Harder–Narasimhan categories

Contents lists available atScienceDirect

Journal of Pure and Applied Algebra

journal homepage:www.elsevier.com/locate/jpaa

Harder–Narasimhan categories

Huayi Chen

Institut de Mathématiques de Jussieu, Université Paris Diderot — Paris VII, 175 rue du Chevaleret, 75013, Paris, France

a r t i c l e i n f o

Article history:

Received 3 December 2007

Received in revised form 15 April 2009 Available online 12 June 2009 Communicated by E.M. Friedlander MSC:

18E10 14H60 14G40

a b s t r a c t

Semistability and the Harder–Narasimhan filtration are important notions in algebraic and arithmetic geometry. Although these notions are associated to mathematical objects of quite different natures, their definition and the proofs of their existence are quite similar. We propose in this article a generalization of Quillen’s exact category and we discuss conditions on such categories under which one can define the notion of the Harder–Narasimhan filtrations and establish its functoriality.

©2009 Elsevier B.V. All rights reserved.

1. Introduction

The notion of theHarder–Narasimhan flagof a vector bundle on a smooth projective curve defined over a field was firstly introduced by Harder and Narasimhan [1]. LetCbe a smooth projective curve defined over a field andEbe a non-zero vector bundle onC. Harder and Narasimhan have proved that there exists a flag 0

=

E₀(E₁(E₂(

· · ·

₍E_n

=

Eof subbundles ofEsuch that each subquotientE_i

/

^Ei−1is a semistable vector bundle and that we have the inequalities of successive slopes

µ

max

(

^E

) := µ(

^E1

/

^E0

) > µ(

^E2

/

^E1

) > · · · > µ(

^En

/

^En−1

) =: µ

min

(

^E

).

The avatar of the above constructions in Arakelov geometry was introduced by Stuhler [2] and Grayson [3]. Similar construc- tions exist also in the theory of filtered isocrystals. In [4], Faltings suggested the functoriality of the Harder–Narasimhan flags, but it is not evident how to state such a functoriality, because the length of the Harder–Narasimhan flag varies.

Previously, the categorical approach for studying semistability problems has been developed in various contexts such as [5–8].

The category of vector bundles is exact in the sense of Quillen [9]. However, it is not the case for the category of Hermitian vector bundles in Arakelov geometry. We shall propose a new notion –geometric exact category– which generalizes them simultaneously. A geometric exact category is a classical exact category equipped with geometric structures. After introducing a rank function and a degree function, the existence of the Harder–Narasimhan filtration results from a supplementary condition that the maximal destabilizing geometric subobject exists. The condition is automatically satisfied if the underlying exact category is an Abelian category. This formalism includes not only standard examples, but also examples like spectrum filtration of positive definite self-adjoint operator which is classical but less often interpreted in this way.

In order to state the functoriality, we need to take into account the successive slopes in the filtration. That leads to the notion ofR-indexed Harder–Narasimhan filtration. We expect that the subobject morphism, quotient morphism, and their compositions will be compatible withR-indexed Harder–Narasimhan filtrations. However, we observe that the functoriality of the Harder–Narasimhan filtration does not follow from the existence of maximal destabilizing subobject. This is shown

E-mail address:chenhuayi@math.jussieu.fr.

0022-4049/$ – see front matter©2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.jpaa.2009.05.009

by a concrete example (see Section4 Example 2, see alsoRemark 5.10). The functoriality is actually equivalent to the validity of a slope inequality, which compares the slopes of semistable objects.

The rest of the article is organized as follows. We recall the notions ofR-filtration and of Quillen’s exact category in the second section. In the third section, we present the geometric exact categories. In the fourth section, we develop the formalism of the Harder–Narasimhan category, followed by several examples. In the fifth section, we discuss slope inequalities and the functoriality of the Harder–Narasimhan filtrations.

