Some remarks on Beilinson adeles Academic Article uri icon

abstract

  • Let X be a scheme of finite type over a field k. Denote by A' the sheaf of Beilinson adeles with values in the algebraic De Rham complex QX/k* Then QX/k 24 A is a flasque resolution. So if X is smooth, A' calculates De Rham cohomology. In this note we rewrite the proof of Deligne-Illusie for the degeneration of the Hodge spectral sequence in terms of adeles. We also give a counterexample to show that the filtration A,>q does not induce Hodge decomposition. 0. INTRODUCTION In this note we consider two aspects of Beilinson adeles on schemes. Let X be a scheme of finite type over a field k. Given a quasi-coherent sheaf M let Aqed (M) be the sheaf of reduced Beilinson adeles of degree q (see [Be], [Hr], [HYl]). It is known that ArqAd(M) qA d(Ox) ca M. For any open set U C X (0.1) F(U, Ared(M)) C fJ M (ES(U)red where S(U)red is the set of reduced chains of points in U of length q, and Me is the Beilinson completion of M along the chain ( (cf. [Yel]). For q = 0 and M coherent one has M(X) = MX, the mx-adic completion, and (0.1) is an equality. Let QX be the De Rham complex on X, relative to k. As shown in [HYl], setting -A A X/ and A' : p+q=i AP we get a differential graded algebra (DGA) which is quasi-isomorphic to Qx/k and is flasque. Thus H (X, Qx/k) = H F(X, Ak). In particular if X is smooth, we get the De Rham cohomology HbDR(X/k) More generally, let X be a formal scheme, of formally finite type (f.f.t.) over k (see [Ye2]). Then applying the adelic construction to the complete De Rham complex ox/k we get a DGA A'. If X C X is a smooth formal embedding (op. cit.) and chark = 0, then H r(X,A) = HDiR(X/k). There is an analogy between the sheaf Apjq on a smooth n-dimensional variety X and the sheaf of smooth (p, q)-forms on a complex manifold. The coboundary operator D of Ax is defined as a sum D :=D' + D", and D" AP -? VX plays Received by the editors May 24, 1995. 1991 Mathematics Subject Classification. Primary 14F40; Secondary 14C30, 13J10. This research was partially supported by an Allon Fellowship. The author is an incumbent of the Anna and Maurice Boukstein Career Development Chair. ?)1996 American Mathematical Society 3613 This content downloaded from 157.55.39.112 on Wed, 07 Sep 2016 05:31:44 UTC All use subject to http://about.jstor.org/terms 3614 AMNON YEKUTIELI the role of the anti-holomorphic derivative. The map fx ARese tF(X, A :2) k is the counterpart of the integral (Rest is the Parshin-Lomadze residue along the maximal chain ( in X, see [Yel]). This analogy to the complex manifold picture is quite solid; for example, in [HY2] there is an algebraic proof of the Bott residue formula, which in some parts is just a translation of the original proof of Bott to the setting of adeles. The main purpose of this note is to examine the potential applicability of adeles for the study of algebraic De Rham cohomology. In ?1 the construction of Deligne and Illusie [DI] is rewritten in terms of adeles. In ?2 we consider a possibility to relate adeles to Hodge theory, and show by example its failure. 1. LIFTING MODULO p2 We interpret, in terms of adeles, the result of Deligne and Illusie on the decomposition of the De Rham complex in characteristic p. In this section we shall follow closely the ideas and notation of [DI]. Let k be a perfect field of characteristic p. Write k W2 (k). Let Fk: Spec k Spec k be the Frobenius morphism, i.e. Fk* (a) = aP for a C k. By pullback along Fk we get a scheme X' := X Xk k and a finite, bijective k-morphism F = Fx/k X -X'. Assume we are given some lifting X of X to k. By this we mean a smooth scheme X over k s.t. X _ X xk k. Using the Irobenius Fk we also define a scheme X', and a k-morphism F X -* X'. For any point x C X the relative Frobenius homomorphism F* -X*,F(x) OX,x can be lifted to a k-algebra homomorphism Fx 0kF(x) 3* O,,x (cf. [DI]) In view of (0.1), the collection {Fx*}xcx induces a homomorphism of sheaves of DG k-algebras F*: Ared(Q>/ ) 3 F*-red(Q'/j) Lemma 1.1. The liftings {FX* }xEx determine Ox' -linear homomorphisms f:Q, '/k F*A h: Q'/k k F*AO such that D(f + h) = 0. Proof. Let p Q'/k P4 be multiplication by p. This extends to an AOed((x)linear isomorphism P A>=Ard (QX/k) PAOed (Q'/) Just as in [DI] we get a homomorphism f making the diagram Q1 ~~~F*AOdQ/~ iX'/k red(QX/k) commu i*e.F*Aed(Q/k) commutative. This content downloaded from 157.55.39.112 on Wed, 07 Sep 2016 05:31:44 UTC All use subject to http://about.jstor.org/terms SOME REMARKS ON BEILINSON ADELES 3615 Next, for any chain of points (Xo, x1) in X and a local section a E Ox, we have D"F* (a) = Fx*0 (a) Fx1 (a) C Ppx,(xo xl)

publication date

  • January 1, 1996