DERIVED CATEGORIES: LECTURE 2
EVGENY SHINDER
References
[Bo] Alexey Bondal, Representations of associative algebras and coherent sheaves, Izv. Akad. Nauk SSSR
Ser. Mat., 53:1 (1989),
[BK] Alexey Bondal, Mikhail Kapranov, Representable functors, Serre functors, and mutations, Izv.
Akad. Nauk SSSR Ser. Mat., 53:6 (1989),
[Har] R.Hartshorne, Algebraic Geometry
Proposition 0.1. Let E1 , . . . Er be an exceptional collection of sheaves on X. Assume
that there exists a resolution of the diagonal on X × X of the form
0 → p∗1 E1 ⊗ p∗2 F1 → · · · → p∗1 Er ⊗ p∗2 Fr → O∆ → 0
where Fi , i = 1, . . . , r is also a sequence of sheaves on X (not necessarily exceptional).
Then the collection Ei is full.
Lemma 0.2. We have the following resolution:
0 → Ωn (n) O(−n) → · · · → Ω1 (1) O(−1) → OPn ×Pn → O∆ → 0
Proof. It suffices to find a regular section s ∈ Γ(Pn × Pn , p∗1 (T (−1)) ⊗ p∗2 (O(1))) which
vanishes precisely on the diagonal ∆ ⊂ Pn × Pn . Indeed, in this case the Koszul complex
gives a resolution of the diagonal as required. Indeed if E = p∗1 (Ω1 (1)) ⊗ p∗2 (O(−1)), we
may consider the following complex:
0 → ∧n E → · · · ∧2 E → E → O∆ → 0
where each differential is a contraction with s ∈ Γ(Pn × Pn , E ∨ ). The standard theorem
of homological algebra says that the complex above is exact if the section s is regular
(that is, generic in a certain sense).
To find such a regular section s we start with the Euler exact sequence on Pn :
0 → O(−1) → V ⊗ OPn → T (−1) → 0
and then consider the following composition:
φ : p∗2 O(−1) → V ⊗ OPn ×Pn → p∗1 T (−1).
At the point (l1 , l2 ) ∈ Pn × Pn we have
φ(l1 ,l2 ) : l1 ⊂ V → V /l1 .
In particular, φ(l1 ,l2 ) = 0 if and only if l1 = l2 .
Let s ∈ Γ(Pn × Pn , p∗1 (T (−1)) ⊗ p∗2 O(1)) be the section corresponding to φ. Then s
vanishes precisely along the diagonal and s is regular since of the codimension of the
diagonal is equal to n which is the rank of the bundle.
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Triangulated categories in this lecture are assumed to be k-linear over some field k
and of finite type, that is
X
dim Hom(A, B[i]) < +∞
i∈Z
for all objects A, B. Derived categories of coherent sheaves Db (X) for smooth projective
X are examples of such categories.
We call a subcategory of a triangulated category admissible if it is both left and right
admissible.
1. Subcategories generated by exceptional collections are admissible
Lemma 1.1. Let E ∈ Db (X) be an exceptional object. Then E = hEi ⊂ Db (X) is
admissible. The adjoint functors to i : E → Db (X) are given as
i! (F ) := E ⊗ RHom(E, F )
i∗ (F ) := E ⊗ RHom(F, E)∗
Proof. Let E ⊗ K • ∈ E, K • ∈ Db (V ect/k). We have
HomE (E ⊗ K • , i! F ) = HomE (E ⊗ K • , E ⊗ RHom(E, F )) '
' HomDb (V ect/k) (K • , RHom(E, F )) '
' HomDb (V ect/k) (K • , RΓ(RHom(E, F ))) '
' HomDb (X) (K • ⊗ OX , RHom(E, F )) '
' HomDb (X) (K • ⊗ E, F ).
Similarly
HomE (i∗ (F ), E ⊗ K • ) = HomE (E ⊗ RHom(F, E)∗ , E ⊗ K • ) '
' HomDb (V ect/k) (RHom(F, E)∗ , K • ) '
' H 0 (RHom(RHom(F, E)∗ , K • )) '
' H 0 (RHom(F, E) ⊗ K • ) '
' H 0 (RHom(F, E ⊗ K • )) '
' HomDb (X) (F, E ⊗ K • ).
If E ∈ Db (X) be an exceptional object. Then for any F ∈ Db (X), we can consider
projections of F onto the left and right ortohogonals to E.
