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• 697
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### 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
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### 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
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### 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
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### 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
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• 1,714
1 vote
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### 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
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### 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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