Weighted Projective Spaces, Coherent Sheaves and
Homological Projective Duality
Karim El Haloui
Supervised by Dr. Dmitriy Rumynin
Abstract
In this report, we shall present a generalisation of projective spaces known as weighted projective spaces. These spaces naturally t into the work of various geometers. We will present
various ways of constructing these spaces and describe their coherent sheaves in the case where we
view them as algebraic stacks. Finally, we will present how the homological projective duality of
smooth algebraic varieties is dened.
Contents
1 Elementary denition of Weighted Projective Spaces
2
1.1
First denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2 Weighted Projective Spaces as Quotient Varieties
3
2.1
Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.2
Group action on a variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.3
Categorical quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
3 Weighted Projective Spaces as Projective Schemes over
k
7
3.1
Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
3.2
Standard weighted projective spaces results . . . . . . . . . . . . . . . . . . . . . . . . .
8
3.3
Pathologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Weighted Projective Spaces as Quotient Stacks
11
4.1
Short description of stacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2
Coherent sheaves on wps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5 Homological Projective Duality
15
5.1
Triangulated categories
5.2
Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3
Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1
Let k be an algebraically closed eld of characteristic 0. One can assume k = C throughout this paper.
1 Elementary denition of Weighted Projective Spaces
1.1 First denitions
Denition 1.1.
Let An denote the n-dimensional ane space for n a positive integer (basically one
can view A as k ). Dene an equivalence relation ∼ on An+1 \ {0} as follows:
n
n
x ∼ y i ∃λ ∈ k × such that x = λ.y
Coordinatewise, we have (x0 , ..., xn ) ∼ (y0 , ..., yn ) i xi = λyi for i = 0, ..., n. Now dene the ndimensional projective space Pn to be the set of equivalent classes with respect to ∼ on An+1 \ {0} .
Formally,
Pn = (An+1 \ {0})/ ∼
One could very naturally extend this denition to a more general case where the powers of λ are
general positive integers.
P
Denition 1.2. Fix Q = (q0 , ..., qn ) to be n + 1-uple of positive integers and let |Q| = ni=0 qi . The
weighted projective space (w.p.s or wps) of type Q, denoted by P(Q), is the set of equivalence classes
on An+1 \ {0} under the equivalence relation ∼Q dened by:
(x0 , ..., xn ) ∼Q (y0 , ..., yn ) i ∃λ ∈ k × such that xi = λqi yi for i = 0, ..., n
The relation ∼Q denes an equivalence relation on An+1 \ {0}. The n-dimensional projective space is
a particular case of a wps of type (1, ..., 1). For each xi , we dene its degree by letting deg(xi ) = qi .
Denote by [x0 , ..., xn ] the set of equivalence classes corresponding to the point (x0 , ..., xn ) ∈ An+1 \ {0}.
Dene
Ui = {[x0 , ..., xn ] ∈ P(Q) | xi 6= 0}
to be the ane charts of P(Q) and
Hi = {[x0 , ..., xn ] ∈ P(Q) | xi = 0}
its hyperplanes. We have:
P(Q) =
n
[
Ui
i=0
Denition 1.3.
of degree d if
Fix Q as above. A polynomial f (x0 , ..., xn ) ∈ k[x0 , ..., xn ] is weighted homogeneous
f (λq0 x0 , ..., λqn xn ) = λd f (x0 , ..., xn ) ∀λ ∈ k ×
Denition 1.4.
Fix Q. If for (x0 , ..., xn ) ∈ An+1 \ {0} we have f (x0 , ..., xn ) = 0 for a given weighted
homogeneous polynomial, then f is well dened on equivalence classes of ∼Q . The set
V({fi |i ∈ I}) = {[x0 , ..., xn ] ∈ P(Q) | fi (x0 , ..., xn ) = 0, ∀i ∈ I} ⊂ P(Q)
is dened to be the weighted projective variety with respect to the family {fi |i ∈ I} of weighted
homogeneous polynomials.
1.2 Examples
Example 1.5. Let X = P(1, 1, 2). To understand X , a good strategy is to look at the dierent
ane pieces Ui for i = 0, 1, 2. We have U0 ∼
= A2 via the map [x0 , x1 , x2 ] 7→ (x1 /x0 , x2 /x20 ) and
2
U1 ∼
= A via the map [x0 , x1 , x2 ] 7→ (x0 /x1 , x2 /x21 ). This is similar to what we get with the usual
projective space P2 . The dierence comes with U2 . Consider the ane variety V = V(xz − y 2 ) ⊂ C3 .
We get U2 ∼
= V via the map [x0 , x1 , x2 ] 7→ (x20 /x2 , x0 x1 /x2 , x21 /x2 ). Now let us consider the map
φ : [x0 , x1 , x2 ] 7→ [x20 , x0 x1 , x21 , x2 ] from X → P3 .The map φ is injective and its image is the surface
V(y0 y2 − y12 ) ⊂ P3 with homogeneous coordinates y0 , y1 , y2 , y3 . Via this map we can consider X to be
a projective variety. Is this a general result? Are all weighted projective spaces projective varieties?
2
Example 1.6.
[Rei02] Consider the equation y 2 = f (x1 , x2 ) where f is a homogeneous polynomial
having 4 distinct roots in P1 . We can dene the hypersurface C4 = V(y 2 − f (x1 , x2 )) ⊂ P(1, 1, 2) with
homogeneous coordinates x1 , x2 , y and respective degree 1, 1, 2. This hypersurface doesn't pass through
the point [0, 0, 1] and it can be decomposed as the union of two ane charts U1 and U2 respectively
when x1 6= 0 and x2 6= 0 (y 6= 0 implies x1 6= 0 or x2 6= 0) glued together in the obvious way. Hence, we
get a double cover C4 → P1 with 4 branch points given by f = 0. One could have taken the projective
closure and worked directly in P2 but the surface naturally sits in P(1, 1, 2).
Remark 1.7. We can see from the previous examples that wps behave quite dierently to straight
projective spaces. Many surfaces come up naturally with a natural embedding into a wps. However,
so far, we have only used an elementary way of dening them: the next step would be to dene it
using modern techniques such as schemes and establish some general properties for them.
2 Weighted Projective Spaces as Quotient Varieties
2.1 Generalities
Denition 2.1.
Let X be a non-empty set and G a group with identity e. The map α : G × X → X
given by (g, x) 7→ g.x is an action on X (or that G acts on X ) if it satises both:
1. g1 .(g2 .x) = (g1 .g2 ).x ∀g1 , g2 ∈ G ∀x ∈ X
2. e.x = x ∀x ∈ X
We say as well that G acts freely on X if moreover g.x 6= x for every g 6= e and every x ∈ X .
Proposition 2.2. An action on X induces a group homomorphism
G → Aut(X) given by g 7→ Tg
where Tg : X → X, x 7→ g.x. We can see by the denition of an action that Tg is a bijection of X with
inverse Tg−1 . We shall call Tg a translation for a given g ∈ G,. Conversely any group automorphism
φ : G → Aut(X) gives rise to a group action given by (g, x) 7→ φ(g)(x) from G × X → X .
Proof. Easy. (See [Rot95])
Denition 2.3.
Let G act on X . For a given x ∈ X , we dene the orbit of x in G to be the set
OrbG (x) = {y ∈ X | ∃g ∈ G, y = g.x}
Remark 2.4. The set of orbits denes a partition on X .
