Rouquiers cocovering theorem and well-generated triangulated categories

Let T be a triangulated category with coproducts, and recall that an object Y of T is compact if the functor T (Y, -) commutes with coproducts. When T = D(X) is the unbounded derived category of quasi-coherent sheaves on a reasonable scheme X, the condition of compactness in T is local: given an open cover {U 1 , . . . , U n } of X, an object F is compact in D(X) if and only if F | U i is compact in D(U i ) for 1 i n. For arbitrary T , Rouquier introduces in [Rou08,§5] a suitable generalisation: he defines a cocovering of T to be a special family of Bousfield subcategories F = {I 1 , . . . , I n } (the precise definition is recalled below). The analogue of restriction to U i is then passage to the quotient T -→ T /I i , and under some natural hypotheses on F, compactness in T is local: an object Y is compact in T if and only if the image of Y is compact in T /I i for 1 i n.

Finally, a cocovering of T is a finite family of Bousfield subcategories F = {I 1 , . . . , I n } of T which are pairwise properly intersecting, such that n i=1 I i = 0; see [Rou08,(5.3.3)]. The α = ℵ 0 case of the following theorem is the aforementioned result of Rouquier,namely [Rou08,Theorem 5.15].

Theorem 1. Let T be a triangulated category with coproducts and α a regular cardinal. Suppose that F = {I 1 , . . . , I n } is a cocovering of T with the following properties:

(1) T /I is α-compactly generated for every I ∈ F.

(2) For every I ∈ F and nonempty subset F ′ ⊆ F \ {I} the essential image of the composite

Then T is α-compactly generated, and an object X ∈ T is α-compact if and only if the image of X is α-compact in T /I for every I ∈ F. Let S be a Bousfield subcategory of T intersecting properly with each I ∈ F, such that:

(3) S/(S ∩ I) is α-compactly generated in T /I for every I ∈ F.

(4) For every I ∈ F and nonempty subset F ′ ⊆ F \ {I} the essential image of the composite

Then S is α-compactly generated in T .

To return to the geometric example: if T = D(X) and we are given an open cover as above, then for each 1 i n denote by I i = D X\U i (X) the full subcategory of D(X) consisting of complexes with cohomology supported on X \ U i . There is a canonical equivalence T /I i ∼ = D(U i ), the quotient functor T -→ T /I i corresponds to restriction, and the family F = {I 1 , . . . , I n } is a cocovering of D(X) satisfying the hypotheses (1), (2) of the theorem for α = ℵ 0 [Rou08,§6.2]. For this choice of T and F the hypotheses are very natural, and easily verified; for the full elaboration, see Section 5.

Applying the theorem (recall that, since α = ℵ 0 , this is just Rouquier’s [Rou08,Theorem 5.15]) one obtains a proof of the fact, due originally to Neeman [Nee96], that the compact objects in D(X) are precisely the perfect complexes; see [Rou08,Theorem 6.8]. Using the α > ℵ 0 case of the theorem we obtain in Section 5 a description of the α-compact objects in D(X).

We have another application in mind, which will appear in the forthcoming [Mur08]. Let A be an associative ring with identity, K(Proj A) and K(Flat A) the homotopy categories of projective and flat left A-modules, respectively. A complex of left A-modules F is pure acyclic if it is acyclic, and N ⊗ A F is acyclic for every right A-module N . Let K pac (Flat A) denote the full subcategory of pure acyclic complexes in K(Flat A). This is a triangulated subcategory, and Neeman proves in [Nee08] that the composite

is an equivalence. Now let X be a quasi-compact semi-separated scheme. Unless X is affine, projective quasi-coherent sheaves on X are rare, and the homotopy category of projective quasi-coherent sheaves on X is often the zero category. In this case, the equivalence (1) suggests a suitable replacement. Let K(Flat X) be the homotopy category of flat quasi-coherent sheaves on X, and denote by K pac (Flat X) the full subcategory of acyclic complexes F with the property that F ⊗ O X A is acyclic for every quasi-coherent sheaf A . Define

and let {U 1 , . . . , U n } be an affine open cover of X, with say U i ∼ = Spec(A i ) for 1 i n. We show in [Mur08] that there is a cocovering of N(Flat X) by Bousfield subcategories {N X\U i (Flat X)} 1 i n , where N X\U i (Flat X) is the kernel of a natural restriction functor N(Flat X) -→ N(Flat U i ). Moreover, there are canonical equivalences

Neeman proves in loc.cit. that K(Proj A i ) is ℵ 1 -compactly generated, and even compactly generated when A i is coherent. In [Mur08] we combine Neeman’s results with Theorem 1 to see that the global category N(Flat X) is ℵ 1 -compactly generated, and compactly generated when X is noetherian. The proof of Theorem 1 is by induction on the size n = |F| of the cocovering. The real content is in the initial step of the induction, which we separate into its own section. The proof of the theorem is completed in Section 3. Our basic reference for triangulated categories is [Nee01], whose notation we follow with one exception: given a class C of objects in T , we write

for the ortho

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