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Register a new userComparing Dualities in the $K(n)$-local Category
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In their work on the period map and the dualizing sheaf for Lubin-Tate space, Gross and the second author wrote down an equivalence between the Spanier-Whitehead and Brown-Comenetz duals of certain type $n$-complexes in the $K(n)$-local category at large primes. In the culture of the time, these results were accessible to educated readers, but this seems no longer to be the case; therefore, in this note we give the details. Because we are at large primes, the key result is algebraic: in the Picard group of Lubin-Tate space, two important invertible sheaves become isomorphic modulo $p$.
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Following a suggestion of Hovey and Strickland, we study the category of $K(k) vee K(k+1) vee cdots vee K(n)$-local spectra. When $k = 0$, this is equivalent to the category of $E(n)$-local spectra, while for $k = n$, this is the category of $K(n)$-local spectra, both of which have been studied in detail by Hovey and Strickland. Based on their ideas, we classify the localizing and colocalizing subcategories, and give characterizations of compact and dualizable objects. We construct an Adams type spectral sequence and show that when $p gg n$ it collapses with a horizontal vanishing line above filtration degree $n^2+n-k$ at the $E_2$-page for the sphere spectrum. We then study the Picard group of $K(k) vee K(k+1) vee cdots vee K(n)$-local spectra, showing that this group is algebraic, in a suitable sense, when $p gg n$. We also consider a version of Gross--Hopkins duality in this category. A key concept throughout is the use of descent.
Let n geq 1 and let p be any prime. Also, let E_n be the Lubin-Tate spectrum, G_n the extended Morava stabilizer group, and K(n) the nth Morava K-theory spectrum. Then work of Devinatz and Hopkins and some results due to Behrens and the first author of this note, show that if X is a finite spectrum, then the localization L_{K(n)}(X) is equivalent to the homotopy fixed point spectrum (L_{K(n)}(E_n wedge X))^{hG_n}, which is formed with respect to the continuous action of G_n on L_{K(n)}(E_n wedge X). In this note, we show that this equivalence holds for any (S-cofibrant) spectrum X. Also, we show that for all such X, the strongly convergent Adams-type spectral sequence abutting to pi_ast(L_{K(n)}(X)) is isomorphic to the descent spectral sequence that abuts to pi_ast((L_{K(n)}(E_n wedge X))^{hG_n}).
Let n be any positive integer and p any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment pi_*(L_{K(n)}(X)) and E_2-term equal to the continuous cohomology of G_n, the extended Morava stabilizer group, with coefficients in a certain discrete G_n-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local E_n-Adams spectral sequence for pi_*(L_{K(n)}(X)), whose E_2-term is not known to always be equal to a continuous cohomology group.
We discuss some aspects of the relation between dualities and gauge symmetries. Both of these ideas are of course multi-faceted, and we confine ourselves to making two points. Both points are about dualities in string theory, and both have the flavour that two dual theories are closer in content than you might think. For both points, we adopt a simple conception of a duality as an isomorphism between theories: more precisely, as appropriate bijections between the two theories sets of states and sets of quantities. The first point (Section 3) is that this conception of duality meshes with two dual theories being gauge related in the general philosophical sense of being physically equivalent. For a string duality, such as T-duality and gauge/gravity duality, this means taking such features as the radius of a compact dimension, and the dimensionality of spacetime, to be gauge. The second point (Sections 4, 5 and 6) is much more specific. We give a result about gauge/gravity duality that shows its relation to gauge symmetries (in the physical sense of symmetry transformations that are spacetime-dependent) to be subtler than you might expect. For gauge theories, you might expect that the duality bijections relate only gauge-invariant quantities and states, in the sense that gauge symmetries in one theory will be unrelated to any symmetries in the other theory. This may be so in general; and indeed, it is suggested by discussions of Polchinski and Horowitz. But we show that in gauge/gravity duality, each of a certain class of gauge symmetries in the gravity/bulk theory, viz. diffeomorphisms, is related by the duality to a position-dependent symmetry of the gauge/boundary theory.
In this note, we compute the image of the $alpha$-family in the homotopy of the $K(2)$-local sphere at the prime $p=2$ by locating its image in the algebraic duality spectral sequence. This is a stepping stone for the computation of the homotopy groups of the $K(2)$-local sphere at the prime $2$ using the duality spectral sequences.
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