ...

**14**

votes

**2**answers

2k 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

**1**answer

570 views

### About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will ...

**16**

votes

**3**answers

2k views

### Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

Is there an example of a variety over the complex numbers with no embedding into a smooth variety?

**28**

votes

**2**answers

2k views

### Do Grothendieck universes matter for an algebraic geometer?

I recently learned that some parts of SGA require axioms beyond ZFC. I am just a simple algebraic geometer so I am trying to understand how can this fact impact my life (you may have engaged in a ...

**22**

votes

**1**answer

777 views

### Motivation for relative schemes: why should one work with schemes over a ringed topos?

Recently I've been trying to learn more about relative schemes. These were developed in M. Hakim's thesis Topos annelés et schémas relatifs under Grothendieck's guidance and appear in many of later ...

**11**

votes

**4**answers

1k views

### The category of finite locally-free commutative group schemes

I'm trying to understand the properties of the category $\mathcal{FL}/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian ...

**9**

votes

**3**answers

2k views

### Do Disjoint Unions and Fiber Products Commute?

Do disjoint unions and fiber products commute?
In other words, is the following statement true?
Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...

**9**

votes

**1**answer

2k 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 ...

**8**

votes

**1**answer

337 views

### Simple examples of colimits of affine schemes (evaluated in the presheaf category) which are not affine schemes

Notation and Setting: let $\operatorname{Aff}$ denote the category of affine schemes whose objects are covariant representable functors $\operatorname{X}:\operatorname{Ring}\rightarrow\operatorname{...

**8**

votes

**1**answer

549 views

### Are higher etale homotopy groups topological groups in a natural way?

Since etale fundamental group of a scheme $X$ is the group of natural automorphisms of the fibre functor of the category of finite etale covers of $X$, it comes with structure of a topological group. ...

**7**

votes

**0**answers

229 views

### Weil homotopy theory

In algebraic geometry, we have something called Weil cohomology theories, which formalize the notion of a "good" cohomology theory of smooth projective varieties. I believe that for $l$-adic ...

**11**

votes

**2**answers

1k views

### Is a scheme Noetherian if its topological space and its stalks are?

Is a scheme being Noetherian equivalent to the underlying topological space being Noetherian and all its stalks being Noetherian?

**8**

votes

**1**answer

217 views

### When are free modules on sheaves of sets quasicoherent?

This question was previously asked over at math.SE.
Let $X$ be a scheme. Let $\mathcal{E}$ be a sheaf of sets on $X$. Then we can define $\mathcal{O}_X\langle\mathcal{E}\rangle$, the free module over ...

**9**

votes

**2**answers

506 views

### Is the category of schemes wellpowered? regularly wellpowered?

Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice ...

**8**

votes

**1**answer

307 views

### Spin structures on schemes

This is a very naive question, but I have been wondering about the role of spin geometry and spinor structures in the context of algebraic geometry. I know the definition of spin structures and ...

**6**

votes

**2**answers

791 views

### The underlying space of a scheme remembers its affineness?

Let $f:X\rightarrow Y$ be a morphism of schemes. We know that if $Y$ is affine and $f$ induces homeomorphism on the underlying spaces then $X$ is affine. Is it true that if $X$ is affine and $f$ ...

**4**

votes

**2**answers

438 views

### Infinitesimal deformations of a fibration

Let $f:X\rightarrow Y$ be a morphism of normal projective varieties over an algebraically closed field with connected fibers.
Assume that both $Y$ and the general fiber of $f$ admit a non-trivial ...

**3**

votes

**0**answers

894 views

### Given a morphism of schemes, when does bijective + isomorphic tangent spaces = isomorphism?

Let $f: X \to Y$ be a morphism of schemes over a field $k$ such that $f$ induces (1) a bijection between their closed points, and (2) an isomorphism of their Zariski tangent spaces.
Under these ...

**3**

votes

**0**answers

133 views

### Other interesting notions when we change topology on $\text{Sch}/S$

Let $\text{Sch}$ be the category of schemes. Let $S$ be an object of $\text{Sch}$. Consider the category $\text{Sch}/S$.
Some interesting topologies on $\text{Sch}/S$ are Zariski, fpqc, étale, fppf.....

