Topological p-adic vector spaces and index theory

This report is part of a work developed from Robba’s ideas whose ultimate goal would be to obtain a general finiteness theorem for p-adic cohomology. The basic question is to prove existence of index for ordinary differential operators. Here we expose continuity properties of index. Although it is apparently of an algebraic nature, the difficulties of index theory are mainly analytic. In particular, it involves a great deal of topological vector spaces, far beyond mere Banach spaces theory. The aim of this report is to illustrate this fact. 1991 Mathematics subject classification : 12H25 I Index and duality. Let k be a complete ultrametric field, for instance k = ~ p, let E be a k-vector space and u : E --> E a linear map. Definition. The map u is said to have an index if both ker u and coker u = E/ Im u are finite-dimensional. If so, set : x(u) = x( u, E) = dim(keru) dim(cokeru) Let E be a (locally convex) topological k-vector space, let u be continuous and let E’ be the (strong) dual of E. Question. If u has an index does it has one ? If true, compare x( u) and ). The following facts are easy to verify : ker(’u ) = is isomorphic to the dual of the space coker(u) endowed with the quotient topology.


Gilles Christol and Zoghman Mebkhout
Ann. Math. Blaise Pascal, Vol. 2, N° 1, 1995, pp.93-98 Abstract. This report is part of a work developed from Robba's ideas whose ultimate goal would be to obtain a general finiteness theorem for p-adic cohomology. The basic question is to prove existence of index for ordinary differential operators. Here we expose continuity properties of index. Although it is apparently of an algebraic nature, the difficulties of index theory are mainly analytic. In particular, it involves a great deal of topological vector spaces, far beyond mere Banach spaces theory. The aim of this report is to illustrate this fact. 1991 Mathematics subject classification : 12H25 I Index and duality.
Let k be a complete ultrametric field, for instance k = ~ p, let E be a k-vector space and u : E --> E a linear map.
Definition. The map u is said to have an index if both ker u and coker u = E/ Im u are finite-dimensional. If so, set : Let E be a (locally convex) topological k-vector space, let u be continuous and let E' be the (strong) dual of E. Question. If u has an index does it has one ? If true, compare x( u) and ).
The following facts are easy to verify : ker('u ) = is isomorphic to the dual of the space coker(u) endowed with the quotient topology. The pairing {x, y) def (x, y) gives a canonical map coker(tu ) (keru)' . .
To go further, a definition is needed : Definition. The space E is said to have Banach's property if both conditions u continuous and coker u finite-dimensional imply that Im u is closed.
For instance, as already noticed by L. Schwarz, every Banach space has Banach's property (this is a straightforward consequence of the open map theorem applied to the map E x F -~ E for some algebraic, but a priori not topological, supplementary F of Im u ). Proposition 1 : : If E has Banach '3 property and if u has an index then dim(ker tu ) = dim(coker u) and dim(coker tu ) dim(ker u). . In other words, tu has an index and x(u) + Proof : By hypothesis, Im u is closed then 1 ) coker u is Hausdorff hence : dim(ker = dim(coker u)' = dim(coker u) 2) The map coker(u ) ~ (ker u)' is injective. Actually, let x in E. If its image x in coker % belongs to ker i one has (x, y) = 0 for y in ker u. Hence one can define z in (Im u)' by {z, u(y)) = {x, y). As Im u is closed, z is the restriction of some element of E' also denoted by z. Then one has z = ~(~) and .r=0.
Remarque. If Hahn-Banach theorem were true for k-vector spaces, one could also prove that i is onto. To bypass this difliculty, additional conditions are needed.
shows that ,A(r) is a countable inverse limit of Banach spaces. Then it is a Frechet space when endowed with inverse limit topology. This is the "usual" topology defined by the family of and we will use it. .. shows that is a countable direct limit of Banach spaces. We will endow it with (locally convex) direct limit topology. Now Theorem 3 [5] : : The space A(r) is reflexive and its dual is H~(r) for the pairing ( 0 3 A 3 ñ 0 a n x n , 0 3 A 3 ñ 0 b n x -n -1 ) = 0 3 A 3 ñ 0 anbn. So is both an space and a D,~ space (dual of Frechet [4] ). Moreover, the space R(r) is its one dual.
To conclude this section we recall two "classical" results : Theorem 4 : : Every Frechet space has Banach's property. In fact, the second one is written for real or complexe spaces. It is possible but rather tedious to verify that Hahn-Banach is not used in this long proof. Let take the opportunity to express the wish that this basic theorem and related ones take their deserved places in future account of topological spaces over an ultrametric field.