2.2 Group action on a variety
Denition 2.5.
Let G be a group which is a variety. We have two maps
1. m : G × G → G, (g, h) 7→ g.h
2. i : G → G, g 7→ g −1
G is an algebraic group if m and i are morphisms of varieties.
Denition 2.6.
Let G be an algebraic group and X a variety. We say that G acts algebraically on X
if the map G × X → X , given by the action of G on X , is a morphism of varieties.
Proposition 2.7. Any algebraic group is a smooth variety.
Proof. From the above denition, G acts algebraically on itself by left multiplication. It is a basic
fact that any non-trivial variety posseses a non-empty open dense subset U consisting of non-singular
points. The automorphism group Aut(G) is transitive as it contains a transitive subgroup consisting
of translations of G. Now let x, y ∈ G, then there is a g ∈ G such that y = g.x, it induces a linear
map on tangent spaces dg : Tx G → Ty G. Since g is invertible, we get an isomorphism Tx G ∼
= Ty G. So
if x ∈ U , then y is a non-singular point by transitivity. Hence G is a smooth variety.
3
Denition 2.8. Let X and Y be two varieties and G a group acting on both of them. A map
φ : X → Y is said to be G-equivariant if φ is a morphism of varieties and if it commutes with the
action of G i.e.
φ(g.x) = g.φ(x) ∀x ∈ X, ∀g ∈ G
A G-equivariant map is called G-invariant if moreover G acts trivially on Y , in particular
φ(g.x) = φ(x) ∀x ∈ X, ∀g ∈ G
So φ is constant on the orbits.
Example 2.9.
The multiplicative group Gm = (k × , .), the set µm of dth roots of unity of Gm are
algebraic groups. For a given n + 1-uple of positive integers Q = (q0 , ..., qn ), Gm acts algebraically on
An+1 \ {0} in the following way
(λ, (x0 , ..., xn )) 7→ (λq0 x0 , ..., λqn xn )
In the particular case where Q = (1, ..., 1), the orbits are the punctured 1-dimensional line through the
origin.
Remark 2.10. We shall consider the special case where X is an ane variety with coordinate ring
k[X] (or O(X)), and when G is an algebraic group acting algebraically on X . This way, we have an
equivalence of categories: {category of ane varieties} ←→{category of nitely generated reduced k algebras} and in particular {category of irreducible ane varieties} ←→{category of nitely generated
k -algebras which are integral domains}. When X and G are both ane varieties, we have
k[G × X] ∼
= k[G] ⊗k k[X]
as a k -algebra isomorphism. So any morphism G × X → X corresponds to a k -algebra homomorphism
k[X] → k[G] ⊗k k[X]
and vice-versa (recall that k is an algebraically closed eld).
Lemma 2.11.
G acts on k[X] via the group homomorphism τ : G → Autk (k[X]) given by g 7→ τg
where τg : k[X] → k[X], f 7→ f ◦ Tg−1 .
Proof. Easy. (See [Rot95])
Denition 2.12.
We dene the G-invariant subring of k[X] to be the subset of k[X] that is invariant
under the action of G. More explicitly,
k[X]G = {f ∈ k[X] | g.f = f, ∀g ∈ G}
Remark 2.13. It can be checked that k[X]G is a subring of k[X] and that its elements consist of
polynomial functions on X that are constant on the orbits of X under the action of G. When moreover
X is an irreducible ane variety, we denote by k(X)G the subring of invariants of k(X).
Now an important result but which only applies to nite algebraic groups that act algebraically on
ane irreducible varieties.
Theorem 2.14. Let X be an irreducible ane variety and G a nite group acting algebraically on X .
The set of equivalence classes of X under the action of G (also known as the orbit space of X under
the action of G):
X/G = {OrbG (x) | x ∈ X}
endowed with the quotient topology has a natural structure of an ane variety with coordinate ring
k[X/G] ∼
= k[X]G
and eld of fraction
k(X/G) ∼
= k(X)G
4
Proof. See [BR86].
Example 2.15.
Let X = A2 with coordinates u, v and G = Z2 . G acts on X by
(−1).(u, v) = (−u, −v).
The polynomial ring of X is k[u, v] and its G-invariant subring is given by
k[X]G = k[u2 , uv, v 2 ] ∼
= k[x, y, z]/(xz − y 2 )
From 1.5, V = V(xz − y 2 ) ⊂ C3 and hence the quotient X/G is isomorphic to V . So we have U0 ∼
= A2 ,
2
2
U1 ∼
= A /Z2 for P(1, 1, 2).
= A and U2 ∼
Theorem 2.16. [BR86] Let X be a variety (non-necessarily ane) and G a nite group. Suppose
that for every x ∈ X , OrbG (x) is contained in an ane open set, then X/G has a natural structure of
an algebraic variety.
Proof. Assume rst that X is ane. Let x ∈ X , by assumption OrbG (x) ⊂ Ũ where Ũ is an ane
open set. Dene
U=
\
g.Ũ
g∈G
This is a G-invariant ane open set containing OrbG (x) (ane and open since we have a nite
intersection). The variety X is covered by its orbits so, it follows at once that X is covered by
G-invariant ane open subsets.
Corollary 2.17. If X is a projective variety and G a nite group. Then X/G has a natural structure
of an algebraic variety.
Proof. Let x ∈ X , we can assume that X/G does not reduce to one point as otherwise the result holds
trivially. So consider the nite set OrbG (x) 6= X . Let f be a polynomial which doesn't vanish at any
point of OrbG (x). It follows that OrbG (x) ⊂ X\V (f ). Moreover, X\V (f ) is an ane open subset of
X since the complement of a hypersurface in a projective space is an open ane subset. Now apply
the theorem above.
Remark 2.18. More generally, one can show that for P(Q) with Q = (q0 , ..., qn ) the ane charts are
given by Ui ∼
= An /Zqn . It can also be shown that wps have a natural structure of an orbifold. (See
[RT11]).
The problem we face here, is that Gm isn't a nite group, therefore we can't use the above theorem
directly to prove that wps are projective varieties. The method of taking the orbit space and get a
natural structure of a variety isn't always convenient to work with. However it is proven that there
always exists a Zariski dense open G-invariant subset U of X for which the orbit space U/G is a variety
(see [Dol03]). Another approach is to dene our quotient as a universal property.
2.3 Categorical quotients
The main reference for this part is [Dol03].
Denition 2.19.
A categorical quotient of a variety X , where G acts algebraically on X via α (call it a
G-variety ), is a G-invariant morphism p : X → Y such that for any G-invariant morphism g : X → Z ,
there exists a unique morphism ḡ : Y → Z satisfying ḡ ◦ p = g .
X
g
/Z
>
p
Y
∃!ḡ
A categorical quotient is called a geometric quotient if the image of the morphism (α, pr2 ) : G ×
X → X × X , dened by (g, x) 7→ (α(g, x), x) = (g.x, x), equals X ×Y X = (p × p)−1 (∆Y ) with
∆Y = {(y, y) | y ∈ Y } ⊂ Y × Y . We shall denote the categorical quotient (resp. geometric quotient)
by p : X → X //G (resp. p : X → X/G). If it exists then it is dened uniquely up to isomorphism by
this universal property.
5
Remark 2.20. One should regard the morphism (α, pr2 ) as an equivalence relation dened on X .
