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We develop the theory of equivariant sheaves over profinite spaces, where the group is also taken to be profinite. We construct a good notion of equivariant presheaves, with a suitable sheafification functor. Using these results on equivariant presheaves, we give explicit constructions of products of equivariant sheaves of R-modules. We introduce an equivariant analogue of skyscraper sheaves, which allows us to show that the category of equivariant sheaves of R-modules over a profinite space has enough injectives. This paper also provides the basic theory for results by the authors on giving an algebraic model for rational G-spectra in terms of equivariant sheaves over profinite spaces. For those results, we need a notion of Weyl-G-sheaves over the space of closed subgroups of G. We show that Weyl-G-sheaves of R-modules form an abelian category, with enough injectives, that is a full subcategory of equivariant sheaves of R-modules. Moreover, we show that the inclusion functor has a right adjoint.
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The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This is not a conceptual problem, but in order to be clear on this point and to write down a self-contained narrative, I have included a great deal of discussion of the geometry of the stack itself, giving various equivalent descriptions.
If K is a discrete group and Z is a K-spectrum, then the homotopy fixed point spectrum Z^{hK} is Map_*(EK_+, Z)^K, the fixed points of a familiar expression. Similarly, if G is a profinite group and X is a discrete G-spectrum, then X^{hG} is often given by (H_{G,X})^G, where H_{G,X} is a certain explicit construction given by a homotopy limit in the category of discrete G-spectra. Thus, in each of two common equivariant settings, the homotopy fixed point spectrum is equal to the fixed points of an explicit object in the ambient equivariant category. We enrich this pattern by proving in a precise sense that the discrete G-spectrum H_{G,X} is just a profinite version of Map_*(EK_+, Z): at each stage of its construction, H_{G,X} replicates in the setting of discrete G-spectra the corresponding stage in the formation of Map_*(EK_+, Z) (up to a certain natural identification).
For G a profinite group, we construct an equivalence between rational G-Mackey functors and a certain full subcategory of $G$-sheaves over the space of closed subgroups of G called Weyl-G-sheaves. This subcategory consists of those sheaves whose stalk over a subgroup K is K-fixed. This extends the classification of rational G-Mackey functors for finite G of Th{e}venaz and Webb, and Greenlees and May to a new class of examples. Moreover, this equivalence is instrumental in the classification of rational G-spectra for profinite G, as given in the second authors thesis.
We determine systematic regions in which the bigraded homotopy sheaves of the motivic sphere spectrum vanish.
Let $G=Sp_{2n}(mathbb{C})$, and $mathfrak{N}$ be Katos exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $Lambda^+$, the dominant weights for a simple algebraic group $H$, and $textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $widetilde{mathfrak{N}}$ have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(Coh^G(mathfrak{N}))$, and deduce that the resulting $t$-structure on $D^b(Coh^G(mathfrak{N}))$ coincides with the perverse coherent $t$-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of this $t$-structure on $D^b(Coh^G(mathfrak{N}))$.
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