CHOW S LEMMA. Matthew Emerton

Transcription

1 CHOW LEMMA Matthew Emerton The aim o this note is to prove the ollowing orm o Chow s Lemma: uppose that : is a separated inite type morphism o Noetherian schemes. Then (or some suiciently large n) there exists a diagram o the ollowing type i P n p in which the right-hand square is the Cartesian diagram exhibiting P n as the base-change o P n via the morphism, and i is a closed immersion, such that the composition p i is surjective and induces an isomorphism over a dense open subset o, and such that the composition i is an immersion. In other words, we can write any separated scheme o inite type over as the image under a birational projective map o a quasi-projective -scheme. We argue by induction on the number o irreducible components o. Let us complete the inductive step irst: suppose that is reducible, and write = with each o and a non-empty proper closed subset o. Let U = \ and V = \. I we choose and each to be a union o irreducible components o, having no irreducible component in common, then we see that U is dense in and that V is dense i, and that in act the intersection o U (respectively V ) with any dense open subset o (respectively ) is again dense in (respectively ). We assume that we have chosen and in this ashion. Then each o and has ewer irreducible components then. Let I be the ideal shea in O which cuts out the reduced induced structure on. Then when we restrict I to the open subset (o both and ) U it is a nil ideal (i.e. every section is locally nilpotent) and thus is nilpotent, since U is open in a Noetherian scheme and thus is itsel a Noetherian scheme. uppose that IU M = 0. Then give the closed subscheme strucutre corresponding to the shea o ideal I M, and let j : the corresponding closed immersion. This choice o scheme structure has the nice property that the open set U has the same scheme structure whether we regard it as an open subset o or o. imilarly, i we write J or the ideal shea cutting out the reduced induced scheme structure o, then JV N = 0 or some N. We give the closed subscheme structure corresponding to the ideal shea J N, and let j : denote the corresponding closed immersion. Then V has the same scheme structure whether we regard it as an open subset o or o. ince closed immersions are inite type and separated, we see that the composisitions j : and j : are inite type and separated. By the P n p 1 Typeset by AM-TE

2 2 MATTHEW EMERTON inductive hypothesis, Chow s Lemma holds or each o and. o we may ind diagrams i P n ( j ) P n p p j and i P p ( j ) j P having the properties listed in the statement o Chow s Lemma. In particular, combining the conclusions o Chow s Lemma with our above observation, we see that there is a dense open subset U o, which we may assume to be contained in U (by intersecting it with U i necessary), over which p i is an isomorphism, and that there is a dense open subset V o, which we may assume to be contained in V (by intersecting it with V i necessary), over which p i is an isomorphism. The closed immersion j : induces a closed immersion P n P n. imilarly the closed immersion j : induces a closed immersion P n j P. Thus we can actor each o the above diagrams in the ollowing way: and i P n j p P n P n p p p j i j P P P n p p p j. We may take the disjoint unions o the two composite closed immersions j i and j i to obtain the ollowing diagram: (j i ) (j i ) P n P P n P p p j j. Now (j i ) (j i ) is a disjoint union o closed immersions, and so is a closed immersion, while ((j i ) (j i )) = (( i) i ) (( j) i ) is the disjoint union o two immersions and so is an immersion. Also, p ((j i ) (j i )) = j p i j p i induces an isomorphism over U and V. (It is here that we are using the act that U and V, and hence U and V, have the same scheme structure whether we regard them as open subsets o and or o. We are o course also using the act that U and V, and hence U and V, are disjoint in.) ince U is dense in and V is dense i, we see that U V is dense in. Thus in order to prove Chow s Lemma or, it suices to ind a closed immersion j

