Math 248B. Base change morphisms

Transcription

1 Math 248B. Base change morphisms 1. Motivation A basic operation with shea cohomology is pullback. For a continuous map o topological spaces : X X and an abelian shea F on X with (topological) pullback 1 F on X, there is a unique map o δ-unctors (1.1) H i (X, F ) H i (X, 1 F ) that is the usual pullback or i = 0. Example 1.1. When F is a constant shea A on X associated to an abelian group A, so the same holds or 1 F on X, then (or reasonable spaces) the general pullback construction in shea cohomology recovers the traditional pullback map H i (X, A) H i (X, A) as in algebraic topology. Also, i is a map o ringed spaces and F is an O X -module then we get the ringed-space pullback F := O X 1 O X 1 F and so a composite pullback map H i (X, F ) H i (X, 1 F ) H i (X, F ). Remark 1.2. In the topological case (with reasonable spaces) and in the scheme case (with quasi-coherent sheaves) we compute shea cohomology in terms o Čech theory. Sometimes there are natural pullback maps in Čech theory, and it is natural to wonder i the above pullback maps in shea cohomology obtained via abstract δ-unctor considerations are the same as what is computed more explicitly via Čech-theoretic methods. Let us state the desired compatibility in precise terms, and then we give a reerence or its proo. (This compatibility is implicit in some proos in Hartshorne s textbook Algebraic Geometry, although it is never stated there.) Consider a general map o ringed spaces as above; this could be topological spaces equipped with the structure shea Z, or complex maniolds, or schemes, and so on. Let F = F (so this is topological pullback in the topological case). Let U be an open covering o X, and U an open covering o X that reines the pullback open covering 1 (U). Then using pullback o sections, we get a natural map H i (U, F ) H i (U, F ). On the other hand, there are always natural maps rom Čech theory to derived unctor cohomology H i (U, F ) H i (X, F ), H i (U, F ) H i (X, F ) (obtained rom the Čech to derived unctor spectral sequence, or example), and there are isomorphisms in special cases: constant sheaves on maniolds and good covers (contractible open sets with contractible higher overlaps), quasi-coherent sheaves on separated schemes and aine open covers, coherent analytic sheaves on complex maniolds and Stein covers. Consider the diagram H i (U, F ) H i (U, F ) H i (X, F ) H i (X, F ) in which the vertical maps are the natural ones, the bottom is the abstract pullback, and the top is the concrete Čech-theoretic pullback. The compatibility we want to have is that this diagram commutes. This underlies many concrete calculations o pullback maps in shea cohomology. The commutativity o the diagram does hold in general, but the proo is not obvious. We reer the interested reader to EGA, 0 III, 12.1 (see especially the diagram ( )) or a proo. 1

2 2 In this handout, we want to discuss a relative version o the various pullback maps in shea cohomology. Consider a commutative diagram o ringed spaces (1.2) X e h X M h M and an O X -module F with pullback F := h F on X. The three main cases o interest to us are the topological case (using structure sheaves Z on everything, so F is an abelian shea on X and F is its topological pullback on X ), the scheme case (using quasi-coherent sheaves subject to some initeness hypotheses), and the holomorphic case with complex maniolds with a submersion. In this latter case, we are most interested in F locally ree o inite rank (i.e., associated to a vector bundle on X), so F is the corresponding vector bundle pullback along X X. In all three cases, the most important situations will have proper and (1.2) cartesian (i.e., X = X M M ), but we do not make any such assumptions at the outset. Remark 1.3. We use the ringed space language so that we can largely treat the topological and holomorphic cases at the same time, rather than write out everything twice. I G is an O M -module then its stalk G m is an O M,m -module, so in the locally ringed case we can kill m m to get a vector space G (m) := G m /m m G m over the residue ield at m. In the topological setting (with structure shea Z) we only have stalks. The higher direct images R i (F ) are O M -modules, and in class we constructed a natural map (1.3) R i (F ) m H i (X m, F Xm ) in the topological case and a C-linear map (1.4) R i (F )(m) H i (X m, F m ) in the holomorphic case (where F m on X m denotes the ringed-space pullback o F along X m X; this is not a stalk at m!). Note that the construction (1.4) makes sense more generally in the setting o locally ringed spaces (including both schemes and complex maniolds), as a linear map over the residue ield at m. Much as the higher direct image sheaves R i F will be a useul device to glue ibral cohomologies H i (X m, F m ) into a single global structure over M, or m M we seek to glue the ibral pullback cohomology maps H i (X h(m ), F h(m )) H i (X m, F m ) into a map involving higher direct images relative to and. More speciically, the aim o this handout is to deine and analyze the unctorial properties o a vast generalization: to any commutative square (1.2) we will associate canonical O M -linear morphisms (1.5) h R i (F ) R i (F ). These will be called base change morphisms when we are in the topological, scheme, or holomorphic cases and (1.2) is cartesian. Example 1.4. In the special case that h is the inclusion o a point and the let side o (1.2) is the m-iber o in the topological case (resp. scheme or holomorphic cases), the map (1.5) will recover (1.3) (resp. (1.4)). The generality o (1.5), allowing rather arbitrary h, is very useul. In 12 o Chapter III o Hartshorne (as well as the end o Vakil s notes) there is a discussion o (1.5) or certain cartesian diagrams o sheaves (with suitable F and proper ). The main point o that discussion is the understanding o ibral criteria o Grothendieck which imply that both sides o (1.5) are locally ree o inite rank and the map (1.5) is an isomorphism. These are called cohomology and base change theorems. They provide a precise sense in which ibral cohomologies H i (X m, F m ) can be said to vary nicely in m (under suitable ibral hypotheses!).

