Search

We construct an algebraic commutative ring T -spectrum BO which is stably fibrant and (8, 4)-periodic and such that on SmOp/S the cohomology theory (X, U) 7→ BO(X+/U+) and Schlichting’s hermitian K-theory functor (X, U) 7→ KO [q] 2q−p(X, U) are canonically isomorphic. We use the motivic weak equivalence Z×HGr ∼ −→ KSp relating the infinite quaternionic Grassmannian to symplectic K-theory to equip BO with the structure of a commutative monoid in the motivic stable homotopy category. When the base scheme is SpecZ[ 1 2 ], this monoid structure and the induced ring structure on the cohomology theory BO are the unique structures compatible with the products KO [2m] 0 (X)× KO [2n] 0 (Y ) → KO [2m+2n] 0 (X × Y ). on Grothendieck-Witt groups induced by the tensor product of symmetric chain complexes. The cohomology theory is bigraded commutative with the switch map acting on BO(T∧T ) in the same way as multiplication by the Grothendieck-Witt class of the symmetric bilinear space 〈−1〉.

A general duality theorem for the category of motives is established, with a short, simple, and self-contained proof. Introduction Recently, due to the active study of cohomological invariants in algebraic geometry, “transplantation” of classical topological constructions to the algebraic-geometrical “soil” seems to be rather important. In particular, it is very interesting to study topological properties of the category of motives. The concept of a motive was introduced by Alexander Grothendieck in 1964 in order to formalize the notion of universal (co-)homology theory (see the detailed exposition of Grothendieck’s ideas in [5]). For us, the principal example of this type is the category of motives DM−, constructed by Voevodsky [12] for algebraic varieties. The Poincare duality is a classical and fundamental result in algebraic topology that initially appeared in Poincare’s first topological memoir “Analysis Situs” [9] (as a part of the Betti numbers symmetry theorem proof). The proof of the general duality theorem for extraordinary cohomology theories apparently belongs to Adams [1]. Our purpose in this paper is to establish a general duality theorem for the category of motives. Essentially, we extend the main statement of [8] to this category. Many known results can easily be interpreted in these terms. In particular, we get a generalization of the Friedlander–Voevodsky duality theorem [4] to the case of the ground field of arbitrary characteristic. The proof of this fact, involving the main result of [8], was kindly conveyed to the authors by Andrěi Suslin in a private communication. Being inspired by his work and Dold–Puppe’s category approach [2] to the duality phenomenon in topology, we decided to present a short, simple, and self-contained proof of a similar result for the category of motives. Our result might be viewed as a purely abstract theorem and rewritten in the spirit of “abstract nonsense” as a statement about some category with a distinguished class of morphisms. Essentially, what is required for the proof is the existence of finite fiber products and the terminal object in the category of varieties, a small part of the tensor triangulated category structure for motives, and finally, the existence of transfers for the class of morphisms generated by graphs of a special type (of projective morphisms). However, rather, we preferred to formulate all statements for motives of algebraic varieties in order to clarify the geometric nature of the construction and make possible applications easier. This led, in particular, to the appearence of the second (co)homology index responsible for twist with the Tate object Z(1) (see Voevodsky [12]). The only exception is the classical Example 2. 2000 Mathematics Subject Classification. Primary 14F42.