Direct Images of Locally Constant Sheaves on Complements to Plane Line Arrangements
Direct Images of Locally Constant
Sheaves on Complements to Plane
Line Arrangements
©Iara Cristina Alvarinho Gonçalves, Stockholm, 2015
Address: Matematiska Institutionen, Stockholms Universitet, 106 91 Stockholm E-mail address: iara@math.su.se
ISBN 978-91-7649-203-1
Printed by E-Print AB 2015, Stockholm
Abstract
On the complement X = C2−Sn
i =1Li(where each Liis a line passing through
the origin) to a plane line arrangementSn
i =1Li⊂ C2, a locally constant sheaf
of complex vector spaces is given by a multi-indexα ∈ Cn. Using the de-scription of Mac-Pherson and Vilonen ([11] and [12]) we obtain a criterion for the irreducibility of the direct image R j∗Lα as a perverse sheaf, where
Sammanfattning
På komplementet till ett centralt linjearrangemang i komplexa planet X = C2−∪L
i(där Li, i = 1,...,n är linjer genom origo och ∪Lisjälva
linje-arrange-manget), är en lokalt konstant kärve av komplexa vektorrum av dimension 1 Lαbestämd av ett multi-indexα ∈ Cn. Med hjälp av MacPhersons och Vilo-nens beskrivning av perversa kärvar ([11] och [12]) ges i denna avhandling ett kriterium i termer avα för när R j∗Lαär irreducibel som pervers kärve, där j : X → C2är den kanoniska inklusionen.
Contents
Abstract v Sammanfattning vii 1 Introduction 11 2 Introduction to Machinery 13 2.1 Notation . . . 132.1.1 Sheaves and Operations . . . 13
2.1.2 Abelian Categories . . . 14
2.1.3 The Monodromy Representation of Locally Constant Sheaves . . . 15
2.2 Perverse sheaves . . . 18
2.2.1 The category of complexes and the derived category . . 18
2.2.2 Definition of perverse sheaves . . . 19
2.3 Locally constant rank 1 sheaves onC∗ . . . 23
2.3.1 Stalks . . . 24
2.3.2 The cohomology groups of the mapping cone . . . 25
2.3.3 Kernel and cokernel for the morphismθ . . . 27
2.3.4 Kernel and cokernel for the morphismθR . . . 28
3 Direct Images on Lines Configurations inC2 31 3.1 The CategoryC(F,G; T ) . . . . 31
3.1.1 Kernel . . . 32
3.1.2 Cokernel . . . 35
3.1.3 Irreducible Objects . . . 37
3.2 The equivalence betweenM(X ) andC(F,G; T ) . . . . 42
3.2.1 Construction . . . 43
3.2.2 Main Result . . . 47
1. Introduction
The aim of this work is to analyze the irreducible factors in a composition series of a perverse direct image of a rank 1 locally constant sheaf,La. To
defineLa, consider a central line arrangementSni =1Li⊂ C2. A locally
con-stant sheaf (of complex vector spaces) is given by a representation ofπ1(C2−
Sn
i =1Li) which means that to each multi-index a = (a1, . . . , an) there is
asso-ciated such a sheafLa(aiis the result of the action of a loop around Li). The
main result is the establishment of a criterion for the irreducibility in terms of the action of the fundamental group:
Theorem 1.0.1. The perverse sheaf R j∗La, where j :C2− ∪ni =1Li → C2, is
irreducible if, and only if, both of the following conditions are satisfied: • ai6= 1, for all i = 1, . . . , n;
• Πni =1ai6= 1.
The chapter 2, after some basic preliminaries, starts with a presentation of the classification of locally constant sheaves of rank 1 onC2−Sn
i =1Liand
give correspondence to multi-indices inCn(section 2.1.3). Similarly, to each
α ∈ C, there is a locally constant sheafLαonC∗= C − {0}.
In the section 2.2.2,the definition of perverse sheaves (according to [2]) is given. After that, considering the inclusion j :C∗→ C, a computation of j!Lαand j∗Lαis presented. This is expanded to the computation of the
irreducible factors of R j∗Lα.
The results for the case of sheaves inC∗are known, but they appear as a background and introduction for the results presented in the next chapter, concerning perverse direct image of a rank 1 locally constant sheaf inC2− S
iLi, i = 1,...,n. These results represent a new contribution to this subject.
The chapter 2 starts with a description of a certain algebraically defined categoryC(F,G; T ), introduced by R. Macpherson and K. Vilonen ([11]). The objects of this category are the pairs (A, B ) together with diagrams
F A TA //
G A
B
where A and B are objects of categoriesAandB(respectively), F , left ex-act, and G, right exex-act, are functors between these two categories and T is natural transformation from F to G. In [11] and [17] the notions of kernel, cokernel and irreducible objects ofC(F,G; T ) are described without all de-tails of the proofs. In the present work a detailed description of this notions is given and, in order to do so, some assumptions not explicitly present in the original article had to be made.
Then, a result from [12], that establishes the equivalence between the categoryC(F,G; T ) and the category of perverse sheaves, is applied to a con-crete situation. Consider the spaceC2 and let S be a closed stratum: S = S
iLi, i = 1,...,n, where each Li is a line passing through the origin. By
noticing that the action ofπ1(C2− S) on a one-dimensional vector space is
given by a linear character, we have a correspondence between local sys-tems inC2− S and multi-indices a = (a1, . . . , an) inCn. LetLa denote the
rank 1 locally constant sheaf associated to a. The main result consists on deciding in which conditions the direct image ofLavia the maps
C2 − S j 1 −→ C2− {0} j 2 −→ C2
is irreducible. The first step is to translate the information of pj∗iLa into a diagram ofC(F,G; T ). The second step is the computation of the direct image of a locally constant sheafLainC2−{S} (the complement of the closed
stratum), and then of the direct image of the perverse sheafpj∗1LainC2−{0}. The final result is reached by gluing these structures coming from different strata. Finally, the theorem presented earlier is stated and proved.
2. Introduction to Machinery
2.1 Notation
In this section we will introduce the conventions and notations we are going to use, as well as give some results that we need.
2.1.1. Sheaves and Operations As general reference for sheaves on
topo-logical spaces we will use [7]. We will only consider sheaves for which the sections are complex vector spaces.
Let X be a topological space, U an open subset of X and F the closed complement. We will denote by j the inclusion of U in X and by i the in-clusion of F in X . LetSh(X ) (respectivelySh(U ) andSh(F )) denote the
category of sheaves on X (respectively on U and F ).
Then there are the following basic functors that relate the categories above:
• j!:Sh(U ) →Sh(X ): extension by 0 (exact);
• j∗:Sh(X ) →Sh(U ): restriction (exact, also denoted by j!); • j∗:Sh(U ) →Sh(X ): direct image (left exact);
• i∗:Sh(X ) →Sh(F ): restriction (exact);
• i∗:Sh(F ) →Sh(X ): direct image (exact, also denoted by i!);
• i!:Sh(X ) →Sh(F ): sections with support in F (left exact).
It is easy to verify that between these functors we have the relations (see[7], [2], pp.43):
j∗i∗= 0, i∗j!= 0, i!j∗= 0
2.1.2. Abelian Categories Recall that an abelian category is a category in
which all hom-sets are abelian groups and the composition is bilinear, all finite limits and colimits (in particular kernels and cokernels) exist and have certain good properties, some of which we will describe below. As general reference on abelian categories we use [15].
