Pushforwards of tilting sheaves

Abstract

We investigate the behaviour of tilting sheaves under pushforward by a finite Galois morphism. We determine conditions under which such a pushforward of a tilting sheaf is a tilting sheaf. We then produce some examples of Severi-Brauer flag varieties and arithmetic toric varieties in which our method produces a tilting sheaf, adding to the list of positive results in the literature. We also produce some counterexamples to show that such a pushfoward need not be a tilting sheaf. Tilting sheaves and base change2.1. Generation in derived categories. Let D be a triangulated category and S a set of objects in D. We denote by < S > the smallest full triangulated category containing all the objects in S. We denote by < S > κ the smallest thick triangulated containing all the objects in S. Note that thick subcategories are assumed to be full.An object C of D is said to be compact if Hom(C, −) commutes with direct sums. We denote by D c the full subcategory of compact objects.Given a set S of objects of D we define S ⊥ to be the full subcategory of D consisting of objects A with Hom D (E[i], A) = 0 for all E ∈ S and i ∈ Z. We say that S right spans D if S ⊥ = {0}.If D c right spans D we say that D is compactly generated. Theorem 2.1. (Ravenel and Neeman) Let D be a compactly generated triangulated category. Then a set of compact objects S right spans D if and only if < S > κ = D c . Proof. See [BV, Theorem 2.1.2].Let Y be a scheme. We denote the unbounded derived category of quasi-coherent sheaves on Y by D(Qcoh(Y )) and the bounded derived category of coherent sheaves by D b (Y ).Proposition 2.2. Let Y be a quasi-compact, separated scheme. Then D(Qcoh(Y )) is compactly generated.