Indeed, in the case where Y = X/ ∼, where ∼ denotes the equivalence relation induced by the group
action, we precisely require that the image of (α, pr2 ) corresponds to points in X × X lying in the
same equivalence class with canonical map p. This generalises the notion of a quotient in the language
of categories. A geometric quotient is a categorical quotient for which the bers of p are G-orbits.
Denition 2.21.
A good geometric quotient of a G-variety X is a G-invariant morphism p : X → Y
satisfying the following properties:
(i) p is surjective;
(ii) for any open subset U of Y , the inverse image p−1 (U ) is open if and only if U is open;
(iii) for any open subset U of Y , the natural homomorphism p∗ : O(U ) → O(p−1 (U )) is an isomorphism
onto the subring O(p−1 (U ))G of G-invariant functions;
(iv) the image of (α, pr2 ) : G × X → X × X is equal to X ×Y X .
Proposition 2.22. A good geometric quotient is a categorical quotient.
Proof. We have to show that any G-invariant morphism factors through p. So let q : X → Z be a
G-invariant morphism and take an ane open cover {Vi }i∈I of Z . Take a point x ∈ q −1 (Vi ), we have
q(x) = q(g.x) for all g ∈ G. So q −1 (Vi ) is a G-invariant open subset of X .
Dene Ui = p(q −1 (Vi )). we claim that {Ui }i∈I form an open cover of Y . First note that we trivially
have q −1 (Vi ) ⊂ p−1 (Ui ). Moreover by property (iv) of the above denition, the bres of p are orbits
and since q −1 (Vi ) covers X with p−1 (Ui ) ⊂ X , then q −1 (Vi ) = p−1 (Ui ) (both consists of orbits and
one covers the all space so they must coincide). So p−1 (Ui ) is open and hence by (ii), Ui is open in Y .
Now by (i), p is surjective and X is covered by p−1 (Ui ), so {Ui }i∈I form an open cover of Y .
Consider the restriction of q on q −1 (Vi ), ie q −1 (Vi ) → Vi , this is a morphism of varieties, so it corresponds to a k -algebra homomorphism
αi : O(Vi ) → O(q −1 (Vi )) = O(p−1 (Ui ))
Now q is a G-invariant morphism, so αi maps onto O(p−1 (Ui ))G , but by (iii)
O(Ui ) ∼
= O(p−1 (Ui ))G
So we get a k -algebra homomorphism O(Vi ) O(Ui ) which in turn represents a unique morphism
q̄i : Ui → Vi as Vi is ane. To dene a global morphism q̄ : Y → Z from the q¯i they must agree on
their common intersection but this is trivial as αi|Vi ∩Vj = αj |Vi ∩Vj . So there exists a unique morphism
q̄ : Y → Z such that q = q̄ ◦ p.
L
Example 2.23. Let A = i∈N Ai be a nitely generated connected k-algebra. Consider the corresponding action of Gm on X with coordinate ring A (X corresponds to Ank where n isL
the number of
minimal generators of A over k ). Let 0 be the vertex of X dened by the maximal ideal i>0 Ai . Then
the open subset X0 = X\{0} is invariant and the geometric quotient X //Gm exists and is isomorphic
to the wps X0 /Gm .
Proposition 2.24. Wps are projective varieties
Proof. By the Nullstellensatz theorem, points in an ane variety X are in a 1-1 correspondence
with maximal ideals the coordinate ring k[X]. In other words, we have X ∼
= Specm(k[X]) where
Specm(k[X]) denotes the set of maximal ideals of k[X]. Henceforth, we will identify X and Specm(k[X])
in the subsequent section.
Now one would like to build a similar construction for the standard projective space or more generally
for the wps as we did for ane varieties. For general denitions, look at section
L Ÿ3. The procedure
is as follows. Start by a nitely generated N-graded connected k -algebraLS = i∈N Si . This grading
induces a Gm action on X = Specm(S). The irrelevant ideal m0 =
i>0 Si (which is maximal)
corresponds to a point p0 ∈ X . Set
X0 = X\{p0 } = Specm(S)∗
6
Gm acts on X0 and the quotient set is denoted by Projm(S).
Take a minimal set of homogeneous generators of S over k , say {x0 , ..., xn }with respective non-negative
degrees {q0 , ..., qn }, then t ∈ Gm (here Gm is identied with k ∗ ) acts on xi ∈ Aqi as follows:
t.xi = tqi xi
Dene I to be the homogeneous ideal of S describing the relations between x0 , ..., xn . We can therefore
identify X with a closed subset of An+1
and identify p0 with the origin 0 of An+1
. These dierent
k
k
identications induce a natural action of Gm on X\{0} as follows: for t ∈ Gm and (a0 , ..., an ) ∈ X ,
t.(a0 , ..., an ) = (tq0 a0 , ..., tqn an )
And we get a natural bijection
X\{0}/Gm ∼
= Projm(S)
Note that when the homogeneous generators are algebraically independent, we have I = {0} and so
X = Ank and we retrieve our wps P(Q) ∼
= Projm(S) for Q = (q0 , ..., qn ).
Let µl act on S by the induced action of Gm restricted to the closed subgroup µl given by .xi = qi xi
where is a primitive lth root of unity. The invariant subring is given by
M
S µl = S (l) =
Sli
i∈N
Indeed i∈N Sli ⊂ S µl is obvious since for ∈ µl we have l = 1. Conversely, let s =
element (nite sum), then we must have the following equality
X
X
.s =
i si =
si = s
L
i∈N
P
i∈N si
invariant
i∈N
So i si = si and hence (1 − i )si = 0. This implies that si = 0 for i ∈
/ lN and we conclude that s ∈ S (l) .
(l)
The k -algebra homomorphism inclusion S → S induces a surjective morphism Specm(S) → Specm(S (l) )
and in particular we get a map
Specm(S)∗ → Specm(S (l) )∗ , (a0 , ..., an ) 7→ (q0 a0 , ..., qn an )
for a given primitive lth root of unity in k which corresponds precisely to the quotient map of µl on
Specm(S)∗ . Hence,
Specm(S (l) )∗ = Specm(S)∗ /µl
(l)
Let Gm act on S (l) such that the induced graduation on S (l) is given by Si = Sli . We have
Projm(S) = Specm(S)∗ /k ∗ = (Specm(S)∗ /µl )/k ∗ = Specm(S (l) )∗ /k ∗ = Projm(S (l) )
But for any such graduation, there exists an l suciently big enough such that S (l) is generated by
degree 1 elements (see[Rei02] Chap. III, Ÿ1). And hence, we obtain a bijection between a closed subset
of PN
k and we conclude that all wps are projective varieties.
Remark 2.25. In the next section, we prove that wps are projective schemes. In particular, the closed
points of a weighted projective scheme form a projective variety; they correspond precisely to the
elements of Projm(S) (see [Har77] Prop. 2.6).
3 Weighted Projective Spaces as Projective Schemes over k
3.1 Generalities
Denition
3.1.
L
A graded ring is a ring S together with a decomposition into abelian groups S =
S
with
S
.S
i
i
j ⊂ Si+j . The non-zero elements of the subgroup Si are called homogeneous elements
i∈Z
of degree i. This graduation is called a Z − graduation and S is said to be Z-graded.
