5
votes
Accepted
Is the change of base scheme $\operatorname{Spec} K \to \operatorname{Spec} k$ relevant?
This isn't widely used and I also haven't seen it around. But it can lead to some weird things which play with intuition, atleast for me.
Suppose $ X $ is the hyperbola $ \operatorname{Spec} \mathbb{C}...
- 2,005
3
votes
Is there a notion of inverse image for schemes?
The image of the natural morphism
$$
f^*I_Z \to f^*\mathcal{O}_Y \cong \mathcal{O}_X
$$
is an ideal in $\mathcal{O}_X$. The corresponding subscheme of $X$ is a natural candidate for the inverse image ...
- 14.2k
3
votes
Accepted
Do restrictions preserve ring structure of ringed space?
This is baked into the definition of a sheaf of rings, and is not something about algebraic geometry or even about locally ringed spaces. The restriction maps have to be a ring homomorphism, so in ...
- 27.4k
3
votes
Accepted
Diagonal of immersion is an isomorphism
In any category with pullbacks, if $f\colon X\rightarrow Y$ is a morphism and $\Delta\colon X\rightarrow X\times_YX$ the associated diagonal, then $\Delta$ is an isomorphism if and only $f$ is a ...
- 9,165
2
votes
Accepted
The $x$-coordinate map of $X:y^2+y=x^3\to\mathbb{P}^1$
This is an instance of the curve-to-projective extension theorem. The way to solve this is to find a uniformizer $t$ in the local ring of at $[0:1:0]$, then write $x=ut^n$ and $z=vt^m$ in terms of the ...
- 56.7k
2
votes
Accepted
Proof Proposition 4.1 Hartshorne
It is exactly that, and this is proved in Exercise II.2.18 in Hartshorne.
Indeed, let $\varphi:A\to B$ be a surjective homomorphism of rings. Then $B\cong A/\ker\varphi$, so it suffices to treat the ...
- 1,714
2
votes
Accepted
A question on Vakil's Rising Sea (Exercise 4.1.A, version 2022)
This is a general property of commutative unital rings. For any commutative ring $A$ and $f \in A$, if $f^n$ is invertible, say with inverse $g$, then $f^{n + 1}$ is invertible with inverse $f^{n - 1}...
- 697
1
vote
Accepted
The degree of a morphism of schemes
The most general definition I can think of is a finite locally free morphism, that is, a morphism of schemes $f\colon X \to Y$ such that:
$f$ is an affine morphism, that is, if $U \subseteq Y$ is any ...
- 4,929
1
vote
Accepted
An algebro-geometric instance of the inverse function theorem?
If you assume that $X$ is regular at $x$, this is true. For notation fix $\dim_x X = \dim \mathcal O_{X,x}$ and $T_x X = (\mathfrak m_x / \mathfrak m_x^2)^*$, and similarly for $Y$. Now first observe ...
- 6,071
1
vote
Accepted
Sheaf morphism from closed subscheme is a closed immersion
Conceptually, we can think of $\mathcal{O}_Z$ as the sheaf of regular functions on a discrete set of points. Now, on any individual point, the only regular functions are constant, and our space is ...
- 2,750
1
vote
Monomorphism of schemes which is not a an open or closed immersion
As a composition of two monomorphisms is a monomorphism, you can take the composition of an open and a closed immersion. For example
$$\mathbb A^1 \setminus \{0\} \hookrightarrow \mathbb A^2, x \...
- 6,071
1
vote
Accepted
detail in proof that existence of ample invertible sheaf implies separatedness
That is because $A$ is local. Let $x\in X$ be the image of the closed point of $\operatorname{Spec} A$, then you have a morphism on stalks $\mathcal O_{X,x}\to A$. Say $B$ is the coordinate ring of $...
- 6,071
1
vote
Accepted
Question on Hartshorne exercise II 5.17(e)
I'm not 100% sure I understand everything you wrote, but here is a way to simplify the situation:
To show that the sheaves glue, you actually only need to consider distinguished open sets $D_U(g)\...
- 1,714
1
vote
Accepted
When is a closed immersion an isomorphism?
Just to get this off of the answered list, let me just elaborate on what I said in the above comment.
Namely, let us say that $f\colon X\to Y$ is a clopen embedding if $f(X)\subseteq Y$ is a clopen(=...
- 51.4k
1
vote
Is there any literature on geometric invariant theory on algebraic spaces?
Quotients in the category of algebraic spaces are discussed in the following papers.
Kollár, János. Quotient spaces modulo algebraic groups. Ann. of Math. (2) 145 (1997), no. 1, 33--79.
Białynicki-...
- 11
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