We will later be concerned with kernels and cokernels, in abelian cate-gories different from module catecate-gories, and present then their definition. LetCbe an abelian category, A, B ∈Cand f the morphism A → B.
Definition 2.1.1. We say that k : S → A is a kernel for f when firstly f ◦k = 0,
and secondly, k has the property that any morphism g : M → A, with f ◦ g = 0, factors uniquely through k(see the diagram below).
S 0 '' k A f //B M ∃!g0 OO g >> 0 77
Now we describe the dual notion of cokernel:
Definition 2.1.2. We say that p : B → C is a cokernel for f when firstly p ◦
f = 0, and secondly, k has the property that any morphism h : B → N with h ◦ f = 0 factors uniquely through p.
C ∃!h0 A f// 0 77 0 '' B p ?? h N
Definition 2.1.3. We say that the morphism f : A → B is an injection (or
monomorphism) if ker ( f ) = 0 and a surjection (or epimorphism) if cok( f ) = 0. If f is a injection we say that A is a subobject of B , and if f is a surjection that B is a quotient object of A.
Remark 2.1.4. By abuse of notation we are going to use the notions of kernel
and cokernel both for objects and morphisms.
Also important for abelian categories are the notions of image and co-image ([15], pp.129):
Definition 2.1.5. The image of a morphism f : A → B is a subobject g : B0→
B of B , such that f factors through g and, furthermore, if g1: B1 → B is
another subobject of B , such that f factors through g1, then B0 ⊆ B1, i.e.
there exists a morphism h : B0→ B1such that g1◦ h = g .
Definition 2.1.6. The co-image of a morphism f : A → B is a quotient object
q : A → C such that there is a map fC: C → B with f = fC◦ q. Furthermore,
for any epimorphism h : A → D, for which there is a map fD: D → B with
f = fD◦ h, there is a unique map t : D → C such that both q = t ◦ h and
fD= fC◦ t .
Recall that the fundamental defining property of an abelian category is:
cok(ker ( f ) → A)−→ ker (B → cok( f )),∼= (2.1.2) which for categories of modules over a ring reduces to one of the isomor-phism theorems: A/ker ( f ) ∼= i m( f ).
In addition we will have use for some results concerning morphisms and exact functors that will be applied later. Recall that monomorphisms are morphisms whose kernel is 0, and epimorphisms morphisms whose coker-nel is 0.
Lemma 2.1.7. (see [15], pp.7) Let X −→ Yf −→ Z be morphisms in an abeliang
categoryC:
1. if f and g are monomorphisms, then g ◦ f is also a monomorphism; 2. if g ◦ f is a monomorphism, then f is a monomorphism;
3. if f and g are epimorphisms, then g ◦ f is also an epimorphism; 4. if g ◦ f is a epimorphism, then g is an epimorphism.
The following definition is also standard (see [9], pp.201): letAandBbe abelian categories. The additive functor T :A→Bis called left exact, when it preserves kernels, right exact when it preserves cokernels, and exact if it preserves both kernels and cokernels.
2.1.3. The Monodromy Representation of Locally Constant
Sheaves A general reference for this section is [7], in particular pp.
223-225. We recall that we only consider sheaves with sections that are finite-dimensional vector spaces overC. There is a direct equivalence between complex representations of the fundamental group and locally constant
sheaves, through the monodromy representation on stalks. This representa-tion is defined in the following way. First recall that a locally constant sheaf on a simply connected and path-connected space is constant. Let thenLbe a locally constant sheaf on X . For a fix point x0∈ X , consider an element γ ∈ π1(X , x0), represented by a pathγ(t),t ∈ [0,1]. ClearlyLx0 =Lγ(t), 0 ≤ t < 1
and similarlyLx0=Lγ(t), 0 < t ≤ 1. Then v ∈Lx0may first be identified with
w ∈Lγ(1/2)using the first identification, and then this element with an ele-ment v0∈Lx0 using the second identification. The elementγv := v
0will not
depend on the homotopy class ofγ(t), the map v 7→ v0is linear and hence this gives us a representation ofπ1(X , x0) onLx0.
Proposition 2.1.8. Let X be a path-connected topological space andπ1(X , x0) be its fundamental group with base point x0. The monodromy representation ofπ1(X , x0) on the stalkLx0, defines an equivalence between the category of local systemsLon a space X and the category of finite dimensional complex representations of the fundamental group of X .
We will now introduce the sheaves that we are going to work with later.
2.1.3.1 The complement to a point in the complex line
In this case the fundamental group ofC∗= C − {0} is
π1(C∗) = {nγ,n ∈ Z} ∼= Z,
whereγ is a loop going around 0 once starting in x0= 1.
Hence the proposition means the following:
Corollary 2.1.9. Locally constant sheavesLof rank 1 onC∗are classified up to isomorphism by the elementα ∈ C, such that for the monodromy represen-tation
γe = αe (2.1.3)
whereL1= Ce is the stalk at 1 ∈ C∗.
Denote byLαa rank 1 sheaf corresponding to the representation (2.1.3). The sections ofLαon an open neighborhood of the type U = U (x,r ) = {z ∈ C∗: |z − x| < r } will be
Lα(U ) = ½
Ce, if π1(U ) = 1 ⇐⇒ 0 ∉ ¯U
ker (γ − 1), if π1(U ) = Z ⇐⇒ 0 ∈ ¯U
This sheaf occurs naturally. Consider the multi-valued analytic function defined by a power xβ,β ∈ C. It defines a sheafEβwith sections
Eβ(U ) := {Axβ, A ∈ C, such that Axβis defined as a single-valued analytic function on U }.
Then a quick calculation of the monodromy of the analytical function
xβ, gives that
Eβ∼=Lα, whereα = e2πiβ.
2.1.3.2 The complement in the plane to a line configuration
Now consider a union of lines Li, i = 1,...,n, through the origin in C2, given
by equations zi = aix1+ bix2= 0. Let β = (β1,β2, ...,βn) ∈ Cn, be a
multi-index, and define the corresponding function
zβ= zβ1 1 ...z
βn
n ,
which is a multi-valued analytic function defined in the complement to the union ∪ni =1Li, and hence defines, as in the previous example, a locally
con-stant rank 1 sheaf,Eβ, with sections
Eβ(U ) := {Azβ, A ∈ C, such that Azβis defined as a single-valued analytic function on U }.
The fundamental group of the complement to the line configuration has been known for a long time (see [12]). Choose a complex plane M that is transversal to all the Li and does not contain the origin, and a base point
x0∈ M. Then the intersection Li∩ M is a single point Pi, and we letΓi be
a loop in M around Pi, starting and ending in x0. These loops generate the
fundamental group, which has the following presentation:
π1(C2\ ∪ni =1Li) = 〈Γ1, . . . ,Γn〉/R,
where R is the group generated by the (cyclic) relations Γ1Γ2. . .Γn= Γ2. . .ΓnΓ1= ΓnΓ1. . .Γn−1.
Corollary 2.1.10. Locally constant sheavesLof rank 1 onC2\∪ni =1Liare
clas-sified up to isomorphism by the elementα = (α1, ...,αn) ∈ Cn, such that for the
monodromy representation
Γie = αie (2.1.4)
whereLx0= Ce is the stalk at x0.