That is, we consider the following two triangles:
E ⊗ RHom(E, F ) → F → LE (F )
RE (F ) → F → E ⊗ RHom(F, E)∗ .
Definition 1.2. LE (F ) ' cone(E ⊗ RHom(E, F ) → F ) is called the left mutation of
F with respect to E.
RE (F ) ' cone(F → E ⊗ RHom(F, E)∗ )[−1] is called the right mutation of F with
respect to E.
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Note that LE (F ) lies in the right orthogonal to E whereas RE (F ) lies in the left
orthogonal to E.
Proposition 1.3. Let E = hE1 , . . . , Er i ⊂ Db (X) be generated by an exceptional collection. Then E is admissible.
Proof. Let F ∈ Db (X). Let Rj F = REj REj−1 . . . RE1 F be the composition of j right
mutations which together give rise to the following filtration of F :
ur
Rr F d
/ Rr−1 F
/ ...
y
[1]
u2
/ R2 F
c
[1]
i∗r Rr−1 F
u1
/ R1 F
b
{
/F
[1]
i∗2 R1 F
i∗1 F
~
(here ij denotes the embedding hEj i ⊂ Db (X) and i∗j is the corresponding left adjoint
functor).
We prove by induction that Rj F ∈ ⊥ hE1 , . . . , Ej i. Indeed, for j = 1 we get R1 F =
RE1 F ∈ ⊥ hE1 i by definition. Now since Rj F = cone(Rj−1 F → i∗j Rj−1 F )[−1] and both
terms Rj−1 F and i∗j Rj−1 F lie in ⊥ hE1 , . . . , Ej−1 i it follows that
Rj F ∈ ⊥ hE1 , . . . , Ej−1 i ∩ ⊥ hEj i = ⊥ hE1 , . . . , Ej i .
In particular Rr F ∈ ⊥ E.
Consider u = ur ◦ · · · ◦ u2 ◦ u1 : Rr F → F . It follows from Lemma below that the
term F 0 = cone(u) in the triangle
Rr F → F → F 0 → Rr F [1]
lies in E. Thus E is left admissible.
Similarly we can use left mutations:
F_
ur
i!r F
}
/ Lr F
d
[1]
ur−1
/ Lr−1 F
y
/ ...
/ L2 F
c
u1
[1]
i!r−1 Lr F
i!1 L2 F
{
/ L1 F
[1]
Then cone(u1 ◦ u2 ◦ · · · ◦ ur : F → L1 F ) ∈ E, L1 F ∈ E ⊥ and E is right admissible. Lemma 1.4. Let f : A → B, g : B → C be morphisms in a triangulated category. Then
cone(g ◦ f ) fits into a distinguished triangle
cone(f ) → cone(g ◦ f ) → cone(g) → cone(f )[1]
Proof. We start with a commutative square
BO
g
/C
O
f
A
g◦f
id
3
/A
The octahedron axiom implies that this square can be included into a diagram with
distinguished rows and columns
cone(f )
O
BO
/ cone(g ◦ f )
O
/ cone(g)
O
/C
O
/ cone(g)
O
g
f
A
g◦f
id
/A
/0
2. Mutations of exceptional collections
Let A be an admissible subcategory in C. In the most general form mutations are the
corresponding projections LA : A → A⊥ and RA : A → ⊥ A.
Lemma 2.1. LA ⊥ A and RA A⊥ are mutually inverse equivalences between the two
orthogonals to A.
Proof. We check that LA RA A⊥ is isomorphic to identity, the proof for other composition
goes exactly the same. Let B ∈ A⊥ . Then B 0 = RA (B) is defined from the distinguished
triangle
(2.1)
B 0 → B → A1 → B 0 [1].
To compute B 00 = LA (B 0 ) we need a distinguished triangle
A2 → B 0 → B 00 → A2 [1].
However from the triangle 2.1 we obtain:
A1 [−1] → B 0 → B → A1 .
Thus we get A2 ' A1 [−1], B 00 ' B.
Let E• = (E1 , . . . , Er ) be an exceptional collection in C. For each i = 1, . . . , r − 1 we
define left and right i-th mutation of E• as follows:
Li (E• ) = (E1 , . . . , Ei−1 , LEi Ei+1 , Ei , Ei+2 , . . . , En )
Ri (E• ) = (E1 , . . . , Ei−1 , Ei+1 , REi+1 Ei , Ei+2 , . . . , En )
Proposition 2.2. 1. Li (E• ) and Ri (E• ) are exceptional collections. If E• is full, then
so are Li (E• ) and Ri (E• ).
