Stephen Lichtenbaum

The Weil-étale topology on schemes over finite fields

We introduce an essentially new Grothendieck topology, the Weil-´ etale topology, on schemes over finite fields. The cohomology groups associated with this topology should

New Results on Weight-Two Motivic Cohomology

Among Grothendieck’s manifold contributions to algebraic geometry is his emphasis on the search for a universal cohomology theory for algebraic varieties and a conjectured description of it in terms of motives [Ma]. Various authors have recently set out to describe the properties of and conjecturally define a cohomology theory for algebraic varieties, which has been baptized “motivic cohomology” by Beilinson, MacPherson, and Schechtman ([BMS],[Be],[Bl],[T],[L1],[L2]). It is hoped that this theory, when and if it is fully developed, will in some sense be universal and thus provide at least a partial response to Grothendieck’s question.

Zeta-Functions of Varieties Over Finite Fields at s=1

Let κ be a finite field of cardinality q = p ’ . Let \(\overline \kappa \) be a fixed algebraic closure of κ. Let X be a smooth projective algebraic variety of dimension d over κ such that \(\overline X = X \times \overline \kappa \) is connected.

Suslin Homology and Deligne 1-Motives

Suslin has defined a complex for any algebraic variety X over an algebraically closed field k which computes what he calls the “algebraic homology” of X. If X is an arbitrary curve C, we show that this complex may be viewed as the points of a “homology motive” of C with values in k.

Values of L-Functions of Jacobi-Sum Hecke Characters of Abelian Fields

In this paper we compute the values of L-series of Jacobi-sum Hecke characters in terms of values of the Γ-function at rational numbers. The computation is done only up to algebraic numbers, and we assume that the Hecke character is in the “good range.” We may make a more refined Statement (the Γ-hypothesis), which actually predicts the values up to rational numbers, and which has been verified in the totally real case ([B]) and in the case of imaginary quadratie fields with odd class number ([L], [B]). Here we content ourselves with the algebraicity Statement, but prove it for all abelian fields.

A Nullstellensatz for higher derivations

Suppose dim R, = 1. (Since the image of Y in R, is a non-nilpotent element in the maximal ideal we have that dim RA > 1.) Let R be the integral closure of R in Q its quotient field. Then by ([2] Corollary 3, 4, p. 13) there is a maximal ideal A? C R with A! = R n 2. By ([6] Theorem 9, p. 267) i? is still finitely generated over k, and so L = R/J? is a finite algebraic extension of k (by the Hilbert Nullstellensatz) so that L is a finite extension of K. The local ring Rx is normal (integrally closed) by ([5] remark, p. 22), and is the integral closure of R, in Q. Since 1 = dim R, = dim Rx we have by ([4] Proposition 8, p. 111-12) that R, is a discrete valuation ring. Then the completion of R& is a discrete valuation ring and is isomorphic to L[[X]] by ([5] Theorem 2 and Remark just below, p. 43). If p is the natural injection of Rx into its completion L[[X]] an d u is the composite of the injective maps R -+ RA -+ Rx 5 L[[X]] then u has the desired properties. Now suppose dim R, = N > 1. Let R be the integral closure of R in Q its quotient field. Then by ([2] Corollary 3, 4 p. 13) there is a maximal ideal J? C R with A = R n J?. The local ring RJ is normal and is the integral closure of RM in Q (and has the same dimension as R&4(). We consider and if JV denotes the unique maximal ideal of R,- then r E Jfr. Since dim R& > 2 it follows from ([4] Theorem 11, ii, p. IV-44) and ([4] Proposition and Definition 6, p. IV-14) that maximal R-sequences in X have at least two elements. Complete