# 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.

471
questions

**4**

votes

**2**answers

518 views

### Relationship between Line Bundles with isomorphic ring of sections

Suppose two positive holomorphic line bundles $L_1 \to X_1, L_2\to X_2$ over two projective complex manifold $X_1, X_2$ have isomorphic ring of sections $R=R_1=R_2$ where $R_i=\oplus_{m=0}^\infty\...

**5**

votes

**3**answers

818 views

### Affine morphisms in different settings coincide?

1.If we identify two schemes $X$ and $Y$ as two presheaves of set on category of affine schemes.($Aff:=\text{CRing}^{op}$) If there is a morphism as natural transformations $f:X\to Y$, then, how to ...

**31**

votes

**6**answers

6k 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 ...

**14**

votes

**3**answers

3k 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 ...

**23**

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 ...

**11**

votes

**4**answers

2k views

### Nonalgebraic complex manifolds

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 ...

**38**

votes

**1**answer

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 ...

**23**

votes

**5**answers

4k 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 $p \in X$...

**34**

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 (...

**14**

votes

**3**answers

1k 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 ...

**37**

votes

**6**answers

4k 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 ...

**10**

votes

**4**answers

1k 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

**1**answer

431 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

**3**answers

619 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?

**18**

votes

**4**answers

1k 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 ...

**6**

votes

**4**answers

609 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 is ...

**16**

votes

**1**answer

1k 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 ...

**29**

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 ...

**14**

votes

**3**answers

1k 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

**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 ...

**9**

votes

**1**answer

488 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 ...