algebraic geometry and homotopy theory
Over the past century, cohomology operations have played a crucial role in homotopy theory and its applications. A powerful framework for constructing such operations is the theory of commutative algebras in spectra. In this article, we discuss an algebro-geometric analogue of this framework, called the theory of normed algebras in motivic spectra. Specifically, we show that the motivic spectrum ko representing very effective hermitian K-theory can be equipped with a normed algebra structure, and that the orientation map from MSL to ko respects this structure. The main step will be showing that the motivic infinite loop space machine is compatible with norms.
Expository article submitted to the IAS/Park City Mathematics Series
Assume k is a field and R is a smooth k-algebra of dimension d. If P is a projective module of rank r, then it is well-known that P can be generated by r + d elements (Forster–Swan). Under suitable assumptions on r and d, we investigate obstructions to generation of P by fewer than r + d elements using motivic homotopy theory. For example, we observe that a quadratic enhancement of the classical Segre class obstructs generation by r + d – 1 elements, whether or not k is algebraically closed, generalizing old results of M.P. Murthy. Along the way, we also establish efficient generation results for symplectic modules.
In this article, we establish the compatibility between norms and transfers in motivic homotopy theory. More precisely, we construct norm functors for motivic spaces equipped with various flavours of transfer. This yields a norm monoidal refinement of the infinite P1-delooping machine of Elmanto-Hoyois-Khan-Sosnilo-Yakerson. We apply this refinement to construct a normed algebra structure on the very effective Hermitian K-theory spectrum.
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