algebraic geometry and homotopy theory
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.
submitted
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.