nLab
quasicompact

The modifier ‘quasicompact’ (or even more often ‘quasi-compact’) is used to denote a compactness property in the relative setup (that is, for morphisms) and in the setups emphasising non-Hausdorff topology.

For many topologists, analysts and so on, the word ‘compact’ means “a compact Hausdorff space”. Some topological schools distinguish a “compact space” (not necessarily including the Hausdorff property) and a “compactum” (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. Indeed, 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).

Mike: To accord with terminological conventions, this page should probably be either “quasicompact space” or “quasicompact object.”

Zoran Skoda: I do not know what are the conventions, but it was intentional to look both at quasicompact spaces and quasicompact morphisms (which are according to the dominant point of view in algebraic geometry, more important and basic notion); and aside also for q. objects. Personally I do not understand English-language preference for noun phrases. If one is to choose, quasicompact morphism is the choice.

Toby: By the «Each definition gets its own page.» convention, I'm not even sure that this shouldn't just redirect to compact space or compact object. My impression is that assuming that ‘compact’ implies Hausdorff is either (like assuming that ‘ring’ implies commutative) restricted to fields where it's a common assumption or to languages (I'm thinking mostly of Bourbaki in French here) other than English. On the other hand, if it's used that way by English-writing algebraic geometers, then I would seem to be wrong (since algebraic geometers often have non-Hausdorff spaces).

Zoran Skoda: Convention that ‘compact’ includes Hausdorff is very common also among people working predominantly on nice spaces, particularly differetial geometers, differential topologists, people studying metric spaces and so on. But for “paracompact” the situation is more tricky: in literature, even on general topology there are also competing definitions, which are all equivalent for Hausdorff spaces. All my life I bounce in such people; my own education does not assume Hausdorffness, unless it is said in the form “compactum”. Algebraic geometers always say quasi-compact, it has nothing to do with language; but as I say for algebraic geometers the basic notion is quasi-compact. The emphasis of this entry is on the terminology and morphisms (what should be expanded on: I still did not write the deifnitions of quasi-compact MORPHISM in various setups); so redirection won’t work I think. Plus although from my point of view saying quasicompact and compact is the same for spaces; one would never say compact for the scheme; scheme is said to be quasicompact if its underlying space is (quasi)compact.
There is an additional reason for that: one can consider a nonsingular variety over complexes which is quasicompact, and which itself is not compact in complex topology (under GAGA). But in the same considerations it is often useful to have some arguments in Zariski and some in complex topology; one of the reasons for word quasicompact is that sometimes we have the “same” example which we are used to think as of noncompact space but it is (quasi)compact in Zariski topology. When an algebraic geometer thinks of the difference between compact and quasicompact for complex varieties he has that in mind; in more general setups about Hausdorff vs nonHausdorff. In the same time, when talking about objects in derived categories of qcoh sheaves, even algebaric geometers use moreoften term compact than quasicompact; thus redirecting to compact object and saying this is for algebraic geometry won’t do for all the 3 notions in this entry (on the contrary side, nobody says compact morphism as far as I could confirm, but quasicompact morphism).

Toby: Ah, so when you've got both Zariski and complex topologies around, you can easily distinguish the former by the prefix ‘quasi’; that's cute. Anyway, perhaps we'll move this to quasicompact morphism if you write mostly about that, but I won't try to move anything for now.