Proof. Given a locally constant sheafLwe know that it corresponds to the representation of the fundamental group on the stalkLx0 = Ce. Since the
and by commutativity of multiplication of complex numbers the relations in
R do not add any extra information: Γ1. . .Γne = ... = Γi. . .Γi −1e = Πni =1αie.
This last argument also shows that conversely, any multi-indexα ∈ Cn de-fines a representation of the fundamental group, and hence a locally con-stant sheaf.
We denote the locally constant sheaf correspondent toα ∈ CnbyLα. By calculating the monodromy of the multivalued analytical function xβ, it is easy to see that
Eβ∼=Lα,αi= e2πiβi, i = 1,...,n.
2.2 Perverse sheaves
In this section we present a very general view of some of the definitions and results that we will need, that will later be applied in a more concrete way. Our general reference is [2].
2.2.1. The category of complexes and the derived category We will
con-sider the category of complexes K (A) (respectively, K+(A)) with morphisms up to homotopy built from objects in the categoryAof sheaves on a topo-logical space. LetQbe the class of K (A) (K+(A)) consisting of all quasi-isomorphisms. The category obtained by formally inverting the classQof quasi-isomorphisms is the (bounded below) derived category D(A) (D+(A)) ofA(see [7], pp. 430-435).
Any bounded below complex A•admits a quasi-isomorphism f : A•→
I•into a bounded below complex of injective objects, I•(an injective resolu-tion of A•, that is a right resolution whose all elements are injective)(see [7], pp.40). By working with injective resolutions the derived category becomes more manageable. The definition of derived functors is an example.
Definition 2.2.1. ([16], pp.14) Suppose that F :A→Bis a left exact func-tor between abelian categories Aand B. Let A• be a complex of A and A•∼= I (A•) an injective resolution. Then define RF (A•) := F (I (A•)). This establishes a functor
RF : D+(A) → D+(B)
called a right derived functor of F . The i-th right derived functor
RiF : D+(A) →B is defined by RiF = Hi◦ RF .
If A ∈ K (A), the shifted complex A[1] is defined to be the complex that in degree n is An+1, and has differentials that are those of A multiplied by −1. If
f : A → B is a chain map then the mapping cone is the complex M := A[1]⊕B
with differential dM= µ −dA 0 f dB ¶ . It sits in a sequence A→ B → M,f (2.2.1) which can be continued by a map M → A[1] → B[1] → ..., and therefore often written as
A→ B → Mf +1→ ...
We will denote the mapping cone as above by C one( f ).
Any sequence in the derived category that is homotopy equivalent to a sequence of type (2.2.1) in K (A) is called a distinguished triangle. This descends to the derived category D(A), where again a distinguished triangle is a sequence that is isomorphic to one of the type (2.2.1).
These two features, the translation functor A 7→ A[1] and the set of dis-tinguished triangles, make D(A) into a triangulated category (see the axioms that have to be satisfied in [2]).
2.2.2. Definition of perverse sheaves We start by giving an overview of the
main elements related to the definition of perverse sheaves. The defini-tion of perverse sheaves can be presented in different ways according to the properties we want to explore, but every definition demands a certain level of abstraction and a rather complex previous technical work. We are going to present the definition from [2].
Let X be topological space and D = DX the derived category of sheaves
on it. The definition of perverse sheaves is based on the concept of a t-structure, which comprehends a triangulated category D, two full subcat-egoriespD≤0 andpD≥0 and perverse truncation functors,τ≤0
p : D →pD≤0
andτ≥0p : D →pD≥0. The category of perverse sheavesM(X ) in the case
D = DX is a full subcategory of D, corresponding to its heart (or coeur):
M(X ) :=pD≤0(X ) ∩pD≥0(X ).
M(X ) turns out to be an abelian category and therefore we can find ker-nels and cokerker-nels. Let f : Q•→ R•be a map of perverse sheaves. The ker-nel and cokerker-nel are defined through the perverse truncation functors as:
ker ( f ) = (τ≤−1p (C one( f ))•)[−1] and coker (f ) = τ≥0p (C one( f ))•. InM(X ) is
also defined a cohomological functor H0:= τ≤0p τ≥0p .
We will now describe this more precisely. Note that the most important of the above concepts is the concept of truncation–using it the others may be defined.
2.2.2.1 t-category
Definition 2.2.2. ([4], pp.125) A t-category is a triangulated category D, with
two strictly full subcategories D≤0and D≥0of the category D, such that, by setting D≤n= D≤0[−n] and D≥n= D≥0[−n], one has the following proper-ties:
• Hom(X•; Y•) = 0 if X•∈ D≤0and Y•∈ D≥1; • D≤0⊂ D≤1and D≥1⊂ D≥0;
• for any object X•∈ D, there is a distinguished triangle
A•→ X•→ B• +1−−→ A•[+1] with the object A•in D≤0and the object B•in D≥1.
We say that (D≤0, D≥0) is a t-structure over D. Its heart is the full subcategory
C = D≤0∩ D≥0.
In the above definition A•:= τ
≤0X•and B•:= τ≥1X•. More generally,
truncation functors may be defined asτ≤nX•:= (τ≤0(X•[−n]))[n] and τ≥nX•:= (τ≥0(X•[−n]))[n]. For these functors we have the following
propo-sition.
Proposition 2.2.3. ([2], pp.29) The inclusion of D≤n in D admits a right ad-jointτ≤n, and the inclusion of D≥nadmits a left adjointτ
≥n. For every X•in D, there exists a unique morphism d ∈ Hom1(τ≥1X•,τ≤0X•) such that
τ≤0X•→ X•→ τ≥1X• d−→
is a distinguished triangle. Apart from isomorphism, this is the unique dis-tinguished triangle (A•, X•, B•) with A•in D≤0and B•in D≥1.
The simplest example of a t-structure on DX is the one induced byτ≥0
andτ≤0being the ordinary truncation operators on complexes. In this case the heart is just the category of sheaves on X . By using shifted trunca-tion operators one gets a t-structure with a heart that is the the category of sheaves on X , but now shifted to a fix degree. We call this the shifted trivial
t-structure. The idea of using perversities is now to shift differently according
2.2.2.2 Perversities
We will only use a certain perversity, but it makes the construction clearer if we introduce the concept more generally.
Definition 2.2.4. Let X be a topological space andΣ a stratification of X . A
perversity is a map p :Σ → Z (to each stratum an integer is associated).
Example 2.2.1. Some important perversities are given by:
m = (0,0,1,1,2,2,3,...), lower middle perversity (the one we are going to
use in our work);
n = (0,1,1,2,2,3,3,...), upper middle perversity.
2.2.2.3 Gluing t-categories using perversities
(For the following construction see [2], pp. 48-58.) The truncation in the category DX has to be understood inductively from the truncationτU,
asso-ciated to a t-structure in the category DU, andτF in DF, where U ⊂ X is an
open subset and F = X \U the closed complement. Consider (DU≤0, D≥0U ) a t-structure on DU and (D≤0F , DF≥0) a t-structure on DF. Then define:
D≤0:= {K ∈ D | j∗K ∈ D≤0U and i∗K ∈ D≤0F }
D≥0:= {K ∈ D | j∗K ∈ DU≥0and i!K ∈ D≥0F }
Proposition 2.2.5. (D≤0, D≥0) is a t-structure over D.
We say that we glue the t-structures over U and F .
