# fibre of a scheme of finite type over a field being quasi-compact

Mathematics Asked on January 1, 2022

Let $$X,Y$$ be schemes of finite type over a field $$k$$. In particular, they are quasi-compact. Let $$f: X to Y$$ be a morphism of finite type (for all open affines $$U subset Y$$, $$f^{-1}(U)$$ is quasi-compact and $$Gamma(V, O_X)$$ is finitely generated over $$Gamma(U, O_Y)$$ for all open affine $$V subset f^{-1}(U)$$).

I would like to deduce that the fibre $$f^{-1}(y)$$ as a scheme over $$operatorname{Spec}kappa(y)$$ is quasi-compact, for any $$y in Y$$. How can I prove this? Any comments are appreciated. Thank you!

We know that morphisms of finite type are stable under base extension (see Hartshorne exercise 3.13 (d)). Then $$f:X rightarrow Y$$ being of finite type implies that $$f^{-1}(y) rightarrow$$ Spec $$k(y)$$ is of finite type. Note that morphisms of finite type are necessarily quasi-compact (see Hartshorne exercise 3.3(a)), since the preimage of Spec $$k(y)$$ is $$f^{-1}(y)$$, we must have it is quasi-compact space.

Answered by Rikka on January 1, 2022

A scheme is quasicompact iff it can be covered by finitely many open affine subschemes.

When $$X, Y$$ are affine, it’s easy to see that the fiber is the spectrum of $$mathcal{O}_X(X) otimes_{mathcal{O}_Y(Y)} kappa(y)$$ so is quasicompact.

In general, let $$y in V subset Y$$ be an affine open subset, let $$f^{-1}(V)=bigcup_i{U_i}$$ be a finite reunion of affine open subsets of $$X$$ (as $$X$$ is a Noetherian scheme, all its affine open subsets are quasicompact). Then $$X_y$$ is the (finite) reunion of the $$(U_i)_y$$ which are open affine (they are fibers of $$U_i rightarrow V$$ affine of finite type), so is quasi-compact.

Answered by Mindlack on January 1, 2022

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