2. Consider the action of the group generated by all Li and Rj on the set of isomorphism classes of exceptional collections. Then the following relations are satisfied:
Li Ri = Ri Li = id
Li+1 Li Li+1 = Li Li+1 Li
Ri+1 Ri Ri+1 = Ri Ri+1 Ri
Li Lj = Lj Li , Ri Rj = Rj Ri , |i − j| ≥ 2.
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This gives a braid group action on the set of isomorphism classes of exceptional collections.
Proof. The first claim follows easily from definitions and Lemma 2.1.
The first formula in (2) also follows from Lemma 2.1. The fourth formula in (2)
is obvious. Let us prove the second formula. We assume for simplicity that E• =
(E1 , E2 , E3 ) and i = 1. In this case
L1 L2 L1 E• = (LE1 LE2 E3 , LE1 E2 , E1 )
L2 L1 L2 E• = (LLE1 E2 LE1 E3 , LE1 E2 , E1 ).
On the other hand
LE1 LE2 E3 ' LhE1 ,E2 i E3 ' LhLE E2 ,E1 i E3 ' LLE1 E2 LE1 E3 .
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Example 2.3. Let X = Pn . We prove that
(2.2)
LhO,...,O(k)i O(k + 1) ' Ωk+1 (k + 1)[k + 1].
In particular ”reverting” the collection O, O(1), . . . , O(n) we obtain a collection
Ωn (n)[n], . . . , Ω1 (1)[1], O.
Our computations rely on Euler’s short exact sequence:
(2.3)
0 → Ωk (j) → Λk V ∗ ⊗ O(j − k) → Ωk−1 (j) → 0.
Consider the associated long exact sequence of cohomology groups:
0 → H 0 (Ωk (j)) → H 0 (Λk V ∗ ⊗ O(j − k)) → H 0 (Ωk−1 (j)) →
...
H p (Ωk (j)) → H p (Λk V ∗ ⊗ O(j − k)) → H p (Ωk−1 (j)) → . . .
For 0 ≤ j < k ≤ n all the middle terms vanish. Then descending induction on k
shows that the sheaf Ωk (j) is acyclic for 1 ≤ j ≤ k (the base case is k = n, where
Ωn (j) = O(−n − 1 + j) is indeed acyclic in the give range). It follows then from the
same long exact sequence with k = j that H ∗ (Ωk−1 (k)) = Λk V ∗ [0].
Setting k = j in the sequence (2.3) we get a distinguished triangle in Db (Pn ):
Ωk (k) → Λk V ∗ ⊗ O → Ωk−1 (k) → Ωk (k)[1].
Using this triangle and the definition of left mutation we compute
(2.4)
LO Ωk−1 (k) = cone(RΓ(Ωk−1 (k)) ⊗ O → Ωk−1 (k)) ' Ωk (k)[1].
Finally (2.2) is proven by induction using (2.4):
LhO,...,O(k)i O(k + 1) ' LhOi LhO(1),...,O(k)i O(k + 1) '
' LhOi (LhO,...,O(k−1)i O(k))(1) '
' LhOi Ωk (k + 1)[k] '
' Ωk+1 (k + 1)[k + 1].
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3. The Serre functor
Definition 3.1. Let C be an Ext-finite triangulated category. We say that a covariant
auto-equivalence S : C → C is a Serre functor if we have a bifunctorial isomorphism
Hom(A, B)∗ ' Hom(B, SA), A, B ∈ C
Proposition 3.2. If X is a smooth projective n-dimensional variety, then Db (X) admits
a Serre functor given as
S(F • ) = F • ⊗ ωX [n]
(ωX = ∧n Ω1X is the determinant of the cotangent sheaf ).
Proof. Recall that if F is a coherent sheaf on X, the Serre duality theorem [Har],III.7
tells that there is a natural isomorphism
H i (X, F) ' Extn−i (F, ωX )∗ .
In fact these isomorphisms lift to a natural quasiisomorphism of complexes
RΓ(X, F) ' RHom(F, ωX [n])∗ .
Note that dualizing the complex puts a component of degree i into degree −i.
Now let F1 , F2 ∈ Db (X) and let
F = RHom(F1 , F2 ) ∼
= F2 ⊗ F ∨ ,
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where F1∨ = RHom(F1 , OX ). We have
RHom(F1 , F2 ) ∼
= RHom(O, F) ∼
= RHom(F, ωX [n])∗ ∼
= RHom(F2 , F1 ⊗ ωX [n])∗ .