Now, given a perversity, the idea is to build the t-structure inductively, starting with U1= S0and F1= S1(S0, S1∈ Σ), and using the t-structures on DU1 and DF1, that are trivial t-structures that are shifted according to the
perversities pU1 and pF1, to build a t-structure on X \ X2= S0∪ S1. In the
next step one considers X \ X3= X \X2∪(X2\ X3), where one, by the previous
inductive step, has a t-structure on DU2, with U2= X \ X2and uses the trivial
t-structure on DF2, with F2= X2\ X3= S2shifted according to pS2.
We now describe this equivalently, just using the strata.
Lemma 2.2.6. The subcategorypD≤0(X ) (resp.pD≥0(X )) of D(X ) is the
sub-category given by the complexes K•(resp. K•in D+(X )) such that for each
stratum S, being iSthe inclusion of S in X , one has Hn(iS∗K•) = 0 for n > p(S)
(resp. Hn(iS!K•) = 0 for n < p(S)).
Definition 2.2.7. The categoryM(p, X ) of p-perverse sheaves over X is the categorypD≤0∩pD≥0(the heart of the t-structure (pD≤0,pD≥0)).
2.2.2.4 Truncation
We just describe the truncation for the case of two strata, an open U and a closed F . Let j : U → X and i : F → X . To these strata we assume that we have perversities, where pU is the perversity of U and pF the perversity
of F . ForK•on U ,τU
≥i(K•) in DU is given by the shifted trivial truncation τU
≥i:= τ≥i +pU(K•). ForL•on F ,τ
F
≥i(L•) is given byτ
F
≥i:= τ≥i +pF(L•) in DF.
Similarily with the other truncations.
To get the perverse truncation which is associated to the glued t-structure: • start with an object X•in D, and choose Y•in a way so that we have
the distinguished triangle
(Y•, X•, j∗τU≥1j∗X•);
• then we define A•such that the following is also a distinguished trian-gle
(A•, Y•, i∗τF≥1i∗Y•);
• finally we define B•so that we have the third distinguished triangle (using the composition A•→ Y•→ X•as the first map)
(A•, X•, B•).
Clearly all of this constructions may be done using mapping cones. It is not difficult to prove from Propositions 2.2.2 and 2.2.3 that A•is in D≤0and
B•in D≥1, and so the perverse truncation is determined as A•= τ≤0p X•and
B•= τ≥1p X•.
2.2.2.5 The relation between the perverse truncation and the kernel and cokernel
In the case of the trivial truncation it is easy to relate it with the kernel and cokernel of a morphism in the heart, i.e. a map between sheaves. There is an equivalent result for the perverse truncation and thus this gives a way of computing the kernel and cokernel for a morphism of complexes in the derived category.
Let Z•be the mapping cone of the morphism f : Q•→ R•, between two perverse sheaves. Then we have maps
Q•→ R•→ Z• and
Following the results of [2], pp.27-31, we have that:
ker ( f ) = (τ≤−1p Z•)[−1] , coker ( f ) = τ≥0p Z•,
or more precisely the kernel of f is the composition (τ≤−1p Z•)[−1] → Z•[−1] → Q•, and the cokernel is the composition
R•→ Z•→ τ≥0p Z•.
For our purposes we rewrite this as
ker ( f ) = (τ≤−1p Z•)[−1] = (τ≤0p Z•[−1][1])[−1] = τ≤0p (Z•[−1]) (2.2.2) coker ( f ) = τ≥0p Z•= ³ τ≥1 p (Z•[−1]) ´ [1] (2.2.3) In particular, one should note that the kernel and cokernel are possible to calculate in terms of the truncation operators.
Remark 2.2.8. Let Q•and R•be complexes that are different from zero only in degree 0. Therefore the mapping cone is the complex Z•: Q•→ R•. Let us apply the definitions of kernel and cokernel given above to this Z•, but using the normal truncation, which corresponds to the perversity p ≡ 0.
Z•[−1] will be a complex different from zero in degrees 0, 1 and the result of τ≤0(Z•[−1]) will be precisely ker ( f ) (in degree 0). The result of τ≥1(Z•[−1])
is, according to the definition of (trivial) truncation functors, coker ( f ), in degree 1. After the last shifting,¡
τ≥1(Z•[−1])¢ [1], the result will be coker (f )
in degree 0. Hence the truncation functor gives the kernel and cokernel for the trivial t-structure.
2.3 Locally constant rank 1 sheaves on
C
∗The subject of the thesis is the irreducibility of certain direct images of lo-cally constant sheaves in the derived category. We will now present the eas-iest such result, using the previous machinery to prove (most of ) the follow-ing well known result. The irreducibility follows from general results in [2], but we will instead prove it as a consequence of the algebraic description of the category of perverse sheaves in the next chapter (see Example 3.2.1).
Recall the sheavesLαonC∗from Section 2.1.3. Consider the inclusion
Proposition 2.3.1. 1. Ifα 6= 1 then
j!Lα∼= R0j∗Lα∼= R j∗Lα are isomorphic and irreducible perverse sheaves in DC.
2. Ifα = 1, and soC:=L1is the constant sheaf onC∗, there are short exact sequences of perverse sheaves
S[−1] ,→ j!C R0j∗C (2.3.1) and
R0j∗C,→ R j∗C S[−1]. (2.3.2)
FurthermoreS[−1] and R0j∗Care irreducible perverse sheaves.
The proof proceed in several steps. First we will (for completeness sake) calculate the sheaves involved through describing their stalks, then we will use the technique for calculating kernels and cokernels in the preceding sec-tion from [2].
2.3.1. Stalks We denote the constant sheafL1onC∗byC.
Lemma 2.3.2. We have that:
1. Ifα 6= 1 then j!Lα∼= R j∗Lα. 2. (R0j∗C)p= C, f or al l p ∈ C. (R1j∗C)p= ½ 0, i f p 6= 0 C, i f p = 0
Proof. If p 6= 0, clearly (j!Lα)p= ( j∗Lα)p. Since j!is exact it has no higher
derived images. Also, if p 6= 0, there is a sequence of contractible open neigh-borhoods p ∈ V ⊂ C∗that converge to p, and so lim
V 3pH1(V,Lα) = 0. Hence
(R1j∗C)p = 0. Finally, for p = 0, we calculate Hi(V \ {0},Lα), i ≥ 0, for a
disk V ⊂ C centered around 0. Define the two open contractible subsets,
U1= V \ R+and U2= V \ R−, such that:
• V \ {0} = U1∪U2;
• The intersection U1∩U2has two components O1and O2;
U1and U2is then a cover for V \ {0}. Let U = {U1,U2}. We have that Hi(V − {0},Lα) = ˇHi(U ,Lα),
where the ˘Cech complex (see [3], pp. 89-112), is as in the diagram below. Let
s and t be sections ofLαon U1and U2respectively. The ˘Cech differential, d , from the global sections of U1and U2to the global sections of O1and O2,
looks as follows: Γ(U1,Lα) ⊕ Γ(U2,Lα) d s t (s, t ) d0 Γ(O1,Lα) ⊕ Γ(O2,Lα) (s, s) (−t,−γt) (s − t, s − γt)
sinceΓ(U1∩U2,Lα) = Γ(O1,Lα) ⊕ Γ(O2,Lα). It is easy to observe that each
one of the global sections is equal to the vector spaceC.
SinceC = Γ(U2,Lα), we can identify it withΓ(O1,Lα) and withΓ(U1,Lα).