7
Remark 3.2. In the above denition, when Si = 0 for i < 0, we say that S is non-negatively graded or
N-graded.
Theorem 3.3. [Gro61] For a xed nitely generated k-algebra S , there is a 1-1 correspondence between
{Z-graduation of S } and {Gm action on Spec(S)}.
Proof. To dene a Gm action on Spec(S) , it suces to give a k-algebra homomorphism S → S ⊗k
k[X, X −1 ] ∼
= S[X, X −1 ].
So let {x0 , ..., xr } denote a minimal set of homogeneous generators with degree q0 , ..., qr respectively
so that R = k[x0 , ..., xr ], we dene the k -algebra homomorphism S → S[X, X −1 ], xi 7→ xi X qi ; this
gives us a Gm action on S .
P
For the converse, given a k -algebra homomorphism ψ : S → S[X, X −1 ], we can write ψ as n∈Z ψn X n
where ψn : S → S are S -module homomorphisms with ψn (x) = 0 for all but nitely many n ∈ Z and
x ∈ S . It can then be checked that the images of S under ψn denes a Z − graduation on S .
Denition 3.4. Let S(Q) be the polynomial ring k[x0 , ..., xr ] with graduation given by deg(xi ) = qi
for a xed r + 1-uple of positive integers Q = (q0 , ..., qr ). We dene the weighted projective scheme over
k of type Q to be Proj(S(Q)). It is denoted similarly by P(Q) and is also called weighted projective
space.
Remark 3.5. We shall always assume that gcd(q0 , ..., qr ) = 1 and that q0 6 ... 6 qr . Indeed, one can
show (see [Rei02]) that for d > 0 we have a canonical isomorphism
Proj(S) ∼
= Proj S (d)
(d)
where S (d) is a graded ring with graduation given by Si
permutation of coordinates.
= Sdi and the order can be set by a
Notation 3.6. Ar+1 = Spec (S(Q)) the r + 1-dimensional ane space over k, 0 denotes the irrelevant
ideal S+ = ⊕r>0 Sr , U = Ar+1 \ {0} the punctured
quasi-cone. We identify the standard ane open set
D+ (xi ) = {p ∈ P(Q) | xi ∈
/ p} with Spec S(xi ) where S(xi ) is the subring of degree 0 in the localised
ring Sxi . The standard ane open sets form an open cover of P(Q). As usual, we denote by S(Q)(n) the
S(Q)-graded module with graduation given by S(Q)(n)i = S(Q)n+i . For every S(Q)-graded module
f. In particular, we denote by OP(Q) (n) the
M , we denote the sheaf associated to M on P(Q) by M
^ .
sheaf S(Q)(n)
3.2 Standard weighted projective spaces results
The main reference for this section is [Dol82].
Theorem 3.7. 1. The universal geometric quotient U //Gm exists and coincides with P(Q).
2. The group scheme µQ = µq0 × ... × µqr acts on Pr and induces an isomorphism P(Q) ∼
= Pr /µQ . In
particular wps are projective varieties.
3. For Q = (1, q1 , ..., qr ), P(Q) are compactications of the ane space Ar .
4.
toric varieties.
Wps are complete
Qr Foreach Q = (q0 , ..., qr ), the polyhedral space P∆ where ∆ =
P
r
(x0 , ..., xr ) ∈ Rr+1 | i=0 qi xi = i=0 qi is isomorphic to P(Q).
Proof. We will give only indications for the proofs
1. U is a Gm -invariant open set. The G-invariant morphism p : U → P(Q) is a good geometric
quotient.
2. µQ acts on k[x0 , ..., xr ] by sending f (x0 , ..., xr ) to f (0 x0 , ..., r xr ) where i is a qith primitive root
of unity. The invariant subring is k[x0 , ..., xr ]µQ = k [xq00 , ..., xqrr]. This action can be extended to
µQ
k[x0 , ..., xr ](xi ) and the invariant subring becomes k[x0 , ..., xr ](xi )
= k [xq00 , ..., xqrr ](xqi ) . So locally
i
q
the standard ane open set D+ (xi ) ⊂ Pr corresponds to k [x00 , ..., xqrr ] (xqi ) . Globally k [xq00 , ..., xqrr ] ∼
=
i
S(Q), hence this isomorphism of graded rings implies that P(Q) = Pr /µQ as µQ is a nite algebraic
group. The niteness of µQ also implies that wps are projective varieties.
8
i
h
3. Note that D+ (x0 ) corresponds to S(Q)(x0 ) ∼
= k[y1 , ..., yr ] which in turn is the
= k xxq11 , ..., xxqrr ∼
0
0
coordinate ring of Ar . Moreover the complement of D+ (x0 ) in P(Q) can be identied to P(q1 , ..., qr ).
4. See [CLS11].
Notation 3.8. Let
di
=
ai
=
gcd(q0 , ..., qbi , ..., qr )
lcm(d0 , ..., dbi , ..., dr )
a
=
lcm(d0 , ..., dr )
Note that ai |qi , gcd(ai , di ) = 1 and ai di = a.
Proposition 3.9. Let Q = (q0 , ..., qr ) and dene Q0 = (q0 /a0 , ..., qr /ar ). Then there exists a natural
isomorphism
P(Q) ∼
= P(Q0 )
Proof. Dene
S0 =
M
S(Q)ai = S(Q)(a)
i∈N
h
i
This a graded subring of S(Q). We claim that S 0 = k xd00 , ..., xdr r as graded rings with deg xdi i =
i
h
qi di = aqi /ai . But since qi /ai ∈ N, we have k xd00 , ..., xdr r ⊂ S 0 . Conversely, let xs00 ...xsrr ∈ S 0 , then
P
s0 q0 +...+sr qr = an for some n ∈ N, so si qi = − j6=i sj qj +ai di n since ai di = a. When i 6= j we have
di |qj by denition. Therefore, di |(si qi ) but gcd(qi , di ) = gcd(q0 , ..., qr )h= 1 by assumption,
so di |si and
i
hence si is a multiple of di . This gives us the reverse inclusion S 0 ⊂ k xd00 , ..., xdr r . Take yi = xdi i and
let deg(yi ) = aqi /ai , it follows that S 0 = S(aQ0 ) with aQ0 = {aq0 /a0 , ..., aqr /ar } as graded rings. We
also have S(aQ0 )(a) ∼
= S(Q0 ) and S 0 = S(Q)(a) as graded rings. Hence
P(Q0 )
=
∼
=
P roj(S(Q0 ))
∼
=
=
P roj(S(aQ0 ))
=
∼
=
P roj(S(Q)(a) )
=
P(Q)
P roj(S(aQ0 )(a) )
P roj(S 0 )
P roj(S(Q))
and all these isomorphisms are natural.
Corollary 3.10.
P(q0 , q1 ) ∼
= P1 .
Proof. We have a0 = d0 = q0 and a1 = d1 = q1 , so Q0 = (1, 1) and the result follows.
It would be interesting to know the kind of induced isomorphism we get between sheaves. Since
gcd(qi , di ) = 1, there exists unique integers bi (n) and ci (n) determined for each n ∈ Z by
Proposition 3.11.
n = bi (n)qi + ci (n)di , 0 6 bi (n) < di
P
Dene φ(n) = (n − ri=0 bi (n)qi ) /a.
1. φ(n) is an integer for all n ∈ Z.