2. Notation and preliminaries

2.1. Filtrations in a category

We fix a categoryCwhich has an initial object. LetXbe an object inC. ByfiltrationofX, we mean a familyF

= (

^Xt

)

t∈_R of subobjects ofX, which satisfies the following conditions:

(1) (decreasing property) ifs6tare two real numbers, then the canonical monomorphismX_t

→

Xfactorizes throughX_s; (2) (separation property) for sufficiently positivet,X_tis an initial object;

(3) (exhaustivity) for sufficiently negativet,X_t

=

X;

(4) (left locally constant property) for anyt

∈

_R, there exists

δ >

0 such that, for anys

∈ (

^t

− δ,

^t

]

, the canonical morphism X_t

→

X_sis an isomorphism;

(5) (finite jump) the followingjump setis finite:

J

(

^F

) := {

s

∈

_R

| ∀

t

>

^s

,

the canonical mapX_t

→

X_sis not an isomorphism

} .

Assume thatXandYare two objects inC, andF

= (

^Xt

)

t∈_RandG

= (

^Yt

)

t∈_Rare respectively filtrations ofXand ofY. We call morphismfromF toGany morphismfinCfromXtoYsuch that, for anyt

∈

_R, the composed map X_t ^it //_X ^f //_Y factorizes throughY_t, wherei_t

:

X_t

→

Xdenotes the canonical morphism. Alternatively, we say that the morphismf is compatiblewith the filtrationsF andG. All filtrations inCand all morphisms of filtrations form a category which we denote byFil

(

^C

)

^.

Suppose that all fiber products exist inC. For any monomorphismf

:

X

→

YinCand any filtrationG

= (

^Yt

)

t∈RofY, the familyf^∗G

:= (

^Yt

×

_YX

)

t∈_Ris a filtration ofX, called theinduced filtration.

Letf

:

X

→

Ybe an epimorphism inCandF

= (

^Xt

)

t∈_Rbe a filtration ofX. For anyt

∈

_R, denote byi_t

:

X_t

→

Xthe canonical monomorphism. Assume that, for anyt

∈

_R,fi_t

:

X_t

→

Yhas an imageY_t. Then

(

^Yt

)

t∈Ris a filtration ofY, denoted byf∗F, called thequotient filtration.

2.2. Exact categories

The notion of exact category is defined by Quillen [9]. LetCbe an essentially small additive category and letEbe a class of diagrams of morphisms inCof the form

0 //_X⁰ //_X //_X⁰⁰ //₀

.

If 0 //_X^{0 f} //_X ^g //_X⁰⁰ //0 is a diagram inE, we say thatf is anadmissible monomorphismand thatgis anadmissible epimorphism. We shall use the symbol to denote an admissible monomorphism, and for an admissible epimorphism.

If 0 //_X⁰ //_X //_X⁰⁰ //₀ and 0 //_Y⁰ //_Y //_Y⁰⁰ //₀ are two diagrams of morphisms inC, we callmorphismfrom the first diagram to the second one any commutative diagram

0

(

Φ

) :

//_X⁰ //

ϕ⁰

X //

ϕ

X⁰⁰ //

ϕ⁰⁰

0

0 //_Y⁰ //_Y //_Y⁰⁰ //₀

.

We say that

(

Φ

)

^{is an}isomorphismif

ϕ

^0,

ϕ

^and

ϕ

⁰⁰are all isomorphisms inC.

Definition 2.1 (Quillen). We say that

(

C

,

E

)

^{is an}exact categoryif the following axioms are verified:

(Ex1) For any diagram 0 //_X^{0 ϕ} //_X ^ψ //_X⁰⁰ //_{0 inE,}

ϕ

^{is akernelof}

ψ

^and

ψ

^{is acokernelof}

ϕ

^. (Ex2) IfXandYare two objects inC, then the following diagram is inE:

0 //_X ^(Id,0) //_X

⊕

Y ^pr2 //_Y //₀

(Ex3) Any diagram which is isomorphic to a diagram inElies also inE.

(Ex4) Iff

:

X

→

Yandg

:

Y

→

Zare two admissible monomorphisms (resp. admissible epimorphisms), then so isgf. (Ex5) For any admissible monomorphismf

:

X⁰ Xand any morphismu

:

X⁰

→

Y inC, the pushout offanduexists.

Furthermore, if the diagram X⁰ // ^f //

u

X

v

Y _g //_Z

is cocartesian, thengis an admissible monomorphism.

(Ex6) For any admissible epimorphismf

:

_XX⁰⁰and any morphismu

:

_Y

→

_X⁰⁰_inC, the fiber product offanduexists.