In the relation betweenΓ(U2,Lα) andΓ(O2,Lα) however, we have to take
in account the fact of completing a loop around 0 and so the effect of the generatorγ of the fundamental group of C∗in the differential above. Clearly, ifα 6= 1, γt = αt 6= t and then d0is an isomorphism, and both the kernel and cokernel are zero. So Hi(V − {0},Lα) = 0 for all i , and so also (Rij
∗Lα)0=
0. This concludes the proof of (1). Ifα = 1, then both the kernel and the cokernel of d0are isomorphic toC, and this concludes the proof of (2).
Remark 2.3.3. In the case where i : {0} → C is the inclusion of a locally closed
subspace andC{0}is the constant sheaf on {0}, the result of applying i!= i∗
toC{0}is the skyscraper sheaf,S, at 0:
(i!(C{0}))p= (S)p=
½
0, p ∈ C∗ C, p = 0
The results above then says that, while whenα 6= 1, j!Lα∼= R j∗Lα, for
α = 1, we have the short exact sequence
j!C,→ R0j∗C S. (2.3.3)
2.3.2. The cohomology groups of the mapping cone We just know R j∗L•α,
by the cohomology of its stalks, but we can observe that it is a complex dif-ferent from zero only in degrees 0 and 1 represent it as:
R j∗L•α: . . . → 0 → E0→ E1→ 0 → . . . (2.3.4)
• the (cohomological) degree zero case, for sheaves and morphismθ
j!C−→ Rθ 0j∗C→ MT• (2.3.5)
• the full cohomological case, for complexes of sheaves and morphism
θR
j!C−→ R jθR ∗C→ M• (2.3.6)
The mapping cone MT• is a complex different from zero only in degrees -1 and 0, given by the sequence:
MT•: . . .−−→ 0d−3 −−→ jd−2 !C
d−1
−−→ R0j∗C−→ 0d0 −→ . . .d1
The mapping cone M•is a complex different from zero in degrees -1,0 and 1, given by the sequence:
M•: . . . → 0 → j!C d−1 −−→ E0 d 0 −→ E1 d 1 −→ 0 → . . .
The sequence of complexes (2.3.6) can now be described by the follow-ing diagram: 0 //0 //0 0 // OO E1 // OO E1 OO j!C // OO E0 // OO E0 OO 0 OO //0 OO // j!C OO 0 OO //0 OO //0 OO
Recalling the result of the preceding section, and using the long coho-mological sequence, associated to the mapping cone, we get the following description.
Lemma 2.3.4. The cohomology groups of the mapping cone are
Hn(MT•) = ½ S , n = 0 0, n 6= 0 Hn(M•) = ½ S , n = 0,1 0, n 6= 0,1
2.3.3. Kernel and cokernel for the morphismθ In this section we will prove
the first part of Proposition 2.3.1 (2), in the following form. Observe that there is a quasi-isomorphism MT•[−1] ∼=S[−1], as a consequence of Lemma 2.3.4. Hence sequence (2.3.1) follows from the following result.
Lemma 2.3.5. The morphismθ : j!C→ R0j∗Chas kernel MT•[−1] and
coker-nel 0.
Proof. Our complex MT•[−1] is
MT•[−1] : ...−−→ 0d−2 −−→ jd−1 !C d0 −→ R0j∗C d1 −→ 0 d 2 −→ . . . The cohomology groups are (see Lemma 2.3.4):
Hn(MT•[−1]) = ½ S
, n = 1 0, n 6= 1
We want to apply the construction described in Section 2.2.2.5 to our com-plex MT•[−1]. So, X will correspond to MT•[−1], the open set is U = C∗, with
p(C∗) = 0 and the closed set is F = {0}, with p({0}) = 1 (see Section 2.2.2.2). We start by determining Y•. As j∗ is an exact functor we know that, for every complexF•, Hi( j∗F•) = j∗Hi(F•), for all i ∈ Z. Being obvious
that Hi(MT•[−1])p= 0 for every p 6= 0, we have that: j∗Hi(MT•[−1]) ∼= 0 ⇔
Hi( j∗MT•[−1]) ∼= 0 ⇒ j∗MT•[−1] ∼= 0 ⇒ τ≥1( j∗MT•[−1]) ∼= 0. Therefore Y•=
MT•[−1].
Now we want to construct A•. First we observe that i∗τ≥1p i∗MT•[−1] =
i∗τ≥2i∗M•
T[−1], because i∗(MT•[−1]) is in DFand p({0}) = 1. By applying i∗
to the complex MT•[−1], we get that i∗( j!C) = 0 (see relations in (2.1.1)) and i∗( j∗C) = i∗C=S, therefore: i∗(MT•[−1]) : ...−−→ 0d−1 d 0 −→S−→ 0d1 −→ . . .d2 and Hn(i∗MT•[−1]) = ½ S , n = 1 0, n 6= 1
From this we get: τ≥2i∗MT•[−1] = 0 ⇒ i∗τ≥2i∗MT•[−1] = 0. Therefore A•=
Y•= X•, implying that B•= 0.
Finally we conclude that the kernel A•is just the complex MT•[−1] and that the cokernel B•is zero. Thus the morphismθ is a surjection:
R0j∗C //0 //0 //0 j!C // OO j!C θ // OO R0j∗C OO //0 OO
Note that this is precisely the opposite result to what we get for the trivial t-structure, when the kernel is taken as for morphisms between sheaves and it is the cokernel that isS, and the map is injective.
2.3.4. Kernel and cokernel for the morphismθR In this section we will
prove the remaining part of Proposition 2.3.1 (2). Again since by the quasi-isomorphism MT•[−1] ∼=S[−1] (Lemma 2.3.4), the exactness of the sequence (2.3.2) will be shown to follow from the next result.
Lemma 2.3.6. For the morphismθR : j!C,→ R j∗C, the kernel is M•[−1] and the cokernel is B•[1], where B•is the complex given by B•= i∗τ≥2i∗M•[−1], with cohomology groups
Hn(B•) = ½ S
, n = 2 0, n 6= 2
Proof. Our complex M•[−1] comes from the morphism (2.3.6):
M•[−1] : ...−−→ 0d−2 −−→ jd−1 !C d0 −→ E0 d1 −→ E1 d2 −→ 0 d 3 −→ . . . The cohomology groups are (see Lemma 2.3.4):
Hn(M•[−1]) = ½ S
, n = 1,2 0, n 6= 1,2
Like before X•= M•[−1]. The method used in the previous situation to determine Y•can be replicated: since j∗τ≥1p j∗M•[−1] = 0, we get the same
result, Y•= M•[−1]. Note that
i∗τ≥1p i∗M•[−1] = i∗τ≥2i∗M•[−1].
Then A•is defined by the distinguished triangle
(A•, Y•, i∗τ≥1p i∗Y•) = (A•, M•[−1],i∗τ≥2i∗M•[−1]),
and consequently, by uniqueness of distinguished triangles, we must have
B•= i
∗τ≥2i∗M•[−1]. We now calculate the cohomology groups of B•.
The functors i∗ and i∗ are exact functors. Applying these functors to Hn(M•[−1]) will not affect it because they represent, respectively, the re-striction to the closed set, {0}, and the direct image to the setC; given that
Hn(M•[−1])p= 0 for p 6= 0, the cohomology groups remain the same, being
only affected by the truncation functor. Then we get
Hn(B•) = ½ S
, n = 2 0, n 6= 2
Recalling that coker (θR) = τ≥1p (M•[−1])[1] (see (2.2.3)) we can finally say
that coker (θR) = B•[1].
