Cohomological Bousfield classes

Cohomological Bousfield Classes

In this paper, we begin the study of Bousfield classes for cohomology theories defined on spectra. Our main result is that a map f : X → Y induces an isomorphism on E(n)-cohomology if and only if it induces an isomorphism on E(n)-homology. We also prove this for variants of E(n) such as elliptic cohomology and real K-theory. We also show that there is a nontrivial map from a spectrum Z to the K...

Tate Cohomology Lowers Chromatic Bousfield Classes

Let G be a finite group. We use the results of [5] to show that the Tate homology of E(n) local spectra with respect to G produces E(n− 1) local spectra. We also show that the Bousfield class of the Tate homology of LnX (for X finite) is the same as that of Ln−1X. To be precise, recall that Tate homology is a functor from G-spectra to G-spectra. To produce a functor PG from spectra to spectra, ...

Juvitop on Bousfield Localization

Homotopy theory studies topological spaces up to homotopy, so it must study the functor π∗ : Top → Π−algebras. Passage to the homotopy category is declaring π∗-isomorphisms to be isomorphisms. Because this functor is difficult to compute, one way to do homotopy theory is to study simpler functors which also map to graded abelian categories, e.g. πQ ∗ , H∗(−;Z), K∗, or more generally E∗ for E so...

Bousfield Localisations along Quillen Bifunctors

Bousfield Localization on Formal Schemes

Let (X,OX) be a noetherian separated formal scheme and consider D(Aqct(X)), its associated derived category of quasi-coherent torsion sheaves. We show that there is a bijection between the set of rigid localizations in D(Aqct(X)) and subsets in X. For a stable for specialization subset Z ⊂ X, the associated acyclization is RΓZ . If Z ⊂ X is generically stable, we provide a description of the as...