3 3 Remark 1.5. A conusing aspect o the discussion o cohomology and base change in Hartshorne s textbook is that the construction and unctorial properties o (1.5) are never systematically discussed there, yet they pervade the (omitted) rigorous details o many o the proos. Hence, in eect the present discussion logically precedes any treatment o the cohomology and base change in the scheme or holomorphic cases. Strictly speaking, Hartshorne only discusses the cohomology and base change theorems or projective ; the general case is in Vakil s notes (and also in EGA III). There are analogous results in the holomorphic setting, due to Kodaira Spencer and Grauert. In class we discussed some reasons why these results are useul in the analytic setting, and in our later study o elliptic curves over schemes we will ind them to be equally useul. Briely, whenever an argument with elliptic curves in the classical setting over a ield uses Riemann Roch, the proo o a corresponding result in the relative setting in either the holomorphic or scheme cases will use the theorems on cohomology and base change to reduce diicult problems in the relative setting to solved problems in the classical setting on ibers. 2. Constructions and examples To construct (1.5), irst observe that or any open U in M with preimage U = h 1 (U) in M, the preimage o X U := 1 (U) in X is X U := 1 (U ) due to the commutative o (1.2). Hence, there is a natural pullback map H i (X U, F ) H i (X U, F ). But in view o the description o higher direct images as sheaiications, there is a natural map H i (X U, F ) = H i ( 1 (U ), F ) (R i F )(U ) whose target is Γ(h 1 (U), R i F ) = Γ(U, h (R i F )). Summarizing, or open U in M we have constructed natural maps (2.1) H i ( 1 (U), F ) Γ(U, h (R i F )) which are easily checked to respect restriction in U. Since h (R i F ) is a shea whereas U H i ( 1 (U), F ) is merely a preshea whose sheaiication is R i F, by the universal property o sheaiication we obtain rom the maps (2.1) a natural map R i F h (R i F ) o sheaves on M. By the construction this is seen to be O M -linear, so by the adjointness o h and h we obtain an O M -linear map as in (1.5). This completes the construction. Example 2.1. Suppose that (1.2) is a diagram o topological spaces or locally ringed spaces (such as complex maniolds or schemes) with h the inclusion o a point m M. Then h G is the stalk G m in the topological case and is G (m) := G m /m m G m in the locally ringed space case. In this way, one sees upon unraveling the construction that (1.5) recovers (1.3) in the topological case and (1.4) in the locally ringed case. Example 2.2. I h is an open immersion and (1.2) is cartesian (i.e., X is the preimage o an open M M under the map : X M), then (1.5) is an isomorphism expressing the act that the ormation o higher direct image sheaves is local on the base. Example 2.3. In general, what is (1.5) saying at the level o stalks? For m M, the map on m -stalks has the orm O M,m O M,m lim H i (X U, F ) lim H i (X U, F ) where U varies through opens in M around h(m ) and U varies through opens in M around m. A special class o such U are the opens h 1 (U), and in act the m -stalk map is induced by the natural pullback maps (2.2) H i (X U, F ) H i (X h 1 (U), F ) linear over h : O M (U) (h O M )(U) = O M (h 1 (U)), as is checked by reviewing how (1.5) was constructed. This description is a reason that we regard (1.5) as a relativization o ordinary pullback maps in shea cohomology.

