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Let A be an A1 ring spectrum. We use the description from our preprint [1] of the cyclic bar and cobar construction to give a direct definition of topological Hochschild homology and cohomology of A… (More)

The topological Hochschild homology THH(R) of a commutative Salgebra (E∞ ring spectrum) R naturally has the structure of a Hopf algebra over R, in the homotopy category. We show that under a flatness… (More)

We consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, K.kŒx1;:::;xn!=.x a1 1 ;:::;x an n //. This naturally leads to a new generalization of the big… (More)

We show that there is an essentially unique S-algebra structure on the Morava K-theory spectrum K(n), while K(n) has uncountably many M U or E(n)-algebra structures. Here E(n) is the K(n)-localized… (More)

We describe a construction of the cyclotomic structure on topological Hochschild homology (THH) of a ring spectrum using the Hill–Hopkins– Ravenel multiplicative norm. Our analysis takes place… (More)

Fix a prime number p and a non-negative integer n. We prove that if a p-complete spectrum X satisfying a mild finiteness condition has the same mod p cohomology as BP<n>as a module over the Steenrod… (More)

We give an algorithm for calculating the RO(S^1)-graded TR-groups of F_p, completing the calculation started by the second author. We also calculate the RO(S^1)-graded TR-groups of Z with mod p… (More)

We discuss a multiplicative version of the Verschiebung map of Witt vectors that we call the norm.

We show that K2 i (Z[x]/(x ), (x)) is finite of order (mi)!(i!) −2 and that K2 i+1 (Z[x]/(x ), (x)) is free abelian of rank m − 1. This is accomplished by showing that the equivariant homotopy groups… (More)

The construction of various bordism theories as Thom spectra served as a motivating example for the development of highly structured ring spectra. Various other examples of Thom spectra followed; for… (More)