**141**

votes

**14**answers

10k 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 ...

**35**

votes

**1**answer

2k views

### Connections between various generalized algebraic geometries (Toen-Vaquié, Durov, Diers, Lurie)?

As far as I know, there are four possible ways to generalize algebraic geometry by 'simply' replacing the basic category of rings with something similar but more general:
$\bullet$ In the approach by ...

**30**

votes

**2**answers

2k views

### A closed subscheme of an open subscheme that is not an open subscheme of a closed subscheme?

A morphism $f: V \rightarrow X$ of schemes is a locally closed immersion if it can be factored into a closed immersion followed by an open immersion. It is not hard to show that if $f$ is an open ...

**29**

votes

**7**answers

3k views

### Arbitrary products of schemes don't exist, do they?

Thinking of arbitrary tensor products of rings, $A=\otimes_i A_i$ ($i\in I$, an arbitrary index set), I have recently realized that $Spec(A)$ should be the product of the schemes $Spec(A_i)$, a ...

**26**

votes

**2**answers

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

**26**

votes

**0**answers

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 ...

**24**

votes

**6**answers

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 ...

**24**

votes

**0**answers

1k 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 ...

**23**

votes

**6**answers

4k views

### How much of scheme theory can you visualize?

I am just starting to learn about schemes and algebraic geometry in general, but I am finding it very hard to visualize things. For example, affine schemes that look like varieties are easily ...

**23**

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 ...

**21**

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 ...

**20**

votes

**4**answers

2k views

### Spectrum of the Grothendieck ring of varieties

Here's a problem that may ultimately require just simple algebraic-geometry skills to be solved, or perhaps it's very deep and will never be solved at all. From the comments, some literature and my ...

**20**

votes

**3**answers

3k views

### Reduced scheme and closed points

In The Geometry of Schemes by Eisenbud and Harris, Exercise I-32 asks one to show that a scheme $X$ is reduced if and only if every local ring $\mathcal{O}_{X,p}$ is reduced for closed points $p \in ...

**19**

votes

**3**answers

2k views

### A book on locally ringed spaces?

Are there enough interesting results that hold for general locally ringed spaces for a book to have been written? If there are, do you know of a book? If you do, pelase post it, one per answer and a ...

**19**

votes

**1**answer

977 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 ...

**19**

votes

**1**answer

495 views

### Why would one “attempt” to define points of a motive as $\operatorname{Ext}^1(\mathbb{Q}(0),M)$?

I'm a novice when it comes to motives. (I've read multiple introductory texts.)
I'm attempting to read Galois Theory and Diophantine geometry by Minhyong Kim. In it, he says that "One might attempt, ...

**18**

votes

**6**answers

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" ...

**18**

votes

**2**answers

896 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

**1**answer

477 views

### Does every relative curve have a Picard scheme?

More precisely:
Let $X \to S$ be a smooth proper morphism of schemes such that the geometric fibers
are integral curves of genus $g$. Must the fppf relative Picard functor
$\operatorname{\bf ...

**17**

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. ...

**17**

votes

**1**answer

476 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 ...

**16**

votes

**2**answers

686 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?

**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 ...

**16**

votes

**2**answers

2k 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 ...

**15**

votes

**4**answers

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 ...

**15**

votes

**4**answers

645 views

### What are the Benefits of Using Algebraic Spaces over Schemes?

I have heard that algebraic spaces have better formal properties than schemes. What are these benefits? Also, is there a natural way to go straight from affine schemes to algebraic spaces bypassing ...

**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. ...

**15**

votes

**2**answers

554 views

### Minimal number of generators for $A^n$

Let $A$ be a commutative ring and $n \in \mathbb{N}$. What is the minimal number $e_A(n)$ of generators of the $A$-algebra $A^n$? Here is what I already know (I can add proofs if necessary) from a ...

**14**

votes

**1**answer

425 views

### Geometric generic fibre

This is a pretty elementary question about schemes, but it came up in the course of research, so let's try it here rather than MSE.
Question 1: Are the fibres of a family of complex varieties ...

**13**

votes

**2**answers

742 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 ...

**13**

votes

**3**answers

976 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?

**12**

votes

**3**answers

556 views

### Constructing a degeneration (as a group scheme) of G_m to G_a

SGA 3, expose 12, remark 1.6 says that one can easily construct a group scheme over a discrete valuation ring with generic fiber Gm and special fiber Ga.
What is such an example?

**12**

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 ...

**12**

votes

**1**answer

936 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 $\mathscr F^+$ ...

**12**

votes

**1**answer

826 views

### Functorial characterization of open subschemes?

Given a morphism of schemes f: U → X, can one determine when f is an isomorphism of U onto an open subscheme of X in terms of some induced functors between the categories of quasicoherent modules ...

**12**

votes

**1**answer

1k 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, ...

**11**

votes

**3**answers

2k views

### Closed vs Rational Points on Schemes

Background: When Ueno builds the fully faithful functor from Var/k to Sch/k he mentions that the variety $V$ can be identified with the rational points of $t(V)$ over $k$. I know how to prove this on ...

**11**

votes

**3**answers

943 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

**2**answers

1k 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

**4**answers

975 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, ...

**11**

votes

**3**answers

984 views

### Non-simply-connected smooth proper scheme over Z?

Source
This question came up in the discussion between Kevin Buzzard and Minhyong Kim in the comments to Smooth proper scheme over Z. It was 2 weeks ago, so I took the liberty of posting it as ...

**10**

votes

**3**answers

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 ...

**10**

votes

**1**answer

716 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 ...

**9**

votes

**2**answers

398 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 ...

**9**

votes

**4**answers

1k views

### Nonalgebraic complex varieties

I'm turning here (a variation of) a question asked by a friend of mine. For the purposes of this question I will say that a compact complex manifold is projective if it is isomorphic to a subvariety ...

**9**

votes

**1**answer

233 views

### Torsors trivializing over a fixed finite etale cover

Let $S$ be an integral regular scheme and let $T\to S$ be a finite etale morphism. Let $G$ be a smooth affine finite type group scheme over $S$.
Is the set of $S$-isomorphism classes of $G$-torsors ...

**9**

votes

**1**answer

427 views

### Is there a direct proof that affine schemes are fppf quasi-compact?

Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...

**9**

votes

**1**answer

915 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 $\psi_{ij}: V_{ij} \to V_{ji}$ ...

**9**

votes

**1**answer

361 views

### Can an algebraic space fail to have a unviersal map to a scheme?

Let $\mathcal{X}$ be an algebraic space. Can it happen that there does not exist a map $\mathcal{X} \to X$ with $X$ a scheme that is initial for maps from $\mathcal{X}$ to schemes? Are there ...

**9**

votes

**1**answer

446 views

### Can one check formal smoothness using only one-variable Artin rings?

Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...