4 4 3. Functorial properties We conclude by discussing how (1.5) behaves with respect to δ-unctoriality in F and with respect to compositions in the horizontal and vertical directions: loosely speaking, unctional behavior with respect to composition in h and in. For the δ-unctoriality, suppose that M M and X X are lat maps o ringed spaces, such as submersions o maniolds, lat maps o schemes, or maps o topological spaces equipped with the structure shea Z. In such cases F is an exact unctor in F, so the target o (1.5) is a δ-unctor in F. Likewise, h is exact, so the source o (1.5) is δ-unctorial in F. This leads to: Proposition 3.1. When M M and X X are lat then (1.5) is a morphism o δ-unctors in F ; that is, it respects the δ-unctorial structure o both sides as we vary F in a short exact sequence o O X -modules. Proo. It suices to check the δ-unctorial compatibility ater passing to stalks. In view o the description on stalks given in Example 2.3, it suices to prove that (2.2) is a morphism o δ-unctors or each open U M. Renaming U as M (as we may clearly do), we are reduced to checking that the composite map H i (X, F ) H i (X, 1 F ) H i (X, F ) is a morphism o δ-unctors. This holds or the irst step by the very deinition o topological pullback in shea cohomology, and or the second step (which is deined to be the cohomology unctor applied to the natural map o sheaves 1 F F ) are use that cohomology is deined as a δ-unctor. The compatibility with respect to composition in h works out as ollows. Consider a concatenated pair o commutative squares X X X M h M h M and let h : M M denote h h. We then get the pullbacks F on X and F on X, so F is also naturally identiied with the pullback o F along X X. Thus, there are natural morphisms θ : h R i (F ) R i (F ), θ : h R i (F ) R i (F ), θ : h R i (F ) R i (F ). The consistency among these morphisms is given by: Proposition 3.2. The composition θ h (θ) is equal to θ. Proo. Again passing to stalks and arguing as in the preceding proo, we are reduced to the general unctoriality o shea cohomology pullback. That is, or a pair o maps o topological spaces h : X X and h : X X with composite h = h h and any abelian shea F on X, we claim that the composition o pullback maps H i (X, F ) H i (X, h 1 F ) H i (X, h 1 (h 1 F )) = H i (X, h 1 F ) is the pullback along h. (This is a act one should have really veriied at the beginning o the entire story, when irst introducing shea cohomology pullback and contemplating its unctorial behavior.) In view o the δ-unctoriality o everything in sight (as we vary F ), this problem reduces to the case i = 0, which is easily veriied. The case o behavior in the vertical direction is somewhat more subtle to even ormulate, and or our purposes will not be nearly as important as the horizontal compatibility in the preceding proposition.

5 5 Consider a commutative diagram o ringed spaces Y Y g X h X M and an O Y -module G. Thus, (1.5) provides a natural morphism h M h R i ( g) (G ) R i ( g ) (G ) where G is the ringed-space pullback o G along Y Y. There are Leray spectral sequences R p R q g (G ) R p+q ( g) (G ), R p g R q g (G ) R p+q ( g ) (G ), and when h and h are lat (so h and h are exact and hence commute with the ormation o kernels and images, as arise in the ormation o spectral sequences) it makes sense to ask i these spectral sequences are compatible with (1.5) relative to the smaller commutative squares. This is the content o: Proposition 3.3. When h and h are lat, the natural maps h (R p R q g (G )) R p h (R q g G ) R p R q g (G ) and h R n ( g) (G ) R n ( g ) (G ) are compatible with the ormation o the associated spectral sequences. Proo. By passing to stalks as in the two preceding proos, this again reduces to an assertion about pullback in topological shea cohomology: it commutes with the ormation o Leray spectral sequences such as H p (X, R q g G ) H p+q (Y, G ) and the analogue or X Y M. This in turn is seen by going back to how spectral sequences are constructed in terms o Cartan Eilenberg resolutions and how shea cohomology pullback may be computed in terms o injective resolutions (due to the characterization o the latter in terms o δ-unctoriality). We leave the details to the interested reader. g