**3**

votes

**1**answer

370 views

### Affine hulls and base change

Let $S$ be a scheme. We consider the functor, called affine hull, from the category of quasicompact and quasiseparated $S$-schemes to the category of affine $S$-schemes, defined as a left adjoint to ...

**3**

votes

**2**answers

503 views

### Quasi-compactifying schemes

Let $X$ be a scheme. Does there exist an open immersion $X\rightarrow Y$ with $Y$ quasi-compact?

**2**

votes

**0**answers

130 views

### Size of the ring of functions on open subschemes

This question consists of two related sub-questions.
Let $X$ be a Noetherian integral affine scheme. Under what assumptions on $X$ does every open subscheme of $X$ have a Noetherian ring of global ...

**2**

votes

**0**answers

129 views

### Application of Galois descent

I not understand an assumption done at the beginning of the proof of Rigidity lemma in Moonens and van der Geers book about Abelian variaties (page 12). Here is it:
Question: Why the assumption $k= \...

**0**

votes

**0**answers

125 views

### Question about the statement of the going-down theorem of Cohen-Seidenberg in Mumford

In Mumford's red book the statement of the Going-Down Theorem (Chapter II Section 8) is as follows.
Let $f: X \to Y$ be a finite morphism. Assume that $Y$ is an irreducible normal scheme. Assume that ...

**6**

votes

**1**answer

260 views

### Irreducible of finite Krull dimension implies quasi-compact?

Let $X$ be the underlying space of a scheme.
If $X$ is irreducible of finite Krull dimension, is it necessarily
quasi-compact?
Is it necessarily Noetherian?
What if we assume not
only that Krull ...

**4**

votes

**0**answers

446 views

### A slightly canonical way to associate a scheme to a Noetherian spectral space

Let $C$ be the category whose objects are Noetherian spectral topological spaces and whose morphisms are homeomorphisms. Let $\mathrm{AffSch}$ be the category of Noetherian affine schemes (morphisms ...

**3**

votes

**0**answers

170 views

### Abelian varieties by Moonen and van der Geer: proof of rigidity lemma

I try to understand a reduction step in the proof of rigidity lemma as proved in Moonen's and van der Geer's Abelian varieties (Lemma 1.11 on page 12- if the link not work the draft version is online ...

**3**

votes

**2**answers

228 views

### Minimal fields of isomorphism for varieties

Let $V$ be an algebraic variety over a field $K$. Is there a constant $d = d(V) \in \mathbb{N}$ such that for any variety $W$ defined over $K$ and isomorphic to $V$ over the algebraic closure of $K$, ...

**3**

votes

**0**answers

128 views

### Examples of subspaces singled out by modular forms

I am wondering what subspaces of modular varieties defined as the zero locus of modular forms have been studied in the literature.
To be more clear let me explain the example I have in mind.
Let $N\...

**3**

votes

**0**answers

158 views

### Topological properties of Noetherian affine schemes that do not hold for general Noetherian spectral spaces

I used to think that the only reason why an affine scheme with a Noetherian space can fail to be Noetherian is nilpotents. It turns out that this is not true.
This leads me to the following question: ...

**3**

votes

**1**answer

342 views

### Krull dimension of the ring of global sections

Let $X$ be an irreducible scheme. Can the Krull dimension of $\mathcal{O}_X(X)$ exceed that of $X$?

**2**

votes

**0**answers

227 views

### infinite dimensional germs of schemes and tangent spaces

(The question of the type "how to define?")
Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as ...

**2**

votes

**1**answer

266 views

### Non-flat locus for smooth schemes

Let $X$, $Y$ be connected smooth schemes of finite type over an algebraically closed field of characteristic $0$. Let $f:X\rightarrow Y$ be a non-birational morphism surjective on the underlying ...

**2**

votes

**1**answer

132 views

### Quasi-compactification of locally spectral spaces

Let $X$ be a locally spectral topological space (i.e. a space admitting an open cover by spectral spaces). Does there necessarily exist a quasi-compact locally spectral space $Y$ and an injective ...