SHUFFLE PRODUCTS IN HIGHER K-THEORY

[摘要] We construct shuffle products in higher K-theory. The fundamental observation for this is that the following assignments can be melted into one another in an entirely natural fashion. On the one hand to any locally free modules V1,..., V(k) we assign the module +sigma (Xr=1(p) V(sigma)(r)) x (Xr=p+1k V(sigma(r)) and on the other hand to any chain V1 hooked arrow pointing right ... hooked arrow pointing right V(k) = V of admissible monomorphisms we assign the submodule SIGMA(sigma) (LAMBDA(r=1)p V(sigma(r)) x LAMBDA(r=p+1)k V(sigma)(r))) of LAMBDA(p)V x LAMBDA(k-p)V. In both cases the (direct) sum is taken over all (p, k - p)-shuffles sigma. By means of these shuffle products we show that the exterior power operations in higher K-theory defined by D. Grayson are compatible (already on the simplicial level) with the direct sum and with the symmetric power operations in the expected way. Furthermore, we investigate the connection between the shuffle products and the usual products in higher K-theory.

[发布日期] 1994-03-25