Furthermore, if the diagram Z ^g //

v

Y

u

X f ////_X⁰⁰

is cartesian, thengis an admissible epimorphism.

Keller [10] has shown that the axioms above imply the following property, which was initially an axiom in Quillen’s definition:

(Ex7) For any morphismf

:

X

→

YinChaving a kernel (resp. cokernel), if there exists an morphismg

:

Z

→

X(resp.

g

:

Y

→

Z) such thatfg(resp.gf) is an admissible epimorphism (resp. admissible monomorphism), then also isf itself.

Iff

:

X

→

Yis an admissible monomorphism, by (Ex1), the morphismf admits a cokernel, which is denoted byY

/

^X.

According to [9], ifCis an Abelian category and ifE is the class of all exact sequences inC, then

(

^C

,

^E

)

is an exact category. Furthermore, any exact category can be naturally embedded (through the additive version of Yoneda’s functor) into an Abelian category.

3. Geometric exact categories

Some natural categories, like the category of Hermitian spaces, are not exact categories. However, a Hermitian space can be considered as a vector space overC(which is an object in an Abelian category), equipped with a Hermitian inner product (which can be considered as a geometric structure). This observation leads to the following notion of geometric exact categories.

Definition 3.1. Let

(

^C

,

^E

)

be an exact category. We callgeometric structureon

(

^C

,

^E

)

^{the data:}

(1) a mappingAfromobjCto the class of sets,

(2) for any admissible monomorphismf

:

Y

→

X, a mapf^∗

:

A

(

^X

) →

A

(

^Y

)

^, (3) for any admissible epimorphismg

:

X

→

Z, a mapg∗

:

A

(

^X

) →

A

(

^Z

)

^, subject to the following axioms:

(A1)A

(

⁰

)

is a one-point set,

(A2) if X // ⁱ //_Y // ^j //_Z are admissible monomorphisms, then

(

^ji

)

^∗

=

i^∗j^∗, (A3) if X ^p ////_Y ^q ////_Z are admissible epimorphisms, then

(

^qp

)

^∗

=

q∗p∗, (A4) for any objectXofC, Id^∗_X

=

Id_X∗

=

Id_A(X),

(A5) iff

:

X

→

Yis an isomorphism, thenf^∗f∗

=

Id_A(X),f∗f^∗

=

Id_A(Y), (A6) for any cartesian or¹cocartesian square

X // ^u //

p

Y

q

Z //

v //_W

(1)

1 Here we can prove that the square is actually cartesianandcocartesian.

in C, whereu and

v

^{(resp. pand} q) are admissible monomorphisms (resp. admissible epimorphisms), we have

v

^∗q∗

=

p∗u^∗,

(A7) if X ^u ////_Y // ^v //_Z is a diagram in C where u (resp.

v

) is an admissible epimorphism (resp. admissible monomorphism) and if

(

^hX

,

^hZ

) ∈

A

(

^X

) ×

A

(

^Z

)

^satisfiesu∗

(

^hX

) = v

^∗

(

^hZ

)

, then there existsh

∈

A

(

^X

⊕

Z

)

^such that

(

^Id

, v

^u

)

^∗

(

^h

) =

h_X and that pr_2∗

(

^h

) =

h_Z(note that

(

^Id

, v

^u

)

is always an admissible monomorphism, since it is the composition of the isomorphism

(

^pr1

, v

^upr1

+

pr₂

) :

X

⊕

Z

→

X

⊕

Zwith the admissible monomorphism

(

^Id

,

⁰

) :

X

→

X

⊕

Z).

The triplet

(

^C

,

^E

,

^A

)

is called ageometric exact category. For any objectXofC, we call any elementhinA

(

^X

)

^ageometric structureonX. The pair

(

^X

,

^h

)

is called ageometric objectin

(

C

,

E

,

^A

)

^{. Ifp}

:

X

→

Zis an admissible epimorphism,p∗

(

^h

)

^is called thequotient geometric structureonZ. Ifi

:

Y

→

Xis an admissible monomorphism,i^∗his called theinduced geometric structureonY.

(

^Z

,

^p∗

(

^h

))

is called ageometric quotientof

(

^X

,

^h

)

^and

(

^Y

,

^i∗

(

^h

))

is called ageometric subobjectof

(

^X

,

^h

)

^.