2. The isomorphims P(Q) ∼
= P(Q0 ) induces an isomorphism of sheaves OP(Q) (n) ∼
= OP(Q0 ) (φ(n)).
9
Proof. 1. n −
Pr
P
P
bi (n)qi = n − bj (n)qj − i6=j bi (n)qi = cj (n)dj − i6=j bi (n)qi but dj |qi for i 6= j .
i=0
Pr
Pr
Hence dj | (n − i=0 bi (n)qi ), it follows that a| (n − i=0 bi (n)qi ).
2. Let xs00 ...xsrr be a monomial of degree n + ka. In particular its degree can be written n + hdi . Hence
s0 q0 + ... + sr qr = n + hdi . Since di |qj for i 6= j , then n = si qi + ldi . However, by denition of bi (n),
b (n)
b (n)
we must have si > bi (n). Hence, xs00 ...xsrr is a multiple of x00 ...xrr . It follows that:
S0
=
=
S(Q)(a)
M
S(Q)(n)ai
i∈N
=
=
M
b (n)
x00 ...xbrr (n) S(Q)
n−
r
X
i∈N
i=0
M
b (n)
x00 ...xbrr (n) S(Q) (aφ(n))ai
!
bi (n)qi
ai
i∈N
L
0
Thus S 0 ∼
=
i∈N S(Q) (aφ(n))ai as S -graded modules. This induces an isomorphism of sheaves
0
OP roj(S 0 ) (n) ∼
O
(aφ(n))
(recall
that ∼ is functorial). But since P(Q) ∼
= P roj(S )
= Proj(S 0 ), we
0(a) ∼
0
∼
get OP(Q) (n) = (aφ(n)). Now since S
= S(Q ) through the isomorphism f : P(Q) → P(Q0 ),
OP(Q) (aφ(n)) corresponds to OP(Q0 ) (φ(n)).
3.3 Pathologies
Let's recall some well-known properties of the standard projective space.
Theorem 3.12. The r-dimensional projective space Pr has the following properties:
1. For all n ∈ Z, OPr (n) is an invertible sheaf.
2. For all n ∈ N\{0}, OPr (n) is ample.
3. The graded ring homomorphism S(n) ⊗S(Q) S(m) → S(Q)(n + m) where S = k[x0 , ..., xr ] induces
the following isomorphism of sheaves: OPr (n) ⊗OPr OPr (m) ∼
= OPr (n + m).
^
f ⊗O r OPr (n).
4. For any graded S -module M and for all n ∈ Z, M
(n) ∼
=M
P
Proof. See [Har77] p.117 proposition 5.12.
None of the above properties hold for all wps. We shall give counter examples for each of the 4
properties given in the above theorem.
1. Recall that an invertible sheaf is a locally free sheaf of rank 1. Clearly, as Pr is connected, so is
P(Q) = Pr /µQ as a topological quotient space. Let Q = (1, 1, 2), consider the sheaf OP(Q) (1) restricted
to the open set D+ (x2 ). It corresponds to the ring
a
a
S(Q)(1)(x2 ) =
|
a
∈
S(Q)(1)
=
|
a
∈
S(Q)
2k
2k+1
xk2
xk2
But OP(Q) restricted to the open set D+ (x2 ) corresponds to the ring
a
S(Q)(x2 ) =
|
a
∈
S(Q)
2k
xk2
The monomial xa0 xb1 xc2 has degree a + b + 2c. As a consequence S(Q)(1)(x2 ) as an S(Q)(x2 ) -module is
x2
minimaly generated by x0 and x1 . Moreover, it isn't a free module as we have x02 .x1 − xx1 x2 0 .x0 = 0.
2. The above example shows that OP(Q) (1) isn't invertible, hence not ample. But we might have an
invertible sheaf which isn't ample. Take Q = (3, 5), we have P(Q) ∼
= P1 which induces an ismorphism
∼
of sheaves OP(Q) (n) = OP1 (φ(n)). Take n = 2, then φ(n) = −1. Hence OP(Q) (2) ∼
= OP1 (−1) which
isn't ample.
∼ OP1 , OP(Q) (4) =
∼ OP1 and OP(Q) (6) ∼
3. Take Q = (2, 3), we get OP(Q) (2) =
= OP1 (1). Clearly,
∼ OP1 OP1 (1).
OP1 ⊗OP1 OP1 =
10
^
f ⊗O
4. Take M = S(Q)(4), M
(2) = OP(Q) (6) ∼
OP(Q) (2) = OP(Q) (4) ⊗OP(Q)
= OP1 (1) and M
P(Q)
∼
OP(Q) (2) = OP1 .
A more sophisticated way to study wps is to see them as quotient stacks. The study of their coherent
sheaves will result in only studying the Gm -equivariant coherent sheaves on the open set U = An+1 \{0}.
This way, we can show that the above properties remain true for wps as quotient stacks.
4 Weighted Projective Spaces as Quotient Stacks
The main references for this section is [Ful] and [LMB00].
4.1 Short description of stacks
There are two ways to think of what an algebraic stack is:
1. A category bred in groupoid X
2. An atlas (or groupoid presentation) R ⇒ U where R and U are schemes, and R determines an
equivalence relation on U
Denition 4.1.
A category bered in groupoids over a base category S is a category X with functor
p : X → S satisfying the following two axioms:
1. For every morphism f : T → S in S , and object s in X with p(s) = S , there is an object t in X ,
with p(t) = T , and a morphism φ : t → s in X such that p(φ) = f .
2. Given a commutative diagram in S
U
h
g
T
/S
f
with φ : t → s in X mapping to f : T → S , and η : u → s in X mapping to h : U → S , there is a
unique morphism γ : u → t in X mapping to g : U → T such that η = φ ◦ γ
u
∃!γ
η
t
φ
/ s
Axiom (2), applied with U = T , h = f , and g = IdT , implies that the object t with φ : t → s
guaranteed by the rst axiom is determined up to canonical isomorphism. So Axiom (1) can
be regarded as saying that pullbacks of objects exist, and Axiom (2) then tells us that these
pullbacks are unique up to canonical isomorphism.
For an object S in S , we denote by XS the subcategory of X whose objects map to S , and whose
morphisms map to the identity map IdS . It follows from Axiom (2) that every morphism in XS is an
isomorphism. (Given a morphism φ : t → s in XS , take u = s, and η = Ids to get an inverse for φ.)
Recall that a groupoid is a category in which every morphism is an isomorphism. This explains the
terminology category bered in groupoids
For a such a category to be a stack, it has to satisfy two descent (sheafy) properties in étale topology
that we won't detail here. Moreover, to get an algebraic stack, it has to satisfy some additional
representability conditions. For a precise denition see [Ful] or [LMB00].
All of our schemes will be given over a xed scheme Λ, in particular we can take it to be Spec(k). In
the category of Λ-Schemes, the bre product exists and is unique up to isomorphim. For two schemes
X and Y (over Λ), we denote their bre product (over Λ) by X × Y . The projection morphisms are
p1 : X × Y → X and p2 : X × Y → Y .
11
Denition 4.2.
We say that an ane group scheme G acts (on the right) on a scheme X if there
exists a morphism X × G → X that induces a right action X(U ) × G(U ) → X(U ) of the group G(U )
on the set X(U ) for every open U .
