separated and proper morphisms

In ordinary topology two properties make spaces well-behaved: Hausdorffness (limits are unique — a sequence cannot converge to two different points) and compactness (a space cannot 'leak out to infinity'). The Zariski topology is too coarse for these to be useful directly, so scheme theory replaces them with relative, scheme-theoretic versions: separatedness is the Hausdorff analogue and properness is the compactness analogue. Both are conditions on a MORPHISM f: X -> S rather than on a single space, because everything in modern algebraic geometry is relative to a base.

Separatedness: a morphism f: X -> S is separated if the diagonal morphism delta: X -> X times_S X is a closed immersion. This is the exact algebraic shadow of 'the diagonal is closed', which for topological spaces is precisely the Hausdorff condition; concretely it forbids the pathology of gluing two copies of a scheme along an open set in a way that creates 'two origins' that cannot be separated. The clean test is the valuative criterion of separatedness: a map from the punctured spectrum of a valuation ring (a 'curve with one point removed') has AT MOST one extension across the missing point — limits, when they exist, are unique. Properness: a morphism is proper if it is separated, of finite type, and UNIVERSALLY closed (stays closed after any base change). The matching valuative criterion of properness demands that such a limit always EXISTS and is unique — every map from the punctured curve extends uniquely across the missing point. So separated = 'limits unique', proper = 'limits exist and are unique' = compactness, relatively.

These conditions are the gatekeepers of good behavior. Properness is what guarantees the finite-dimensionality of cohomology and the existence of intersection numbers and degrees; projective morphisms (and in particular projective varieties) are proper, which is why projective space is the natural home for the strongest theorems. Separatedness is so basic that it is usually assumed throughout. Honest cautions: the standard counterexample, the affine line with a doubled origin (two copies of A^1 glued away from 0), is a non-separated scheme — a perfectly good scheme that is not Hausdorff-like, where the origin has no unique limit. Proper is NOT the same as 'compact' in any naive sense (the Zariski topology is quasi-compact for almost everything, even the affine line), so quasi-compactness of the space does not give properness; properness is genuinely about the universally-closed, limit-existence condition on the morphism. And affine morphisms are separated but almost never proper — A^1 -> point is separated but not proper, since A^1 does run off to infinity.