Operadic Tensor Products and Smash Products

Let k be a commutative ring. E∞ k-algebras are associative and commutative k-algebras up to homotopy, as codified in the action of an E∞ operad; A∞ k-algebras are obtained by ignoring permutations. Using a particularly well-behaved E∞ algebra, we explain an associative and commutative operadic tensor product that effectively hides the operad: an A∞ algebra or E∞ algebra A is defined in terms of maps k −→ A and A A −→ A such that the obvious diagrams commute, and similarly for modules over A. This makes it little more difficult to study these algebraic objects than it is to study their classical counterparts. We also explain a topological analogue of the theory. This gives a symmetric monoidal category of modules over the sphere spectrum S whose derived category is equivalent to the classical stable homotopy category. The existence of this category allows the wholesale importation of algebraic techniques into stable homotopy theory. There will not be time to go into this, but the algebraic theory has applications to mixed Tate motives in algebraic geometry and the topological theory has applications to the construction and study of MU -module spectra, the construction of generalized Künneth and universal coefficient spectral sequences, a construction of the algebraic K-theory of S-algebras that includes Quillen’s algebraic K-theory of discrete rings and Waldhausen’s algebraic K-theory of spaces, a construction of the topological Hochschild homology of an S-algebra that generalizes Bökstedt’s THH, and a completion theorem for equivariant complex cobordism and any of its modules analogous to the Atiyah-Segal completion theorem in equivariant K-theory. 1. The category of C-modules and the product £ Let C be an operad in a cocomplete symmetric monoidal category S with product ⊗ and unit κ. We are thinking of the category of differential graded modules over a commutative ring k and will restrict to it shortly. In general, C = C (1) is a monoid in S . In our algebraic context, this means that C is a DGA. We call left C-objects C-modules in any case. In the algebraic situation, if C is unital and the augmentation 2 : C→ k is a quasi-isomorphism, then the derived categories Dk and DC are equivalent. Via instances of the structural maps γ, we have a left action of C and a right action of C⊗C on C (2), and these actions commute with each other. Thus we have a bimodule structure on C (2). Let M and N be left C-modules. Clearly M ⊗ N is a left C ⊗ C-module via the given actions. This makes sense of the following definition of the “operadic tensor product £”. 1991 Mathematics Subject Classification. Primary 18-02, 55-02; Secondary 18C99, 55P42. This talk is largely based on Part V of [7] and on [5], to which the interested reader is referred for details and more complete references. The author was supported in part by NSF Grant #DMS-9423300.