Example 4.3. Let X be an scheme and G an ane group scheme. The projection morphism p1 :
X × G → X is called the trivial action and gives an action of G on X .
Denition 4.4.
The (right) trivial torsor over an scheme S is the scheme E = S × G together with
the trivial action of G on S : E → S and the projection map on the rst factor E × G → E . More
generally a (right) G-torsor over a scheme S is a pair of schemes (X, S) together with a morphism
X → S and a right action of G on X which is locally trivial in the given topology on S . Given any
morphism f : T → S , we have a pullback f ∗ E = T × E over T . It induces two maps, one to T given
by T × E → T and one to the action of G on X given by T × E → E .
Remark 4.5. The locally trivial property required for G-torsors is given by the existence of a covering
map f : T → S such that the pullback f ∗ E is isomorphic to the trivial G-torsor on T in particular
T ×E ∼
= T × G.
Example 4.6.
Suppose that an ane group scheme G acts on the right on a scheme X . There is
a category denoted [X/G], whose objects are right G-torsors E → S together with an equivariant
morphism from E → X . A morphism of [X/G] is a morphism from a G-torsor E → S to a G-torsor
E → S which is given by a morphism S 0 → S and a G-equivariant morphism E → E such that the
diagram
/E
E0
´
´
is cartesian and that
S0
/S
E0
/E
´
X
commutes.
The functor is given by p : [X/G] → S, (E, S, E → S, E × G → E, E → X) 7→ S and (S 0 → S, E 0 →
E) 7→ S 0 → S where S is the category of Λ-schemes.
Denition 4.7.
Fix Λ = Spec(k). Take R = k[x0 , ..., xn ] a graded k -algebra with deg(xi ) = qi for
some positive integers. Let An+1 = Spec(R) and let 0 ∈ An+1 correspond to the irrelevant ideal.
The multiplicative ane scheme Gm action on An+1 \{0} is given by the graduation given to R. We
can dene a category bred in groupoid [An+1 \{0}/Gm ] called the weighted projective stack or more
commonly weighted projective space and is denoted by P(Q) where Q = (q0 , ..., qn ). The canonical
functor is given by p : [An+1 \{0}/Gm ] → S, (E, S, E → S, E × G → E, E → X) 7→ S where S is the
category of Λ-schemes and (S 0 → S, E 0 → E) 7→ S 0 → S . In particular, the weighted projective stack
is an algebraic stack.
Denition 4.8.
Given a scheme E , the canonical category bred in groupoid over the category of
schemes denoted by X is the category of E -schemes. Its objects are (S, S → E) where S is a scheme
and S → E a morphism of schemes. Its morphisms are S → T such that the following diagram
commutes
/T
S
E
Denition 4.9. A groupoid scheme, or algebraic groupoid, consists of two schemes and ve morphisms,
satisfying several properties. One has a scheme U , a scheme R, two morphisms s : R → U and
t : R → U , a morphism e : U → R, a morphism m : Rt ×s R → R (where Rt ×s R denotes the ber
product R ×U R constructed from the two maps t and s), and a morphism i : R → R, satisfying the
ve properties listed below.
12
/R
e
1. The composites U
R:
/ U and R
s
s
/U
/ R are the identity maps on U and
e
R
U
IdU
e
R
U
/U
s
IdR
s
/R
e
2. If pr1 and pr2 are the two projections from Rt ×s R to R, then s ◦ m = s ◦ pr1 and t ◦ m = t ◦ pr2 :
/R
m
Rt ×s R
pr1
R
/U
s
s
/R
pr2
Rt ×s R
m
R
/U
t
t
3. (Associativity) The maps m ◦ (IdR × m) and m ◦ (m × IdR ) from Rt ×s Rt ×s R to R are equal:
Rt
ÖRÖR
s
t
/ Rt ×s R
IdR ×m
s
m
m×IdR
Rt ×s R
/R
m
4. (Unit) The maps m ◦ (e ◦ s, IdR ) and m ◦ (IdR , e ◦ t) from R to R are equal to the identity on R:
R
(e◦s,IdR )
/ R t ×s R
R
(IdR ,e◦t)
/ Rt ×s R
m
m
( R
IdR
( R
IdR
5. (Inverses) i◦i = IdR , s◦i = t and (therefore) t◦i = s, and m◦(IdR , i) = e◦s and m◦(i, IdR ) = e◦t:
i
/R
IdR
% R
R
i
R
i
t
/R
s
% R
R
(IdR ,i)
s
U
/ Rt ×s R
m
e
/R
R
(i,IdR )
m
t
U
/ R t ×s R
e
/R
We have two important examples, just as we have described it when we viewed a stack as a category
bred in groupoids.
Example 4.10.
identity maps.
Any scheme X determines a transformation groupoid X ⇒ X ; whose morphisms are
Example 4.11.
Suppose that an ane group scheme G acts on the right on a scheme X . It provides
an atlas X × G ⇒ X for the quotient stack [X/G] dened by
s(x, g) = x; t(x, g) = x.g; m((x, g), (x.g, h)) = (x, g.h); e(x) = (x, eG ); i(x, g) = (x.g; , g)
Now that we presented the weighted projective space as a quotient stack, we would like to understand
the category of the wps (quasi)coherent sheaves describing them in terms of another equivalent category,
just as we do for the regular projective space when considered as a scheme.
It can be shown that for a quotient stack [X/G], the category of coherent sheaves is equivalent to
the category of Gm -equivariant sheaves on X due to eective descent for strictly at morphisms of
algebraic stacks (see [LMB00], Thm. 13.5.5). Applying this fact to weighted projective spaces, we
obtain that
m
Coh (P(Q)) ∼
An+1 \{0}
= Coh G
Q
m
Where Coh G
An+1 \{0} is the category of Gm -equivariant sheaves on An+1 \{0}.
Q
13
4.2 Coherent sheaves on wps
Firstly let us recall some denitions. We shall follow a close presentation given in [AKO08]. A
homomorphism f : M → N of graded modules is said to be of degree d if f (Mn ) ⊂ Nn+d for all n ∈ Z.
If no degree is specied, the homomorphism is of degree zero. We introduce the following notation for
a given nitely generated graded commutative noetherian connected k -algebra A. Let
Mod(A) = the category of right A-modules
Gr(A) = the category of graded right A-modules
The morphisms in Gr(A) are the graded module homomorphims of degree zero.
Denition 4.12.
L Let M be a graded A-module. An element x ∈ M is said to be torsion if ∃s ∈
Z such that x. i>s Ai = 0. The torsion elements of M forms a graded A-submodule denoted by
τ (M ). If τ (M ) = 0 then M is torsion-free. If τ (M ) = M then M is torsion. We can form a full
subcategory
Tors(A) = the full subcategory of Gr(A)of torsion modules
We can then make the following quotient category
QGr(A) = the quotient category Gr(A)/ Tors(A)
In the case where we are only interested in nitely generated graded modules over A, we shall denote
the above categories in lower case: mod(A), gr(A), tors(A) and qgr(A).
The quotient category above exists because Tors(A) is a Serre subcategory of the abelian category
Gr(A). This means that it is a full subcategory and that for a given short exact sequence in Gr(A)
0
/ M0
/M
/ M 00
/0
We have the property that M ∈ Tors(A) if and only if M 0 , M 00 ∈ Tors(A). Similarly tors(A) is a Serre
subcategory of gr(A).
