More Properties of Morphisms

More properties of morphisms Note that finite presentation is not discussed in EGA 1; see EGA 4.1 instead. 1 More about separated morphisms Lemma. The composition of closed immersions is a closed immersion. Proof. Let f : X � Y and g : Y � Z be closed immersions. Since the property of being a closed immersion is local on the base, we may assume Z = Spec(A) is affine. Then Y = Spec(B) for B a quotient of A, so X = Spec(C) for C a quotient of B. Hence C is a quotient of A, proving the claim. (A similar argument shows that a composition of finite morphisms is finite.) Lemma. (a) Any closed immersion is separated. (b) A composition of separated morphisms is separated. (c) Separatedness is stable under base change. (d) A product of separated morphisms is separated. (e) If f : X � Y and g : Y � Z are morphisms, g � f is separated, and g is separated, then f is separated. (f ) If f : X � Y is separated, then fred : Xred � Yred is separated. Proof. We know (a) because closed immersions are affine and affine morphisms are separated. We know (c) from the previous handout. Parts (d)-(f) follow once we also have (b); see exercises. It remains to prove (b). Let f : X � Y and g : Y � Z be separated morphisms. Then X ×Y X maps to X ×Z X; in fact, this morphism is the base change of the closed immersion � : Y � Y ×Z Y by f × f : X ×Z X � Y ×Z Y . (To check this: use functor-of-points to reduce to the analogous assertion for sets. This can be checked with Z equal to a singleton set, so we just want to know that for a morphism of sets X � Y , the fibre product of Y and X × X over Y × Y equals X ×Y X. This is obvious.) Hence X ×Y X � X ×Z X is a closed immersion. Since the composition of closed immersions is a closed immersion (previous lemma), we find that X � X ×Y X � X ×Z X is a closed immersion. 2 Quasicompact morphisms A morphism f : Y � X with X affine is quasicompact if Y is quasicompact as a topological space. This definition satisfies the strong collater (exercise), so we get a notion which is local on the base and stable under base change. Exercise. Any affine morphism is quasicompact. 1 3 Finite type and finite presentation Let A be a ring. Recall that an A-algebra B is finitely generated if it is of the form A[x1 , . . . , xn ]/I for some nonnegative integer n and some ideal I of A[x1 , . . . , xn ]. If I can be chosen to be a finitely generated ideal, we say that B is finitely presented; this is of course automatic if A is noetherian (as it will be in most of our examples). Let f : Y � X be a morphism of schemes with X = Spec(A) affine. We say f is locally of finite type/presentation if Y is a union of open subschemes, each of the form Spec(B) with B a finitely generated/presented A-algebra. If only finitely many such open subschemes are needed, we say f is of finite type/presentation. These satisfy the strong collater (exercise). If f : Y � X is of finite type, we sometimes say that Y is of finite type over X. Similarly for the other definitions. Obvious: any finite morphism, including any closed immersion, is of finite type. Exercise. A morphism f : Y � X is of finite type/presentation if and only if it is quasicompact and locally of finite type/presentation. 4 Algebraic varieties We can now give a scheme-theoretic rendition of the theory of abstract algebraic varieties, in the sense of 18.725. (But see below.) Let k be an algebraically closed field. An affine variety is a locally ringed space defined by some data of the following form. Pick a nonnegative integer n and an ideal I of k[x1 , . . . , xn ], and put X = V (I). Equip X with the Zariski topology, i.e., take a basis of open sets of the form D(g) = {x � X : g(x) →= 0} for g � k[x1 , . . . , xn ]. Define a regular function on an open subset U of X to be a function h : U � k such that for each x � U, there exist f, g � k[x1 , . . . , xn ] and a nonnegative integer m such that g vanishes nowhere on U while g m h − f vanishes identically on U . Then the regular functions on U form a sheaf. In the context of schemes, we interpret X to be the set of maximal ideals in Spec(A) for A = (k[x1 , . . . , xn ]/I)red , equipped with the structure of a locally ringed space given by restriction from Spec(A). Now recall that an abstract algebraic variety is a locally ringed space covered by affine varieties. Theorem 1. The category of abstract algebraic varieties over the algebraically closed field k is equivalent to the category of schemes which are reduced and locally of finite type over Spec(k). Proof. Exercise. The key point is to check that if X = Spec(A) and Y = Spec(B) for A, B two reduced finitely generated k-algebras, then the morphisms from X to Y are the same as the morphisms of the corresponding algebraic varieties. But that is because they both correspond to ring homomorphisms B � A. 2 Beware that there is no universal definition of algebraic varieties, because everyone seems to prefer to add additional hypotheses. For instance, Hartshorne (see Chapter I) forces his varieties to be separated (as often do I). Some authors also force their varieties to be irreducible, i.e., not admitting two disjoint open subschemes. And so on. 5 Proper morphisms We would like to have an algebraic analogue of the notion of a compact algebraic variety over the complex numbers. For this, we introduce the notion of properness. A morphism f : Y � X of schemes is proper if it is separated, of finite type, and universally closed. The latter means that any base change of f is a closed map of topological spaces (i.e., carries closed sets to closed sets); this condition comes from the notion of a proper map of topological spaces (see exercises). Since these properties are all local on the base and stable under base change (the last one by fiat), properness is also. The definition of properness is rather hard to check. One easy case: a closed immersion is separated (because it’s affine), of finite type (obvious), and universally closed (because any base change is still a closed immersion, so has closed image), so is proper. Besides this example, and the following slightly fancier example... Exercise. Any finite morphism (including any closed immersion) is proper. ... all examples of properness will ultimately be extracted from the following theorem. Theorem 2. The morphism f : PnZ � Spec Z is proper. Hartshorne proves this using the valuative criterion for properness (under a somewhat mysterious noetherian hypothesis). I’ll ultimately prove this following EGA, but I need to wait until the next lecture so I can say more about projective spaces in the interim. I will point out now that the fact that f is of finite type is evident from the glueing construction, and the separatedness may be obtained by describing the diagonal � : PnZ � PnZ ×Spec Z PnZ explicitly (exercise). As for separated morphisms, we have some properties. Lemma. (a) Any closed immersion is proper. (b) A composition of proper morphisms is proper. (c) Properness is stable under base change. (d) A product of proper morphisms is proper. (e) If f : X � Y and g : Y � Z are morphisms, g � f is proper, and g is separated, then f is proper. (f ) If f : X � Y is proper, then fred : Xred � Yred is proper. 3 Proof. Again, (d)-(f) follow from (a)-(c). We already observed (a) and (c). To check (b), we already checked that separatedness composes. Finite type composes by an argument similar to the proof that closed immersions compose. Universal closedness composes because a composition of closed maps of topological spaces is again closed. Corollary. Any morphism f : X � Y that factors as a closed immersion of X into P nY = PnZ ×Spec Z Y followed by the projection PnY � Y is proper. The converse is not true even over C, as there are compact algebraic varieties which are not closed subvarieties of any projective space. See the appendices to Hartshorne for an example. One can often deal with these using Chow’s lemma, about which more later. 4