Ample line bundles and generation time

Proposition 1.

Let 𝒜 be an abelian category with enough injectives. Then for each K , L ∈ D b ⁢ ( 𝒜 ) the group Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) carries a functorial decreasing filtration F which satisfies for K , L , M ∈ D b ⁢ ( 𝒜 ) :

1. F 0 ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) = Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) and F p ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) = 0 for p ≫ 0 .

2. If f ∈ F p ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) and g ∈ F q ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( L , M ) , then

   g ∘ f ∈ F p + q ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , M ) .

3. F p ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) / F p + 1 ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) is a subquotient of

   ∏ n ∈ ℤ Ext 𝒜 p ⁢ ( H n ⁢ ( K ) , H n - p ⁢ ( L ) ) .

4. F 1 ⁢ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) = { φ ∈ Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) : H n ⁢ ( φ ) = 0 ⁢  for all  ⁢ n ∈ ℤ } .

Proof.

The filtration is the one induced by the spectral sequence

of [1, Equation 3.1.3.4] obtained by endowing K and L with their canonical filtrations. Part (3) follows immediately from this description (note that p + q = 0 implies 2 ⁢ p + q = p ). Since negative Ext groups vanish in 𝒜 , (3) implies the inclusions ⋯ ⊃ F - 2 ⊃ F - 1 ⊃ F 0 are equalities. Convergence of the spectral sequence gives F p = Hom D ⁢ ( 𝒜 ) ⁢ ( K , L ) for p ≪ 0 (hence for all p ≤ 0 by the previous sentence) and F p = 0 for p ≫ 0 , proving (1). Next, note that all differentials going into an E r 0 , 0 are zero for degree reasons, and hence E ∞ 0 , 0 ⊂ E 1 0 , 0 . But the composition F 0 → F 0 / F 1 = E ∞ 0 , 0 → E 1 0 , 0 is a map

which one can show using the definition of the spectral sequence is just the map taking a morphism to its associated morphisms on cohomology. Hence (4) follows. Finally, (2) holds because the spectral sequence is compatible with composition in the usual sense of a spectral sequence with products. We refer to [8, Appendix A] for all omitted details. ∎

Theorem 2.

Let X be a Noetherian regular scheme of dimension d < ∞ . Let

be morphisms in D coh b ⁢ ( X ) whose induced morphisms on cohomology sheaves vanish. Then the composition K 0 → K d + 1 is zero.

Proof.

Note that D coh b ⁢ ( X ) is a full subcategory of the bounded derived category of the category of 𝒪 X -modules, which is an abelian category with enough injectives. Thus we may use the filtration

of Proposition 1. By [14, Tag 0FZ3] we have Ext i ⁢ ( ℱ , 𝒢 ) = 0 for i > d and ℱ , 𝒢 coherent sheaves on X. Therefore by (3) of Proposition 1, F d + 1 = F d + 2 = ⋯ . Since F p = 0 for p ≫ 0 , in fact F d + 1 = 0 . Then by (4) each K i → K i + 1 is in F 1 ⁢ Hom ⁢ ( K i , K i + 1 ) , so that by (2) the composition K 0 → K d + 1 is in F d + 1 ⁢ Hom ⁢ ( K 0 , K d + 1 ) = 0 and we are done. ∎

Lemma 3.

Let X be a Noetherian scheme with an ample invertible sheaf ℒ and let K ∈ D coh b ⁢ ( X ) . Then there exist a finite set I and a morphism ⊕ i ∈ I ℒ ⊗ m i ⁢ [ n i ] → K with n i , m i ∈ ℤ , which is surjective on cohomology sheaves.

Proof.

Represent K by a bounded complex of coherent sheaves

For n sufficiently negative there is a surjection ⊕ i ℒ ⊗ n → Ker ⁢ ( d k ) with the sum finite. This gives rise to a morphism ⊕ i ℒ ⊗ n ⁢ [ - k ] → K which is surjective on H k . Putting together these morphisms for every k proves the result. ∎

Theorem 4.