To understand ker (θ) we observe the long exact sequence of cohomol-ogy groups associated to the distinguished triangle (A•, M•[−1],B•):
. . . → H−1(B•) → H0(A•) → H0(M•[−1]) → H0(B•) → H1(A•) → → H1(M•[−1]) → H1(B•) → H2(A•) → H2(M•[−1]) → H2(B•) → → H3(A•) → H3(M•[−1]) → ... ⇐⇒ . . . → 0 → H0(A•) → 0 → 0 → H1(A•) →S→ 0 → H2(A•) →S→ →S→ H3(A•) → 0 → ...
By observation of the sequence above, we have: • Hn(A•) = 0, for n ≤ 0;
• Hn(A•) = 0, for n ≥ 3;
• H1(A•)uH1(M•[−1]) therefore H1(A•) =S; • H2(M•[−1])uH2(B•), implying that H2(A•) = 0. Then we can present the cohomology groups for A•:
Hn(A•) = ½ S
, n = 1
0, n 6= 1
We observe that, apart from a shifting in the degrees, A•has the same cohomology groups as the mapping cone MT• (see Lemma 2.3.4).
To conclude the proof of Proposition 2.3.1, we want to understand the relation between the two morphisms whose kernels and cokernels we have computed. The relation between the morphismsθ and θR, and their kernel
and cokernels, are described in the following diagram:
ker (θ) d−1 // η−1 j!C θ // η0 R0j∗C d1 // η1 0 η2 A• ∂−1 // << j!C θ R // R j∗C ∂1 //B•[1] (2.3.7)
We claim that ker (θ) : j!C→ R0j∗Cis also a kernel for the morphismθR
and therefore A•= ker (θR) = ker (θ).
• the morphismη0is an isomorphism;
• the compositionη0◦ d−1is injective;
• the composition∂−1◦ η−1will also be injective;
• the morphismη−1is injective (recall Lemma 2.1.7).
From this follows thatη−1is a quasi-isomorphism, since the cone of the morphismη−1is C one•(η−1) : . . . → 0 → j!C t−1 −−→ R0j∗C⊕ j!C t0 −→ R0j∗C t1 −→ 0 → . . . which is then easily seen to have zero cohomology in all degrees. Conse-quently ker (θ) and A•are quasi-isomorphic, and ker (θ) is isomorphic, in the derived category, to ker (θR). As a consequence, j!C/kerθ ∼= R0j∗C,→ R j∗C equals the kernel of R j∗C B•[1]. This means that the sequence
3. Direct Images on Lines
Configurations in
C
2
In this chapter we will apply the results of [10], [11], [12] and [17], We will start by describing an algebraically defined category,C(F,G; T ), which gives a description of the category of perverse sheaves, and a detailed proof about how its irreducible objects are constructed ([17]). Then, a particular case of aC(F,G; T ) category is presented. Following [10] and [11], we consider the situation of a topological stratified space and the case of a complex analytic manifold. Finally we use the more specific results of [12] and rewrite them in the case when the singular space is given by S =S
iLi, i = 1,...,n, where
each Liis a line passing through the origin. For this example we determine
the conditions in which the direct images of locally constant rank 1 sheaves, with a given action ofπ1(C2−SiLi), are irreducible perverse sheaves.
3.1 The Category
C
(F,G; T )
The definition of the category of perverse sheaves as a subcategory of DX is
less suited for calculations and for our work it will be more clear to use a dif-ferent construction. We will describe a categoryC(F,G; T ) that is equivalent to the category of perverse sheaves,M(X ). The idea is to show that the cat-egory of perverse sheaves is glued from the catcat-egory of perverse sheaves on an open strata of maximal dimension and the category of perverse sheaves on the complement.
We present the initial definitions as stated in [11]: consider two cate-goriesAandB, two functors F and G fromAtoB, and a natural transfor-mation T from F to G. Symbolically F,G :A→Band F −→ G. We defineT the categoryC(F,G; T ) to be the category whose objects are pairs (A; B ) ∈
Ob jA× Ob jBtogether with a commutative triangle
F A TA // m G A B n ==
and whose morphisms are pairs (a, b) ∈ MorA× MorBsuch that F A TA // m !! F a G A G a B n == b F A0 TA0 // m0 !! G A0 B0 n0 == (3.1.1)
Proposition 3.1.1. (see [11], pp. 405-407) IfAandBare abelian categories and if F is right exact and if G is left exact then the categoryC(F,G; T ) is
abelian and the functors taking (A, B ) 7→ A and (A,B) 7→ B fromC(F,G; T )
toAandBare exact.
In the following subsection we are going to present the construction of the kernel and cokernel for a morphism (a, b) in the categoryC(F,G; T ). We are also going to prove that the objects presented actually represent a kernel and a cokernel, according to the definition. This forms an expanded version of the descriptions on [11].
3.1.1. Kernel In an abelian category every morphism has a kernel and a
cokernel. We are going to show that for a morphism (a, b) inC(F,G; T ) (as in Diagram (3.1.1)) the kernel is
F (ker (a)) // && G(ker (a)) ker (b) 88 (3.1.2)
(We will shortly define the maps in this diagram.)
In the categoryA, consider the morphism A−→ Aa 0. There is a natural inclusion ker (a) // 0 (( A a // A0 Applying a functor L: L(ker (a)) // L(0)=0 $$ L(A) La //L(A0) (3.1.3)
we observe that there is a canonical morphism: L(ker (a)) −→ ker (L(a)). If
L is a left exact functor then (see Section (2.1.2)), this morphism is an
iso-morphism, thus G(ker (a)) ∼= ker (G(a)).
Hence we have the following commutative diagram
F (ker (a)) j // T ker (F (a)) i // T F (A) F a // T m F (A0) T m0
G(ker (a)) //ker (G(a)) //G(A) G a //G(A0)
B b // n hh B0 n0 bb
First we want to show that the morphisms described in Diagram (3.1.2) actually exist. One part is to see that m ◦ i ◦ j factors through ker (b) → B (as described in the following diagram). To do this it is enough to check that
b ◦ m ◦ i ◦ j = 0, by the definition of kernels. F (ker (a))i ◦j // F (A) m ker (b) //B b //B0
From the diagram we can see that it is possible to define a morphism
F (ker (a)) → ker (b): according to Diagram (3.1.3), we can say that the
com-position F a ◦ i ◦ j = 0. And if we compose it with m0, clearly the result is also zero. On the other side, and because the diagram is commutative, we know that b ◦ m ◦ i ◦ j = 0 as well. And in this way we described the wanted correspondence.
It remains to prove that the map ker (b) → G(A) factors through G(ker (a)) ∼
= ker (G(a)). We look at a detail of the diagram
ker (b) // 0 ++ )) B n //G(A) G a //G(A0) ker (G(a))? OO
If we apply the composition n0◦ b to ker (b) the result will, obviously, be zero. Since n0◦ b = G a ◦ n (recall (3.1.1)) we first notice that, by the property of kernel of G(a), the morphism n, between ker (b) and G(A), can be fac-torized through ker (G(a)). As ker (G(a)) ≈ G(ker (a)), we have shown that
n : ker (b) → G(A) factors through G(ker (a))). We saw that the diagram of
compositions (3.1.2) is valid.