3 CHOW LEMMA 3 P n P Pn or some n, or then the ollowing diagram P n (j i ) (j i ) P n P P n P p p j j. would prove Chow s Lemma or, by taking =, and taking i to be the closed immersion i : = P n P Pn. But i we take n = n then we may ind a closed immersion P n P P n by identiying P n with the linear subspace T n +1 = = T n = 0 and identiying P with the linear subspace T 0 = = T n = 0. (Here T i denote the homogeneous coordinates on P n.) Thus by induction, we are reduced to the case in which is irreducible. Beore continuing, let us remark that i the above argument seems a little complicated, it is only because we have named everything involved. The idea is very simple: we ind mapping onto by writing =, inding mapping onto and mapping onto, and taking =. Let us now assume that is irreducible, so that every non-empty open subset o is dense in. ince is Noetherian and is inite type over, we may cover by initely many aine opens pec A i, and then cover the preimage o each pec A i in by initely many aine opens pec B ij, with each B ij a inite type A i -algebra. uppose that B ij is a quotient o A i [T 1,..., T r ]. (We may take the same r or ever4 B ij with no loss o generality, since the B ij are inite in number.) Then pec B ij is a closed subset o A r pec A i, which is an open subset o A r. To summarize, can be covered by initely many aine opens, each o which admits an immersion into A r or suiciently large r. Let us now orget the A i and B ij notation, and simply reer to these aine open subsets o as U 1,..., U m. A r is an open subset o Pr, and thus each U i immerses into P r. Let P i denote the scheme theoretic image o U i in P r. (This scheme-theoretic image exists because we are in the Noetherian case, so this immersion is automatically quasi-compact. The underlying space o P i is the closure o U i in P j ). Then the map U i P i is an open immersion. Let us write P = P 1 P m, and or each i write P n P i = P 1 ˆPi P m (where ˆP i means omit P i rom the product ). Each P i is a closed subscheme o a projective space over and so is proper over. Thus each o the products P and P i is proper over. Let us also write U = m i=1 U i. ince is irreducible, each U i is dense in, and so U is dense in. Let h be the map U P deined so that the projection onto is simply the inclusion U, while the projection onto P i is the open immersion U U i P i. We are going to deine to be the scheme-theoretic image o h, which is a closed subscheme o P. Each P i is a closed subscheme o a projective space over, so their product P is a closed subscheme o a product o projective spaces. The egre embedding realizes a product o projective spaces as a closed subscheme o a projective space

4 4 MATTHEW EMERTON o suiciently large dimension, say n. Thus we obtain a diagram P P n Pn P. In order to conclude that Chow s Lemma is true or, we have to show that the projection P is an immersion, and that is an isomorphism over a dense open set. It will then ollow that is surjective, because it is closed (being the composition o the closed immersion P and the map P, which is closed because P is proper) and dominant (being an isomorphism over a dense open subset o ). We now turn to proving these two acts about. Beore giving the details o the argument, let us give the gist o it, by imagining that, U, and each o the U i and P i are simply topological spaces rather than schemes. Assume that U is dense in each U i, and that each U i is in turn dense in the respecitve P i, which we assume are compact Hausdor (the topological analogue o being proper). We also assume that is Hausdor (the topological analogue o being separated). The map h is just the diagonal map u h(u) = (u,, u) P 1 P m. We wish to understand, the closure o the image o h. Thus suppose that h(u s ) is a sequence o points in the image, converging to some point v = (x, p 1,..., p m ) in P 1 P m = P. (ince and each o the P i is Hausdor, this limit is uniquely determined.) ince is covered by the open sets U i, x must lie in U i or some i. Then we see that the sequence u s converges to x in U i, and so also in P i (since U i is a subset o P i ). ince P i is Hausdor, the sequence u s has a unique limit in P i, and so we see that x = p i. uppose that v = (x, p 1,..., p m) is the limit o some other sequence h(u s) o points in the image o h such that (p 1,..., p m) = (p 1,..., p m ). Then in particular p i = p i is in in U i, and we see that the u s converge to p i in U i, so that then also x = lim u s = p i = x, and so v = v. Thus the projection P P is a homeomorphism when restricted to. Furthermore, x U i i and only i p i U i, in which case x = p i. This easily implies that projects homeomorphically onto a locally closed subset o P 1 P m (the topological analogue o being an immersion). For i we let i denote that subset o or which x (and thus p i ) is an element o U i, we see that i maps homeomorphically onto the closure o the diagonal image o U in the open subset P 1 U i P m o P 1 P m. In this context, the argument that is a surjection can be phrased very simply: let x be an element o. Then x lies in U i or some i, and by assumption we can ind a sequence u s o points in U converging to x. Then since each P i is compact, we may reine this sequence so that it converges in each P i. Then we see that h(u s ) converges to a point o lying over x. Having illustrated our arguments in this simpler context, let us now give the scheme-theoretic arguments or Chow s Lemma. I encourage the reader to consider how the arguments below are simply reormulations in the language o schemes o the arguments just given.