Let X be a Noetherian regular scheme of dimension d < ∞ . Assume X has an ample invertible sheaf ℒ . Then D coh b ⁢ ( X ) = 〈 { ℒ ⊗ n } n ∈ ℤ 〉 d + 1 .

Proof.

Let K = K 0 ∈ D coh b ⁢ ( X ) . Choose a finite set I and a morphism

as in Lemma 3 and let K 1 be the cone. Note that K 0 → K 1 is zero on cohomology sheaves by construction, and its cone is in 〈 { ℒ ⊗ n } n ∈ ℤ 〉 1 . Now repeat the process with K = K 1 and so forth to obtain a sequence

such that each K i → K i + 1 is zero on cohomology sheaves and has cone in 〈 { ℒ ⊗ n } n ∈ ℤ 〉 1 . Thus K 0 → K d + 1 is zero by Theorem 2. We will prove by induction that the cone of K 0 → K i is in 〈 { ℒ ⊗ n } n ∈ ℤ 〉 i . For i = 1 this is known and for i = d + 1 this proves the theorem: Since K = K 0 → K d + 1 is zero, it follows that the cone is isomorphic to K d + 1 ⊕ K ⁢ [ 1 ] and the category 〈 { ℒ ⊗ n } n ∈ ℤ 〉 d + 1 is closed under direct summands and shifts.

So assume known that the cone of K 0 → K i is in 〈 { ℒ ⊗ n } n ∈ ℤ 〉 i . Then by the octahedral axiom there is a distinguished triangle

with C a cone of K 0 → K i , D a cone of K 0 → K i + 1 , and E a cone of K i → K i + 1 . Since C ∈ 〈 { ℒ ⊗ n } n ∈ ℤ 〉 i and E ∈ 〈 { ℒ ⊗ n } n ∈ ℤ 〉 1 , it follows that D ∈ 〈 { ℒ ⊗ n } n ∈ ℤ 〉 i + 1 , as needed. ∎

Corollary 5.

Let X be a Noetherian regular scheme of dimension d < ∞ . If X is quasi-affine, then D coh b ⁢ ( X ) = 〈 𝒪 X 〉 d + 1 and hence Rdim ⁢ ( D coh b ⁢ ( X ) ) ≤ d . If X is also of finite type over a field, then Rdim ⁢ ( D coh b ⁢ ( X ) ) = d .

Proof.

The structure sheaf is ample on a quasi-affine scheme so the first part follows from Theorem 4. The reverse inequality when X is of finite type over a field is [13, Proposition 7.16]. ∎

Definition 6.

Let 𝒯 be a triangulated category. The countable Rouquier dimension of 𝒯 , denoted CRdim ⁢ ( 𝒯 ) , is the smallest n such that there exists a countable set { E i } i ∈ I of objects of 𝒯 such that 𝒯 = 〈 { E i } i ∈ I 〉 n + 1 , or infinity if no such n exists.

Lemma 7.

Let F : 𝒯 → 𝒯 ′ be an exact functor between triangulated categories which is essentially surjective. Then CRdim ⁢ ( 𝒯 ′ ) ≤ CRdim ⁢ ( 𝒯 ) .

Proof.

If 𝒯 = 〈 { E i } i ∈ I 〉 n + 1 , then 𝒯 ′ = 〈 { F ⁢ ( E i ) } i ∈ I 〉 n + 1 . ∎

Example 8.

Let X be a Noetherian regular scheme with an ample line bundle. Then by Theorem 4, CRdim ⁢ ( D coh b ⁢ ( X ) ) ≤ dim ⁢ ( X ) .

Proposition 9.

Let k be an uncountable field. Let X be a reduced scheme of finite type over k. Then CRdim ⁢ ( D coh b ⁢ ( X ) ) ≥ dim ⁢ ( X ) .

Proof.