Proposition 3.3. 1. C admits a Serre functor if and only if all contravariant functors
Hom(A, •)∗ , A ∈ C and all covariant functors Hom(•, A)∗ , A ∈ C are representable.
2. If C admits a Serre functor S, then S is a triangulated functor and S is unique up
to a canonical isomorphism.
3. If A is an admissible subcategory in C and C admits a Serre functor, then A admits
a Serre functor and it is given by
SA = i ! SC .
−1
SA
= i∗ SC−1 .
4. Let A ⊂ C be a triangulated subcategory and assume that S is a Serre functor on
C. Then
S(⊥ A) = A⊥
S −1 (A⊥ ) = ⊥ A.
5. If E• = (E1 , E2 , . . . , En ) is a full exceptional collection, then
L1 L2 . . . Ln−1 E• = (S(En ), E1 , . . . , En−1 )
Rn−1 Rn−2 . . . R1 E• = (E2 , . . . , En , S −1 (E1 ))
Proof. For (1) and (2) see [BK]. (3) and (4) follow easily from definitions. (5) follows
from (4) applied to the subcategory generated by n − 1 objects.
Definition 3.4. A category C is called an n-Calabi-Yau category if the functor [n] is a
Serre functor.
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Lemma 3.5. 1. Let C be a Calabi-Yau category. Then any semi-orthogonal decomposition C = hA, Bi is fully orthogonal, that is C ∼
= A ⊕ B.
2. If X is a Calabi-Yau variety, then Db (X) admits no non-trivial (left or right)
admissible subcategories.
Proof. By definition of what an admissible subcategory is, it suffices to show that any
semiorthogonal decomposition C = hA, Bi is trivial.
Let A ∈ A, B ∈ B. Then
Hom(A, B) ∼
= Hom(B, S(A))∗ = Hom(B, A[n]) = 0.
This shows the first claim.
To prove the second claim, note that since X is connected OX admits no notrivial
direct summands, thus if Db (X) ∼
= A ⊕ B, then OX must lie in either A or B. Assume
that OX ∈ A. The same reasoning applies to each Ox , x ∈ X as well. However, since
Hom(OX , Ox ) 6= 0 for all x ∈ X, all Ox ∈ A. Now B is orthogonal to all skyscraper
sheaves and their shifts, thus B = 0.
Proposition 3.6. Let F : C → D be a functor between triangulated categories. Assume
that both C and D have Serre functors.
1. If F admits a left adjoint G : D → C, then it also admits a right adjoint H : D → C,
given as
−1
.
H = SC ◦ G ◦ SD
2. If F admits a right adjoint H : D → C, then it also admits a left adjoint G : D → C,
given as
G = SC−1 ◦ H ◦ SD .
Proof. The proof is straightforward using the definitions.
Example 3.7. Let f : X → Y be a morphism of smooth projectvie varieties. Then
f∗ : Db (X) → Db (Y ) admits a left adjoint f ∗ . By the Proposition above f∗ also admits
the right adjoint functor, which is denoted f ! . We have
f ! (G • ) := SX ◦ f ∗ ◦ SY−1 (G • ) '
' SX ◦ f ∗ (G ⊗ ωY∨ [−dimY ]) '
' SX (f ∗ (G) ⊗ f ∗ (ωY∨ )[−dimY ]) '
' f ∗ (G) ⊗ f ∗ (ωY∨ )[−dimY ] ⊗ ωX [dimX] '
' f ∗ (G) ⊗ ωX/Y [dimX − dimY ],
where ωX/Y = ωX ⊗ f ∗ (ωY∨ ).
In particular, if f = i : X → Y is a closed embedding of codimension r, then
∨
i! (G • ) = i∗ (G • ) ⊗ ∧r NX/Y
[−r].
On the other hand if f = p : X → Spec(k) is the projection to a point, then
DX := p! (OSpec(k) ) ' ωX [dim(X)]
is the so-called dualizing complex.
Corollary 3.8. If A is a left (or right) admissible subcategory in C and both A and C
have Serre functors, then A is admissible.
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Remark 3.9. In the situation of the Corollary above, we do not usually have SA = SC A .
Indeed, when this is the case then one can prove that A⊥ = ⊥ A, and thus C ' A ⊕ A⊥
is completely orthogonal.
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