Let

(

^C

,

^E

)

be an exact category. If for any objectX ofC, we denote byA

(

^X

)

a one-point set, and we define induced and quotient geometric structures in the obvious way, then

(

^C

,

^E

,

^A

)

becomes a geometric exact category. The geometric structureAis called thetrivial geometric structureon the exact category

(

^C

,

^E

)

. Therefore, exact categories can be viewed as trivialgeometric exact categories.

Let

(

^C

,

^E

,

^A

)

be a geometric exact category. If

(

^X0

,

^h0

)

^and

(

^X00

,

^h00

)

are two geometric objects in

(

^C

,

^E

,

^A

)

, we say that a morphismf

:

X⁰

→

X⁰⁰inCiscompatiblewith geometric structures if there exists a geometric object

(

^X

,

^h

)

, an admissible monomorphismu

:

X⁰

→

Xand an admissible epimorphism

v :

X

→

X⁰⁰such thath⁰

=

u^∗

(

^h

)

^{and thath00}

= v

^∗

(

^h

)

^. Remark 3.2. From the definition of morphisms compatible with geometric structures, we obtain the following results:

(1) If

(

^X1

,

^h1

)

^and

(

^X2

,

^h2

)

are two geometric objects and iff

:

X₁

→

X₂is an admissible monomorphism (resp. admissible epimorphism) such thatf^∗h₂

=

_{h1(resp.f∗h1}

=

_{h2), thenf} is compatible with geometric structures.

(2) If

(

^X1

,

^h1

)

^and

(

^X2

,

^h2

)

are two geometric objects and iff

:

X₁

→

X₂is the zero morphism, thenf is compatible with geometric structures.

(3) The composition of two morphisms compatible with geometric structure is also compatible with geometric structure.

This is a consequence of axiom (A7).

Let

(

^C

,

^E

,

^A

)

be a geometric exact category. By argument (3) above, all geometric objects in

(

^C

,

^E

,

^A

)

and morphisms compatible with geometric structures form a category which we shall denote byCA.

We give below some examples of geometric exact categories.

3.1. Hermitian spaces

LetVec_Cbe the category of finite dimensional vector spaces overC. It is an Abelian category. LetE be the class of all short exact sequences of finite dimensional vector spaces overC. For anyX

∈

obj

(

^VecC

)

, denote byA

(

^X

)

the set of all Hermitian metrics onX. Leth

∈

A

(

^X

)

^{. Iff}

:

Y

→

Xis a vector subspace ofX, thenf^∗

(

^h

)

denotes the induced metric on Y. If

π :

X

→

Zis a quotient space ofX, then

π

^∗

(

^h

)

denotes the quotient metric onZ. Note that for anyz

∈

Z, one has

k

_z

k

_π∗(h)

:=

_{infx∈X,π(x)=z}

k

_x

k

_h. We claim that

(

^VecC

,

E

,

^A

)

is a geometric exact category. The axioms (A1)–(A6) are easily verified. The verification of (A7) is as follows. Let X ^u ////_Y // ^v //_Z be a diagram inVec_C. Assume thatXandZare respectively equipped with Hermitian metrics

k · k

_Xand

k · k

_Zsuch that the induced metric onYfrom

k · k

_Zcoincides with the quotient metric from

k · k

_X. We equipX

⊕

Zwith Hermitian metric

k · k

such that, for any

(

^x

,

^z

) ∈

X

⊕

Z,

k (

^x

,

^z

) k

²

= k

x

− ϕ

^?

(

^z

) k

²_X

+ k

z

k

²_Z

where

ϕ = v

^{u, and}

ϕ

^?denotes the adjoint of

ϕ

. With this metric, pr₂

:

X

⊕

Z

→

Zis a projection of Hermitian spaces.

Moreover,u^?is the identification ofYto

(

^{Ker u}

)

^{⊥, and}

v

^?is the orthogonal projection ofZontoY. Therefore,

ϕ

^?

ϕ :

X

→

X is the orthogonal projection ofXonto

(

^Ker

ϕ)

^⊥. Hence, for any vectorx

∈

X, we have

k (

^x

, ϕ(

^x

)) k

²

= k

x

− ϕ

^?