We can give an explicit description of tors(A) when furthermore A is N-graded as follows:
1. Obj(qgr(A)) = Obj(gr(A))
0
0
2. Hom qgr(A) (M̃ , N˜) = lim
−→M Hom gr(A) (M , N )
where M 0 runs through all the submodules of M such that M/M 0 is a torsion module. We denoted by
f ∈ qgr(A) the image of M ∈ gr(A) through the canonical functor π : gr(A) → qgr(A).
M
It is clear that the intersection of the categories qgr(A) and Tors(A) in the category QGr(A) coincides
with tors(A). In particular, the category QGr(A) contains qgr(A) as a full subcategory. Sometimes it
is convenient to work with QGr(A) instead of qgr(A).
Proposition 4.13. [AKO08] Let A be a nitely generated N-graded commutative noetherian k-algebra.
Then the category of (quasi)coherent sheaves on the quotient stack P(Q) is equivalent to the quotient
category qgr(A) (resp. QGr(A)).
Proof. The category of (quasi)coherent sheaves on the stack P(Q) is equivalent to the category of
Gm -equivariant (quasi)coherent sheaves on An+1 \{0}. The category of (quasi)coherent sheaves on
An+1 \{0} is equivalent to the quotient of the category of (quasi)coherent sheaves on An+1 by the subcategory of (quasi)coherent sheaves with support on 0: This is also true for the categories of Gm equivariant sheaves. But the category of (quasi)coherent Gm -equivariant sheaves on An+1 \{0} is just
the category gr(A) (resp. Gr(A)) of graded modules over A; and the subcategory of (quasi)coherent
sheaves with support on 0 coincides with the subcategory tors(A) (resp. Tors(A)). Thus, we obtain
that Coh(P(Q)) is equivalent to the quotient category qgr(A) = gr(A)/ tors(A) (and Qcoh(P(Q)) is
equivalent to QGr(A) = Gr(A)/ Tors(A)).
So what this proposition says is that to study the coherent sheaves on the weighted projective stack, it
suces to understand the quotient category of graded nitely generated modules over A (or quotient
category of graded modules over A for quasicoherent sheaves).
14
5 Homological Projective Duality
The main references for this section is [BBHR09] and [Huy06].
5.1 Triangulated categories
We rst briey recall what a triangulated category is.
Denition 5.1. First x an additive category D, the structure of a triangulated category on D is given
by an additive equivalence
T : D → D,
the shift functor, and a set of distinguished triangles
A
/B
/C
/ A[1]
where A[1] = T (A) subject to four axioms.
We invite the reader to [GM96] for an indepth reading.
Denition 5.2.
A subcategory D0 ⊂ D of a triangulated category is a triangulated category if D0
admits the structure of a triangulated category, such that the inclusion is exact. In the case where
D0 is a full subcategory, it is a necessary and sucient condition that D0 is invariant under the shift
functor and for any distinguished triangles
A
/B
/C
/ A[1]
in D with A, B ∈ D0 there exists C 0 ∈ D0 such that C 0 ∼
= C.
Denition 5.3.
A full triangulated subcategory i : D0 ,→ D is called admissible if the inclusion functor
has a right adjoint functor π : D → D0 . Hence we have functorial isomorphisms Hom D (i(A), B) ∼
=
Hom D0 (A, π(B)) for all A ∈ D0 and B ∈ D.
Denition 5.4.
D
0⊥
The (right) orthogonal complement of a subcategory D0 ⊂ D is the full subcategory
of all objects C ∈ D such that Hom D (B, C) = 0 for all B ∈ D0 .
Denition 5.5.
A collection of objects Ω in a triangulated category D is a spanning class of D if for
all B ∈ D we have:
1. If Hom(A, B[i]) = 0 for all A ∈ Ω and all i ∈ Z, then B ∼
=0
2. If Hom(B[i], A) = 0 for all A ∈ Ω and all i ∈ Z, then B ∼
=0
5.2 Derived categories
Throughout the rest of the document, A will denote an abelian category.
Denition 5.6.
We dene the category of complexes Kom(A) whose objects are complexes A• in A
and whose morphisms are morphisms of complexes (chain maps). This category is an abelian category
since A is abelian.
Remark 5.7. We can identify A in Kom(A) since from any object A we can dene a complex A• with
A0 = A and Ai = 0 for all i > 1. A becomes a full subcategory of Kom(A). We can dene in Kom(A)
a shift functor that has an inverse however this won't be enough to obtain a structure of a triangulated
categories.
Denition 5.8.
A morphism of complexes f : A• → B • is said to be a quasi-isomorphism if for all
i ∈ Z, the induced map in cohomology is an isomorphism H i (f ) : H i (A• ) → H i (B • ).
The central idea behind using derived categories is to make all these quasi-isomorphisms become
isomorphisms.
15
Denition 5.9.
Two morphisms of complexes f, g : A• → B • are called homotopically equivalent
and write f ∼ g if there exists a collection of morphisms in A dened by hi : Ai → B i−1 such that
i
f i − g i = hi+1 ◦ diA + di−1
B ◦h
di−1
A
...
hi
i−2
dB
...
x
/ B i−1
diA
/ Ai
di+1
A
/ ...
hi+1
f i gi
/ Bi x
di−1
B
/ Ai+1
diB
/ ...
Let's gather some elementary results about this relation dened on morphisms of complexes.
Proposition 5.10. i) Homotopy equivalence is an equivalence relation on
Obj(Kom(A)).
Hom A (A, B) for A, B ∈
ii) Homotopically trivial morphisms form an `ideal' in the morphisms of Kom(A).
iii) Two homotopic equivalent morphisms induce the same map in cohomology.
iv) if f : A• → B • and g : B • → A• are such that f ◦ g ∼ IdB • and g ◦ f ∼ IdA• then f and g are
quasi-isomorphisms and H i (f )−1 = H i (g).
Denition 5.11.
We dene the homotopy category of complexes K(A) to be a category whose objects
are the objects of Kom(A) and whose morphisms are HomK(A) (A• , B • ) = HomKom(A) (A• , B • )/ ∼.
Remark 5.12. This denition makes sense, as the composition is well-dened thanks to the properties
of the homotopy equivalence dened above.
Just as we can localise rings given a multiplicative set, we are going to do the same with the class of
quasi-isomorphims to be able to inverse them and get an isomorphism in the corresponding category
which will be the derived category for a given abelian category.
Consider these two diagrams of morphisms of complexes denoted respectively by f /φ and g/ψ :
D•
C•
f
A•
x
g
φ
&
B • A•
ψ
x
&
B•
where φ and ψ are quasi-isomorphisms.
They are said to be equivalent if there exists a commutative diagram in K(A)
E•
β
C•
x
α
&
f
x
A• r
φ
ψ
D•
g
&,
B•
with α and β quasi-isomorphisms.
We say that f /φ and g/ψ are equivalent and we can prove that such a relation is an equivalence
relation.
Denition 5.13. The derived category D(A) of A is the category whose objects are the objects of
K(A) (that is, they are complexes of objects of A), and whose morphisms are equivalence classes [f /φ]
of diagrams.
Remark 5.14. The composition of two such equivalence classes of diagrams is given by [g/ψ] ◦ [f /φ] =
[(g◦β)/(φ◦α)], which makes sense because the above construction is independent of the representatives.