Using the definition, we are going to prove that (3.1.2) is a kernel. Let
X , Y , Z and W be objects in a categoryC. We say that (W, h) is a kernel for the morphism f : X → Y if h : W → X is a monomorphism with f h = 0 and such that any morphism g : Z → X with f g = 0 factors through h, g = hg0.
Z 0 $$ g // g0 X f //Y W h >>
Let (A00, B00) be an object inC(F,G; T ) and (a00, b00) : (A00, B00) → (A,B) a morphism inC(F,G; T ) such that:
A00 a−→ A00 B00 b−→ B00 and (A00, B00) (a 00,b00) // (0,0) )) (A, B ) (a,b) //(A0, B0) Let iF A, iG Aand iBdenote the inclusions:
• iF A: F (ker (a)),→ F (A)
• iG A: G(ker (a)),→ G(A)
• iB: ker (b),→ B
In order to make the diagrams understandable, we show only the objects that include the pairs (A, B ), (A00, B00), (ker (a), ker (b)) and their relations.
F (A00) // "" (( F a00 && G(A00) (( G a00 '' F (A) // && G(A) B00 :: ++ b00 .. B 99 F (ker (a)) // && EE G(ker (a)) FF ker (b) 88 EE
We observe that (F a00,G a00, b00) factor through (iF A, iG A, iB), all the
con-ditions are satisfied and we proved that (3.1.2) is the kernel in this case.
3.1.2. Cokernel We are going to show that, for the present situation, the
cokernel is the pair (cok(a), cok(b)) together with the commutative triangle
F (cok(a)) TA // mc && G(cok(a)) cok(b) nc 88 (3.1.4)
First we want to prove that the maps in the following digram exist:
G(A0) //G(cok(a)) F (A0) 33 TA0 ;; m0 ## F (cok(a)) TA 77 mc '' B0 // n0 OO cok(b) nc OO
All horizontal maps exist canonically. TA0and TAare clear to exist by the
description of the category. It remains to explain mc and nc.
To describe the cokernel we have to follow a process similar to the one of the previous subsection. AsAandBare abelian categories we have that:
• every morphism has a kernel and a cokernel;
• given a morphism M−→ N , in the canonical factorization M → coi m( f )f → i m( f ) → N , the map coi m( f ) → i m( f ) is an isomorphism.
From morphism a : A → A0we get the sequence
ker (a),→ A → coim(a)−→ i m(a) → A≈ 0→ cok(a), giving rise to the diagram
F (ker (a)) γ // T F (A) δ// // T m F (i m(a)) ε // T F (A0) ζ // // T m0 F (cok(a)) T G(ker (a)) γ 0 //G(A) δ0// //
G(i m(a)) ε0 //G(A0) ζ
0 //G(cok(a)) B b // n OO B0 n0 99 // //cok(b)
Remark 3.1.2. The morphismζ is a surjection because F is a right exact
functor and therefore takes surjections to surjections (see Section 2.1.2). Having a close look at a detail of the diagram
F (A) δ // // m 0 ** F (A0) m0 ζ// // %% F (cok(a)) B b // 0 33 B0 ψ // //cok(b)
we observe that the compositionψ◦b ◦m = 0. Because the diagram is com-mutative we know thatψ ◦ m0◦ δ is also 0. Then, naturally it exists a mor-phism 0 form F (A) to cok(b) and hence, by the definition of cokernel, the morphism between F (cok(a)) and cok(b) exists. The map mc is then
de-fined.
Now we want to consider the morphism nc. Observe the diagram
G(A) δ0 // 0 $$ G(A0) ζ0 //G(cok(a)) B b // n OO B0 ψ 0 // n0 OO 0 99 cok(b) OO
Observe thatζ0◦δ0◦n = 0. Because we are in a commutative diagram ζ0◦
n0◦b = 0 as well. So, there is a natural morphism 0 from B0to G(cok(a). And again, by the definition of cokernel, we have the desired morphism between
cok(b) and G(cok(a). The map ncis finally defined.
Using the definition, we are going to prove that Diagram (3.1.4) reprsents a cokernel. Let X , Y , Z and K be objects in category C . We say that (K , p) is a cokernel for the morphism f : X → Y if p : Y → K is a epimorphism with
p f = 0 and such that any morphism g : Y → Z with g f = 0 factors through p, such that g = g0p. X f // 0 ## Y g // p Z K g0 ??
Let (A00, B00) be an object inC(F,G; T ) and (a0, b0) : (A0, B0) → (A00, B00) a morphism inC(F,G; T ) such that:
and
(A, B ) (a,b) //
(0,0)
))
(A0, B0) (a0,b0) //(A00, B00)
Let sF A0, sG A0and sB0denote the epimorphims:
• sF A0: F (A0),→ F (cok(a))
• sG A0: G(A0),→ G(cok(a))
• sB0: B0,→ cok(b)
Due to the amount of relations represented by arrows we only present in the diagrams for the pairs (A0, B0), (A000, B00) and (cok(a), cok(b)).
F (A0) // %% )) G(A0) (( F (A00) // $$ G(A00) B00 88 ++ B << F (cok(a)) // && F a0 ?? G(cok(a)) G a0 GG cok(b) 88 b0 OO
The maps (F a0,G a0, b0) factor through (sF A0, sG A0, sB0), the relations are
the required ones, (3.1.4) is a cokernel for the described morphisms.
3.1.3. Irreducible Objects We want to study the irreducible objects in the
categoryC(F,G; T ). In order to do that we have to describe functors de-fined in [17], that allow us to go from the categoryC(F,G; T ) to the cate-goriesAandB. In the categoryC(F,G; T ) we have the restriction functors |A:C(F,G; T ) →Aand |B:C(F,G; T ) →Bwhich are defined as follows. Let
N ∈ Ob j (C(F,G; T )) be given by (A, B ) ∈ Ob j (A) × Ob j (B) and factorization
F A−→ Bm −→ G A. Then we define N |n A= A and N |B= B.
We also have an inclusion functorB→C(F,G; T ). The functor |Ahas a left and a right adjointsF andb G which are given, respectively, by:b
F A TA // i d G A A // F A TA EE F A TA // TA G A A // G A i d EE
Finally there is a functorT :b A→C(F,G; T ) which is given by
F A TA // G A A // I mTA DD
We are going to assume that the functors F and G have the property: • A 6= 0 =⇒ F A 6= 0;
• A 6= 0 =⇒ G A 6= 0.
Let (A, B ) and (A0, B0), together with the correspondent commutative tri-angles, be objects ofC(F,G; T ) and let (a, b) : (A, B ) → (A0, B0) be a morphism inC(F,G; T ). SinceC(F,G; T ) is an abelian category, the morphism (a, b) is injective if and only if ker (a, b) = 0.
The kernel of the morphism (a, b) is zero when
F (ker (a)) // && G(ker (a)) = 0 // 0 ker (b) 88 0 @@
That is, when F (ker (a)) = 0, G(ker (a)) = 0 and ker (b) = 0. Accord-ing with the assumption above, F (ker (a)) = G(ker (a)) = 0 indicates that
ker (a) = 0. Hence:
Corollary 3.1.3. The morphism (a, b) : (A, B ) → (A0, B0) is injective if and only
if the morphisms a : A → A0and b : B → B0are injective.
We are interested in the irreducible objects ofC(F,G; T ).
