We establish fundamental motivic results about hermitian K-theory without assuming that 2 is invertible on the base scheme. In particular, we prove that both quadratic and symmetric Grothendieck-Witt theory satisfy Nisnevich descent, and that symmetric Grothendieck-Witt theory further satisfies a projective bundle formula, as well as dévissage and A^1-invariance over a regular Noetherian base of finite Krull dimension. We use this to show that over a regular Noetherian base, symmetric Grothendieck-Witt theory is represented by a motivic E-infinity-ring spectrum, which we then show is an absolutely pure spectrum, answering a question of Déglise. As with algebraic K-theory, we show that over a general base, one can also construct a hermitian K-theory motivic spectrum, representing this time a suitable homotopy invariant and Karoubi-localising version of Grothendieck-Witt theory.
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) -> BO^{p,q}( X_{+} / U_{+} ) and Schlichting's hermitian K-theory functor (X,U) -> KO^{[q]}_{2q-p}(X,U) are canonically isomorphic. We use the motivic weak equivalence Z x 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 Spec Z[1/2], this monoid structure and the induced ring structure on …
Very effective algebraic and hermitian K-theory of the cyclic group of order two
University of Kentucky（肯塔基大学）；James Madison University（詹姆斯麦迪逊大学）；Duke University（杜克大学）；University of Virginia（弗吉尼亚大学）；University of Pennsylvania（宾夕法尼亚大学）
