Smash products, inner actions and quotient rings

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...

Generalized Quotient Rings

Inner products and Z/p-actions on Poincaré duality spaces

Let Z=p act on an Fp-Poincaré duality space X, where p is an odd prime number. We derive a formula that expresses the Fp-Witt class of the fixed point set X Z=p in terms of the Fp1⁄2Z=p -algebra H ðX ;FpÞ, if H ðX ;Zð pÞÞ does not contain Z=p as a direct summand. This extends previous work of Alexander and Hamrick, where the orientation class of X is supposed to be liftable to an integral class...

Polynomial Identities in Smash Products

Suppose that a group G acts by automorphisms on a (restricted) Lie algebra L over a field K of positive characteristic. This gives rise to smash products U(L)#K[G] and u(L)#K[G] . We find necessary and sufficient conditions for these smash products to satisfy a nontrivial polynomial identity.

Commuting Homotopy Limits and Smash Products

In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization that involves spectra and smash products over the orbit category of a discrete group. Such a situation naturall...