Topological cyclic homology is a refinement of Connes–Tsygan’s cyclic homology which was introduced by Bökstedt–Hsiang–Madsen in 1993 as an approximation to algebraic K-theory. There is a trace map from algebraic K-theory to topological cyclic homology, and a theorem of Dundas–Goodwillie–McCarthy asserts that this induces an equivalence of relative theories for nilpotent immersions, which gives a way for computing K-theory in various situations. The construction of topological cyclic homology is based on genuine equivariant homotopy theory, the use of explicit point-set models, and the elaborate notion of a cyclotomic spectrum.
The goal of this paper is to revisit this theory using only homotopy-invariant notions. In particular, we give a new construction of topological cyclic homology. This is based on a new definition of the ∞-category of cyclotomic spectra: We define a cyclotomic spectrum to be a spectrum X with S1-action (in the most naive sense) together with S1-equivariant maps φp:X→XtCp for all primes p. Here, XtCp=cofib(Nm:XhCp→XhCp) is the Tate construction. On bounded below spectra, we prove that this agrees with previous definitions. As a consequence, we obtain a new and simple formula for topological cyclic homology.
In order to construct the maps φp:X→XtCp in the example of topological Hochschild homology, we introduce and study Tate-diagonals for spectra and Frobenius homomorphisms of commutative ring spectra. In particular, we prove a version of the Segal conjecture for the Tate-diagonals and relate these Frobenius homomorphisms to power operations.
We develop a local cohomology theory for FI^m-modules, and show that it in many ways mimics the classical theory for multi-graded modules over a polynomial ring. In particular, we define an invariant of FI^m-modules using this local cohomology theory which closely resembles an invariant of multi-graded modules over Cox rings defined by Maclagan and Smith. It is then shown that this invariant behaves almost identically to the invariant of Maclagan and Smith.
We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.
In this paper we study representation theory of the category FI^m introduced by Gadish which is a product of copies of the category FI, and show that quite a few interesting representational and homological properties of FI can be generalized to FI^m in a natural way. In particular, we prove the representation stability property of finitely generated FI^m -modules over fields of characteristic 0.