Compare to the proof of [13, Proposition 7.16]. Let n = CRdim ⁢ ( D coh b ⁢ ( X ) ) and let { E i } i ∈ I be a countable family of objects such that D coh b ⁢ ( X ) = 〈 { E i } i ∈ I 〉 n + 1 . Consider the set of closed points x ∈ X such that for every i ∈ I , the cohomology modules of ( E i ) x are free 𝒪 X , x -modules. Since X is not a countable union of closed subsets with dense complement (see [6, Exercise 2.5.10]), the set contains a closed point x such that dim ⁢ ( 𝒪 X , x ) = dim ⁢ ( X ) . We have ( E i ) x ∈ 〈 𝒪 X , x 〉 1 for each i since a complex with projective cohomology modules is decomposable, hence

hence n ≥ dim ⁢ ( X ) by [13, Proposition 7.14]. ∎

Corollary 10.

Let X be a regular, quasi-projective scheme over an uncountable field. Then CRdim ⁢ ( D coh b ⁢ ( X ) ) = dim ⁢ ( X ) .

Proof.

Combine Example 8 with Proposition 9. ∎

Remark 11.

Since the countable Rouquier dimension only gives the expected answer for varieties over a sufficiently large field, Orlov suggests an alternative notion which makes sense for a k-linear, pre-triangulated dg-category 𝒜 with k a field: Take the smallest n such that there exists a countable family { E i } i of objects of 𝒜 such that 𝒜 K = 〈 { ( E i ) K } i 〉 n + 1 for every field extension K / k .

Theorem 12.

Let k be a field. Let X , Y be smooth projective varieties over k. Assume there exists a fully faithful, exact, k-linear functor F : D coh b ⁢ ( X ) → D coh b ⁢ ( Y ) . Then

Proof.

Let us choose any uncountable extension field K / k . By [11, Theorem 2.2] and [2, Theorem 1.1], F is the Fourier–Mukai transform with respect to a kernel E ∈ D coh b ⁢ ( X × k Y ) . Then E K gives rise to a functor F K : D coh b ⁢ ( X K ) → D coh b ⁢ ( Y K ) which remains fully faithful by the calculus of kernels, see [12, Lemma 2.12].

We have CRdim ⁢ ( D coh b ⁢ ( X K ) ) = dim ⁢ ( X K ) = dim ⁢ ( X ) by Corollary 10 and similarly for Y . Thus by Lemma 7 applied to the right adjoint of F K (which exists by [2, Theorem 1.1] and is essentially surjective since F K is fully faithful), we have

as needed. ∎

Example 13.

The object ⊕ i = 0 N 𝒪 𝐏 N ⁢ ( - i ) is a strong generator of D coh b ⁢ ( 𝐏 ℤ N ) . This can be seen for example from the existence of a semiorthogonal decomposition

(see [10, Theorem 2.6]) and the fact that ℤ is a strong generator of D coh b ⁢ ( ℤ ) .

Theorem 14 ([7, Theorem 0.5]).

Let X be a Noetherian regular scheme of finite Krull dimension which possesses an ample line bundle. Then D coh b ⁢ ( X ) has a strong generator.

Proof.

Let ℒ be an ample line bundle on X. After possibly replacing ℒ with a positive tensor power, we may assume ℒ is generated by finitely many global sections so that ℒ = f * ⁢ 𝒪 𝐏 N ⁢ ( 1 ) for some morphism f : X → 𝐏 ℤ N . We claim there is an integer M such that

This suffices since together with Theorem 4 it implies that the object G = ⊕ i = 0 N ℒ ⊗ - i is a strong generator of D coh b ⁢ ( X ) . To prove the claim, note that since L ⁢ f * ⁢ 𝒪 𝐏 N ⁢ ( n ) = ℒ ⊗ n and the subcategories 〈 - 〉 i are preserved by exact functors, it suffices to prove the claim with X replaced by 𝐏 ℤ N and ℒ replaced by 𝒪 𝐏 N ⁢ ( 1 ) . But this follows from Example 13. ∎