16
5.3 Duality
Kuznetsov introduced in [Kuz07] a notion of homological projective duality for smooth algebraic varieties in dual projective spaces, a homological extension of the classical projective duality. He gave
some examples. The idea is to provide another example of such a duality with wps.
Denition 5.15. A semiorthogonal decomposition of T is a sequence of full subcategories A1 , ...
,An in T such that Hom T (Ai , Aj ) = 0 for i > j and for every object T ∈ T there exists a chain
/ ...
/ T1
/ T0 = T such that the
/ Tn−1
of morphisms 0 = Tn
cone of the morphism Tk → Tk−1 is contained in Ak for each k = 1, 2, ..., n.
Assume that X is an algebraic variety with an eective line bundle OX (1) on X .
Denition 5.16.
A Lefschetz decomposition of the derived category Db (X) is a semiorthogonal decomposition of D (X) of the form
b
=
Db (X) = hA0 , A1 (1), ..., Ai=1 (i 1)i and 0 ⊂ Ai−1 ⊂ Ai−2 ⊂ ... ⊂ A1 ⊂ A0 ⊂ Db (X)
where 0 ⊂ Ai−1 ⊂ Ai−2 ⊂ ... ⊂ A1 ⊂ A0 ⊂ Db (X) is a chain of admissible subcategories of Db (X).
Remark 5.17. The bounded derived category Db (X) is the derived category of the bounded complexes
of the abelian category of coherent sheaves on X
In order to state what the homological projective duality is, we will set few notations introduced by
Kuznetsov.
We need a Lefschetz decomposition of the derived category Db (X), a vector space of global sections
V ∗ ⊂ Γ(X, OX (1)) such that the number of terms in the Lefschetz decomposition i < dim V = N .
Assume also that V ∗ generates OX (1), this induces a regular morphism f : X → P(V ).
Denote by X1 ⊂ X × P(V ∗ ) the universal hyperplane section of X , that is the zero locus of the
OX (1) OP(V ∗ ) (1). Denote also by C the right
canonicalb line bundle
orthogonal to be the subcategory
A0 D (P(V ∗ )), A1 (1) Db (P(V ∗ )), ..., Ai=1 (i 1) Db (P(V ∗ )) in Db (X1 ).
=
Assume moreover that C is geometrical, in the sense that it is equivalent to the derived category of
coherent sheaves on some algebraic variety Y .
The category C is a module category over the tensor category Db (P(V ∗ )). For F ∈ C and G ∈
Db (P(V ∗ )), we have F ⊗ f ∗ G ∈ C . Denote by Φ : Db (Y ) → Db (X1 ), the composition of the two
functors Db (Y ) → C and C → Db (X1 ). We assume that the module category structure is geometrical,
in the sense that there exists an algebraic morphism g : Y → P(V ∗ ) such that there is an isomorphism
of bifunctors
Φ(F ⊗ g ∗ G) ∼
= Φ(F ) ⊗ f ∗ G where F ∈ Db (Y ), G ∈ Db (P(V ∗ ))
Finally, denote by Q = {(v, H) ∈ P(V ) × P(V ∗ ) | v ∈ H} then we have
Y ×P(V ∗ ) X1 = Q(X, Y ) = (X × Y ) ×P(V )×P(V ∗ ) Q
Denition 5.18.
An algebraic variety Y with a projective morphism g : Y → P(V ∗ ) is called Homologically Projectively Dual to f : X → P(V ) with respect to a Lefschetz decomposition given above,
if there exists an object E ∈ Db (Q(X, Y )) such that the functor ΦE = Φ : Db (Y ) → Db (X1 ) is fully
faithful and gives the following semiorthogonal decomposition
Φ(Db (Y )), A1 (1) Db (P(V ∗ )), ..., Ai=1 (i 1) Db (P(V ∗ ))
=
The theorem 6.3 given in [Kuz07], is followed by some concrete examples in section 9. The rst problem
in my PhD thesis would be to provide other such concrete examples of homological dual varieties by
the mean of weighted projective spaces and thoroughly study this phenomenon for this particular case.
17
References
[AKO08]
Denis Auroux, Ludmil Katzarkov, and Dmitri Orlov, Mirror symmetry for weighted projective planes and their noncommutative deformations, Ann. of Math. (2) 167 (2008), no. 3,
867943. MR 2415388 (2009f:53142)
[BBHR09] Claudio Bartocci, Ugo Bruzzo, and Daniel Hernández Ruipérez, Fourier-Mukai and Nahm
transforms in geometry and mathematical physics, Progress in Mathematics, vol. 276,
Birkhäuser Boston Inc., Boston, MA, 2009. MR 2511017 (2010k:14020)
[BR86]
Mauro Beltrametti and Lorenzo Robbiano, Introduction to the theory of weighted projective
spaces, Exposition. Math. 4 (1986), no. 2, 111162. MR 879909 (88h:14052)
[CLS11]
David A. Cox, John B. Little, and Henry K. Schenck, Toric varieties, Graduate Studies in
Mathematics, vol. 124, American Mathematical Society, Providence, RI, 2011. MR 2810322
(2012g:14094)
[Dol82]
Igor Dolgachev, Weighted projective varieties, Group actions and vector elds (Vancouver,
B.C., 1981), Lecture Notes in Math., vol. 956, Springer, Berlin, 1982, pp. 3471. MR 704986
(85g:14060)
[Dol03]
, Lectures on invariant theory, London Mathematical Society Lecture Note Series,
vol. 296, Cambridge University Press, Cambridge, 2003. MR 2004511 (2004g:14051)
[Ful]
William Fulton, What is a stack?, to appear.
[GM96]
Sergei I. Gelfand and Yuri I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996, Translated from the 1988 Russian original. MR 1438306 (97j:18001)
[Gro61]
Alexander Grothendieck, Éléments de géométrie algébrique (rédigés avec la collaboration
de jean dieudonné): Ii. étude globale élémentaire de quelques classes de morphismes, Publications Mathématiques de l'IHES 8 (1961), 5222.
[Har77]
Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York, 1977, Graduate Texts
in Mathematics, No. 52. MR 0463157 (57 #3116)
[Huy06]
D. Huybrechts, Fourier-Mukai transforms in algebraic geometry, Oxford Mathematical
Monographs, The Clarendon Press Oxford University Press, Oxford, 2006. MR 2244106
(2007f:14013)
[Kuz07]
Alexander Kuznetsov, Homological projective duality, Publ. Math. Inst. Hautes Études Sci.
(2007), no. 105, 157220. MR 2354207 (2008k:14043)
[LMB00]
Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik
und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in
Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics],
vol. 39, Springer-Verlag, Berlin, 2000. MR 1771927 (2001f:14006)
[Rei02]
Miles Reid, Graded rings and varieties in weighted projective space, Manuscript, M. Reid's
Web page (www.maths.warwick.ac.uk/miles/surf/more/grad.pdf) (2002).
[Rot95]
Joseph J. Rotman, An introduction to the theory of groups, fourth ed., Graduate Texts in
Mathematics, vol. 148, Springer-Verlag, New York, 1995. MR 1307623 (95m:20001)
[RT11]
Julius Ross and Richard Thomas, Weighted projective embeddings, stability of orbifolds, and
constant scalar curvature Kähler metrics, J. Dierential Geom. 88 (2011), no. 1, 109159.
MR 2819757
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