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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 isequivalent to the category of $E(n)$-local spectra, while for $k = n$, this isthe category of $K(n)$-local spectra, both of which have been studied in detailby Hovey and Strickland. Based on their ideas, we classify the localizing andcolocalizing subcategories, and give characterizations of compact anddualizable objects. We construct an Adams type spectral sequence and show thatwhen $p \gg n$ it collapses with a horizontal vanishing line above filtrationdegree $n^2+n-k$ at the $E_2$-page for the sphere spectrum. We then study thePicard group of $K(k) \vee K(k+1) \vee \cdots \vee K(n)$-local spectra, showingthat this group is algebraic, in a suitable sense, when $p \gg n$. We alsoconsider a version of Gross--Hopkins duality in this category. A key conceptthroughout is the use of descent.

The $\mathop{Sp}_{k,n}$-local stable homotopy category