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We define Albanese and Picard 1-motives of smooth (simplicial) schemes over a perfect field. For smooth proper schemes, these are the classical Albanese and Picard varieties. For a curve, these are the homological 1-motive of Lichtenbaum and the motivic H 1 of Deligne. This paper proves a conjecture of Deligne about providing an algebraic description, via 1-motives, of the first homology and cohomology groups of a complex algebraic variety. (L. Barbieri-Viale and V. Srinivas have also proved this independently.) It also contains a purely algebraic proof of Lichtenbaum's conjecture that the Albanese and the Picard 1-motives of a (simplicial) scheme are dual. This gives a new proof of an unpublished theorem of Lichtenbaum that Deligne's 1-motive of a curve is dual to Lichtenbaum's 1-motive.1 In [30], the conjecture of Deligne is proved up to isogeny for H n (V ); here, our theorem proves the integral version of the conjecture for H 1 .

We shall not cease from explorationAnd the end of all our exploring Will be to arrive where we started And know the place for the first time.-T. S. Eliot, Four Quartets Zeta functions play a primordial role in arithmetic geometry. The aim of this paper is to provide some motivation to view zeta functions of varieties over finite fields as elements of the (big) Witt ring W (Z) of Z. Our main inspirations are• Steve Lichtenbaum's philosophy [38,37,39] that special values of arithmetic zeta functions and motivic L-functions are given by suitable Euler characteristics.• Kazuya Kato's idea of zeta elements; Kato-Saito-Kurokawa [33] titled a chapter "ζ". They say "We dropped the word "functions" because we feel more and more as we study ζ functions that ζ functions are more than just functions.". • The suggestion of Minhyong KimIn brief, the current view is that the Iwasawa polynomial=p-adic L-function should be viewed as a path in K-theory space; see MO.37374. and Steve Mitchell [43] It is tempting to think of KR as a sort of homotopical L-function, with L K(1) KR as its analytic continuation and with functional equation given by some kind of Artin-Verdier-Brown-Comenetz duality. (Although in terms of the generalized Lichtenbaum conjecture on values of ℓ-adic L-functions at integer points, the values at negative integers are related to positive homotopy groups of L K(1) KR, while the values at positive integers are related to the negative homotopy groups! that the algebraic K-theory spectrum itself should be considered as a zeta function.• M. Kapranov's [31] motivic zeta function with coefficients in the Grothendieck ring of varieties and the related notion of motivic measures. One basic reason for an Euler-characteristic description of the special values of zeta functions is that the zeta function itself is an Euler characteristic.There is almost nothing original in this paper. Much of this is surely known to the experts. However, except for a passing remark in [34,13,36] mentioning (i) of Theorem 2.1, the close relations between zeta and the Witt ring do not seem to be documented in the literature 1 ; this provides our excuse to write this paper. Still missing is a formulation of the functional equation for the zeta function in terms of the Witt ring. We shall explore the connections with homotopy in future work.

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