III Operators.
Let D be the non-commutative ring K[x, d dx] I of (linear) differential operators with polynomial coefficients. The spaces A(r), and R(r) are stable by derivation. As A(r) and R.(r) contain ~[3?], they are D-modules for the scalar multiplication P f = P(f).
To define a Ð-module structure on one uses the exact sequence : For f in and P in P we define the scalar multiplication by P/ = ~(P(/)). Then (**) becomes an exact sequence of Ð-modules.
Any difserential operator P acts continuously on the Banach spaces H(r) and K(r).
Hence it acts continuously on A(r) and R(r) and then on H~(r). Basic facts : Let P = ~~o a, ( ~)' be a differential operator of D. The following assertion are easy to check : A) ker(P,X(r)) and are finite-dimensional and their dimensions are bounded by c(P) = d + max, deg(a,).
B) One has ~P = ~~o(a, for the three dualities we defined (use the Leibnitz rule to rearrange the terms). Now the following result is a particular case of corollary 2. Corollary 6 : A differential opener P has an index in A(r) if and only if the differential operator tP has an index in H~(r) . If so ~(P,A(r)) If one is only interested in the exitence of index, one can work on R as shown by the following result. Proposition 7 : If an differential opener P has an index in R(r) then : has index both in A(r) and and one has : = ~(P,A(r)) + = ~(tP,A(r)) Proof : The short exact sequence (**) gives rise to a long exact sequence : 2014. ~ coker(P, A(r)) 2014. coker(P,R(r)) 2014. coker(P, H~(r)) ~ 0 where underlined spaces are finite-dimensional by hypothesis or by assertion A). Then the two remaining spaces are also finite-dimensional.

IV The Theorems.
We are now interested by the way the index varies with r.
Theorem 8 : : Let rn be a growing sequence with limit r and let P be a differential operator. If P has an index in H~(rn) for all n then it has an index in H~(r) and (P,H~(r)) = limn~~ ~(P,H~(rn) (index being integers, that means ~(P,H~(r)) = X( P, for n large enough).
Proof : (see [1]). As rn the canonical injection H~(rn) -H~(rn+1) has dense image. Then the map coker(P,H~(rn)) ~ coker(P,H~(rn+1)), between finitedimensional Hausdorf spaces, has dense image hence is onto. Therefore the sequence dim (coker(P, is decreasing and the sequence dim (coker(P, increasing and bounded by assertion A). Hence both are constant for ra large enough. To conclude, suffice it to say that = lim H~(rn) and that lim is an exact functor.
Corollary 9 : : Let rn be a growing sequence with limit r and let P be a differential operator. If P has an index in A(rn) for all n then it has an index in A(r) and x(P, ,A(r)) _ limn~~ ~(P,A(rn)).
Remark : There are two obstructions to obtain a direct proof of the corollary 9. The first one is that lim is not an exact functor. This could be overpassed by means of a sophisticated version of Mittag-Leffler condition due to Grothendieck (~3?, III-o-13.2.4).
The second and deeper one, is that the sequence dim (coker(P, A(rn)) has no a priori reason to be bounded.
We'll explain elsewhere [2] how to define, for each real r, "p-adic exponents for the radius r" of the differential operator P. This definition is far too long to be given here.
Then it will be possible to prove the folowing very deep result conjectured by Robba [6] : Theorem 10 [1,2] : If p-adic exponents for the radius r of P are not Liouville neither have Liouville differences, then I~ P has an index in A(r).