5 CHOW LEMMA 5 The scheme is covered by the open sets U i, and so P is covered by the open sets U i P. Thus is covered by its intersections with each open set U i P. Now since U U i or every i, we see that h : U P actors through the open immersion U i P P as U hi U i P P. Thus the intersection o (which is by deinition the scheme-theoretic image o h) and U i P is precisely the scheme-theoretic image o the map h i. ince P i is projective over, it is separated over, and so the graph o the inclusion o -schemes U i P i is a closed subscheme o U i P i, which maps isomorphically onto U i via projection onto either the irst or second actor (where in the case o projection onto the second actor, we regard U i as an open subset o P i ). We will denote this graph by Γ i. I we base change the closed immersion Γ i U i P i via the map P i we obtain a closed immersion Γ i P i U i P i P i. Now P i P i = P i P 1 ˆPi P m P1 P i P m simply by rearranging the actors. (In the subsequent argument we will sometimes use this rearrangement o the actors without explicitly mentioning it.) We deine k i to be the closed immersion which is the composition k i : Γ i P i U i P i P i Ui P. By contruction h i : U U i P actors through the closed immersion k i : let us write h i : U h i Γ i P i k i Ui P. Then the scheme-theoretic image o h i is equal to the scheme-theoretic image o h i, since k i is a closed immersion. Now the composition Γ i P i ki U i P P (where the second arrow is the natural projection U i P P ) actors as indicated in the ollowing diagram: Γ i P i U i P i P i Ui P U i P i P i P i P. The well-deinedness o the let-most vertical arrow in this diagram ollows rom the act, which we observed when we introduced Γ i, that the projection onto the second actor Γ i P i induces an isomorphism o Γ i with the open subset U i o P i. In particular, this let-most arrow is the base change o an isomorphism, and so is itsel an isomorphism. The let-most arrow o the bottom row is the base change o the open immersion U i P i and so is an open immersion. Thus the composition Γ i P i P is the composition o an open immersion and two isomorphisms, and so is an open immersion. Now the scheme-theoretic image o h i equals the scheme-theoretic image o h i which is a closed subscheme o Γ i P i. Let us denote this by i. Then the composition i Γ i P i P is the composition o an open and closed immersion, and so is an immersion.

6 6 MATTHEW EMERTON We now wish to conclude that the composition P P is an immersion (recall that is the scheme-theoretic image o h). We have seen that is covered by the open sets i, each o which has image in the open subscheme U i P i o P. Now we claim that in act i is equal to the inverse image o U i P i in. I we knew this, we would see that the projection P is an immersion locally on P, and so is indeed an immersion. Let us prove that i is indeed the inverse image o U i P i in. The inverse image o U i P i in P is isomorphic to U i P i. The morphism h actors through the open subscheme U i P i o P : let us actor h as U hi U i P i P. Then the inverse image o U i P i in is equal to the scheme-theoretic image o h i. We wish to show that this is equal to the scheme-theoretic image o h i, which is a closed subscheme o Γ i P i. Thus it suices to show that Γ i P i is a closed subscheme o U i P i. For this, it suices to show that the Γ i is a closed subscheme o U i (since closed immersions are preserved by base change). But the immersion Γ i U i is precisely the graph o the inclusion o U i in, which is indeed a closed immersion, since is separated over. (This is the only point at which the separatedness o is required, but o course it is the crux o the argument.) This completes the task o showing that P is an immersion. It remains to show that is an isomorphism over a non-empty open subset o (necessarily dense, since is irreducible). We will show that indeed this map is an isomorphism over the open subset U o. To see this, we must consider the inverse image o U in. The inverse image o U in P is the open subscheme U P. The map h : U P actors as U h U P. Thus the intersection o and U P, which is the inverse image o U in, is equal to the scheme-theoretic image o h. Thus we would be done i we could show that h is a closed immersion, or then it would equal its own scheme theoretic image, inducing an isomorphism U U. But h is precisely the graph o the morphism U P obtained as the ibre product o the morphisms U U i P i, and so is a closed immersion, since P is separated over. This completes the proo o Chow s Lemma.