Definition 3.1.4. The object N ofC(F,G; T ) is irreducible if and only if one of the following conditions is satisfied:
• for every injection M → N , we have: M = N or M = 0; • for every surjection N → P, we have: P = N or P = 0. According to [17]:
Proposition 3.1.5. All the irreducible objects inC(F,G; T ) are either of the
formT (L), where L is irreducible inb A, or of the form
0 // 0 L @@ (3.1.5) where L is irreducible inB.
We give a detailed proof below, expanding the presentation in [17].
Proof. Part 1: first we are going to show that these two kind of objects are
irreducible.
First case: Suppose that there exists an object (A, B ) such that (a, b) : (A, B ) → bT (L) is an injection: F A TA // $$ F a G A G a B :: b F L TL // $$ $$ GL i m(TL) , ::
As the morphism (a, b) is injective we know that both a and b are injec-tive. In the categoryAwe have a : A → L with a injective and L irreducible, so it means that either A = 0 or A = L. Then we have two possibilities:
1.A=0 : 0 // m $$ F a 0 G a B n :: _ b F L TL // m0 $$ $$ GL i m(TL) , n0 ::
The square with vertex B,O,GL and i m(TL) is commutative. The
com-position (G a ◦ n) from B to GL is zero. Therefore the comcom-position (n0◦ b) is also zero. According to Lemma 2.1.7, point 1, (n0◦ b) is an injection and for the result to be zero we must have B = 0.
2.A=L : F L TL // m $$ F a GL G a B n :: _ b F L TL // m0 $$ $$ GL i m(TL) , n0 ::
The square with vertex F L, B, i m(TL) and F L is commutative. In this case
F a and G a are isomorphisms The composition (m0◦ F a) gives the same re-sult that the composition (b ◦ m). As m0is a surjection and F a an isomor-phism, the composition (m0◦ F a) is a surjection. By Lemma 2.1.7, point 4, we can say that b is an epimorphism, that is, a surjection. But, by hypothe-sis, b was an injection. That means that b is an isomorphism and therefore
B ≡ i m(TL).
We showed that if (a, b) : (A, B ) → bT (L) is an injection then (A, B ) = (0,0)
or (A, B ) = bT (L), proving thatT (L) is irreducible.b
Second case: Suppose that there exists an object (A, B ) such that (a, b) : (A, B ) → (0,L) is an injection: F A TA // F a G A G a B == b 0 // !! 0 L ==
As before, we know that both a and b are injective. In the categoryAwe have a : A → 0 with a injective so we can say that a ∼= 0. By other side, inB we have b : B → L with b injective and L irreducible inB, so it means that either B = 0 or B = L. Therefore we can say that
(A, B ) = (0,0) or (A,B) = (0,L) So, the object in Diagram (3.1.5) is irreducible.
Part 2: now we are going to prove that the only irreducible objects in
C(F,G; T ) are the ones described above. Suppose F A TA // m G A B n ==
is an irreducible object. Let (A, i m(m)) be an object ofC(F,G; T ) and (p, q) a morphism such that (p, q) : (A, i m(m)) → (A,B).
The morphism (p, q) is an injection because: • p : A → A is an isomorphism;
• i m(m) ∈ B and therefore the morphism q : i m(m) → B is an injection. As (A, B ) is irreducible, we know that (A, i m(m)) is either (0, 0) or (A, B ). (A, i m(m)) = (A,B): In this case the morphism (p, q) is an isomorphism and then (A, B ) is
F A TA // m ## ## G A i m(m) n ::
Let K be the kernel of the morphism i m(m)−→ G A. There is an obvi-n ous injective morphism from the object (0, K ) to (A, i m(m)). There are two possibilities:
• K 6= 0: (0,K ) would be a subobject of (A,i m(m)), different from itself or from (0, 0), contradicting the initial assumption of irreducibility of (A, i m(m));
• K = 0: this means that the morphism n is injective; therefore i m(m) ∼=
i m(TA).
In this way we prove that (A, B ) = (A,i m(TA)).
We have left to prove that A is irreducible inA. Suppose that there exists
Consider (a, b) the morphism between (A0, i m(TA0)) and (A, i m(TA)): F A0 T 0 A // m0 ## F a G A0 G a i m(TA0) n0 ;; b F A TA // m $$ G A i m(TA) n ::
Because G is a left exact functor we have G(ker (a)) ≈ ker (G(a)). As
ker (a) = 0 we know that G(ker (a)) = 0 and therefore ker (G(a)) = 0,
mean-ing that G(a) is injective.
According to Lemma 2.1.7, point 1, the composition G(a) ◦n0is injective and as the diagram is commutative n ◦ b has to be also a monomorphism. By Lemma 2.1.7, point 2, we know that b is an injective morphism.
This tells us that (a, b) is also injective. And now we got a contradiction, because (A, i m(TA)) is irreducible and we could define an injective
mor-phism from (A0, i m(TA0)) to (A, i m(TA)) with (A0, i m(TA0)) 6= (A,i m(TA)) and
(A0, i m(TA0)) 6= 0.
We conclude then that our assumption saying that the object A is not irreducible inAisn’t correct.
(A, i m(m)) = (0,0): It means that F (A) = 0, G(A) = 0 and i m(m) = 0. But, because p is an isomorphism the object (A, B ) is given by
0 //
0
B
??
where B has to be an irreducible object inB(so that the initial supposi-tion, of the irreducibility of (A, B ), holds.)
3.2 The equivalence between
M
(X ) and
C
(F,G; T )
We relate the categoriesM(X ) and C(F,G; T ) through a construction that associates perverse sheaves in an open set to local systems in the space de-fined by the closed strata. We constructM(X ) fromM(X −S), by inductionon the strata of X . An object ofM(X ) is an object A•∈M(X − S) together with a commutative triangle
F A• TA // m !! G A• B n ==
M(X ) is obtained by gluing the information collected from the smaller spaces. We are going to quickly present some fundamental concepts and give a general overview of this construction (for more details see [10], [11] and [12]).
3.2.1. Construction Now we are going to apply the theory ofC(F,G; T ) to a concrete situation. For completeness, we include the definitions of the functors F , G and T .
LetS be a stratification of X and S a closed contractible stratum of di-mension 2d . Let TSbe the tubular neighborhood of S,πS: TS→ S the
pro-jection andρSa function measuring the distance to S.
Definition 3.2.1. ([11, Definition 4.1.]) The link bundleπ : L → S and the
normal slice bundleπ0 : D → S are defined as follows: for a small enough positive valued function² : S → R,
L = {x ∈ TS| ρS(x) = ²(πS(x))}
D = {x ∈ TS| ρS(x) ≤ ²(πS(x))}.
The mapsπ and π0are restrictions ofπS.
Definition 3.2.2. ([10, Définition], pp.444) Let L be the link bundle of S, K a
closed subset of L,κ : K → L an inclusion and γ : L − K → L the inclusion of the open complement. We say that K is a perverse link bundle if:
• Ri(π ◦ κ)∗A•| K ∼= 0 for all i ≥ −(di mS)/2 and all A•∈M(X − S); • Riπ∗γ!A•| L − K ∼= 0 for all i < −(di mS)/2 and all A•∈M(X − S); Remark 3.2.3. A perverse link bundle always exists.
Let F be the functor that sends A•to R−1(π ◦ κ)∗(A•| K )[−(di mS)/2]
and G the functor that sends A•to R0π∗γ!A•| (L − K )[−(di mS)/2] and T be
