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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9
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4answers
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 ...
30
votes
2answers
1k 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 ...
19
votes
3answers
2k 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 ...
21
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2answers
1k 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 ...
9
votes
3answers
884 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 ...
27
votes
7answers
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 ...
8
votes
4answers
951 views

The category of finite locally-free commutative group schemes

I'm trying to understand the properties of the category $FL/S$ of finite locally-free commutative group schemes over an arbitrary base-scheme $S$. I know it is not in general an abelian category: Over ...
9
votes
1answer
325 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 ...
12
votes
3answers
537 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?
15
votes
4answers
592 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 ...
5
votes
4answers
399 views

Can isomorphisms of schemes be constructed on formal neighborhoods?

Let (A,m) be a complete local Noetherian ring and let X and Y be two schemes of finite type over A (and flat over A). Let Xn and Yn be the reductions of X and Y mod mn+1. Question: Suppose there ...
11
votes
1answer
744 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 ...
18
votes
4answers
1k 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 ...
13
votes
3answers
855 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?
10
votes
2answers
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 ...
6
votes
1answer
397 views

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

Let f:X -> 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'[t]/t^n, where k' is also a field? Considering cuspidal curves one ...