nLab
quasicompact morphism

The modifier ‘quasi-compact’ (or simply ‘quasicompact’) is used to denote a compactness property in the relative setup (that is, for morphisms) and in the setups emphasising non-Hausdorff topology. For example, schemes over complex numbers have both the complex and the Zariski topology; we use “quasicompact” for Zariski and “compact” for complex topology in that setup.

For many topologists, analysts and so on, the word ‘compact’ means a compact Hausdorff space. Some topological schools distinguish a “compact space” (which is not necessarily Hausdorff) and a “compactum” (which is Hausdorff); and similarly a “paracompact space” and a “paracompactum”. In algebraic geometry, in contrast, one usually says quasicompact space to denote a topological space which is compact but not necessarily Hausdorff. For example, the Zariski topology on an algebraic variety and the topology of an étalé space of a sheaf (even over a Hausdorff topological space) are typically not Hausdorff.

A scheme is quasicompact iff it has a Zariski cover by finitely many open affine subschemes. In particular, any affine scheme is quasicompact. Most important is the relative version of this concept. A morphism $f:X\to Y$ of schemes is a quasicompact morphism if the inverse image of a quasicompact Zariski open subset of $Y$ is quasicompact (EGAI6.6.1). It is straightforward to show EGAI6.6.4 that it is enough to require this for affine subsets of $Y$, or even to require the existence of a single covering $Y={\cup }_i{U}_i$ of $Y$ by open affine subschemes $U_i$, such that the inverse image $U_i{\times }_{Y}X$ of $U_i$ in $X$ is quasicompact. A scheme $X$ over a base scheme $S$ is quasicompact if the canonical morphism $X\to S$ is quasicompact. This is consistent, because if $X$ is a usual scheme (over the spectrum of integers $S=\mathbb{Z}$) or, more generally, a relative scheme over an affine scheme $S$, quasicompactness of the canonical morphism $X\to S$ is by the above criteria clearly equivalent to the usual quasicompactness of $X$.

A composition of quasicompact morphisms is quasicompact, and the pullback of quasicompact morphisms is quasicompact. This enables the definition for algebraic stacks: a morphism of algebraic stacks is quasicompact if the pullback of that morphism to some atlas is quasicompact.

Algebraic geometers sometimes (but more rarely) also talk about quasicompact objects in more general categories, meaning compact objects (object which corepresent covariant functors commuting with filtered colimits); with or without a modifier denoting a cardinal ($\kappa$-quasicompact objects).

An old discussion on the terminological aspects (Mike, Zoran, Toby) is at $n$Forum here.

category: algebraic geometry