ϕ(

^x

) k

²_X

+ k ϕ(

^x

) k

²_Z

= k

x

− ϕ

^?

ϕ(

^x

) k

²_X

+ k ϕ

^?

ϕ(

^x

) k

²_X

= k

x

k

²_X

.

The geometric objects in

(

^VecC

,

^E

,

^A

)

are Hermitian spaces. From definition we see that if a linear mapping

ϕ :

X

→

Y of Hermitian spaces is compatible with geometric structure, then the norm of

ϕ

must be smaller than or equal to 1. The following proposition shows that the converse is also true.

Proposition 3.3. Let

ϕ :

E

→

F be a linear map of Hermitian spaces. If

k ϕ k

₆1, then there exists a Hermitian metric on E

⊕

F such that, in the decomposition E ^(Id,ϕ) //_E

⊕

_F ^pr2 //_{F of}

ϕ

^,

(

^Id

, ϕ)

is an inclusion of Hermitian spaces andpr₂is a projection of Hermitian spaces.

Proof. Since

k ϕ k

₆1, we have

k ϕ

^?

k

₆1. Therefore, we obtain the inequalities

k ϕ

^?

ϕ k

₆1 and

k ϕϕ

^?

k

₆1. Hence Id_E

− ϕ

^?

ϕ

and Id_F

− ϕϕ

^?are Hermitian endomorphisms with non-negative eigenvalues. So there exist two Hermitian endomorphisms with non-negative eigenvaluesPandQ ofEandFrespectively such thatP²

=

Id_E

− ϕ

^?

ϕ

^andQ2

=

Id_F

− ϕϕ

^?.

Ifxis an eigenvector of

ϕϕ

^?associated to the eigenvalue

λ

^{, then}

ϕ

^?xis an eigenvector of

ϕ

^?

ϕ

associated to the same eigenvalue. Therefore

ϕ

^?Qx

=

√

1

− λϕ

^?x

=

P

ϕ

^{?x. AsF}is generated by eigenvectors of

ϕϕ

^{?, we have}

ϕ

^?Q

=

P

ϕ

^{?. For the}

same reason we haveQ

ϕ = ϕ

^{P. LetR}

=

P ϕ^?

ϕ −Q

. AsRis Hermitian, and verifies R²

=

P²

+ ϕ

^?

ϕ

^P

ϕ

^?

− ϕ

^?Q

ϕ

^P

−

Q

ϕ ϕϕ

^?

+

Q²

=

Id_E⊕F

,

it is an isometry for the orthogonal sum metric onE

⊕

F. Letu

:

E

→

E

⊕

Fbe the mapping which sendsxto

(

^x

,

⁰

)

^{. The} diagram

E ^ϕ //

u

F

E

⊕

F

R //_E

⊕

F

pr2

OO

is commutative. The endomorphism

ϕ

^?

ϕ

is auto-adjoint and positive semidefinite, and satisfies

k ϕ

^?

ϕ k

₆ 1. Hence there exists an orthonormal basis

(

^xi

)

16i6nofEsuch that

ϕ

^?

ϕ

^xi

= λ

ix_i

(

⁰⁶

λ

i61

)

. Suppose that 06

λ

j

<

^{1 for anyj}

∈ {

1

, . . . ,

^k

}

and that

λ

j

=

1 for anyj

∈ {

k

+

1

, . . . ,

ⁿ

}

. LetB

:

E

→

Ebe theC-linear map such thatB

(

^xj

) =

p

1

− λ

jx_jforj

∈ {

1

, . . . ,

^k

}

and thatB

(

^xj

) =

x_jotherwise. DefineS

=

B ϕ^? 0 Id_F

:

E

⊕

F

→

E

⊕

F. SinceRu

= (

^P

, ϕ)

^{and since}

(

^BP

+ ϕ

^?

ϕ)(

^xi

) =

p

1

− λ

iBx_i

+ λ

ix_i

=

(

¹

− λ

i

)

^xi

+ λ

ix_i

=

x_i

,

^16i6k

,

0Bx_i

+

x_i

=

x_i

,

^k

<

ⁱ⁶ⁿ

,

the diagram

E

⊕

F

S

pr₂

))S

SS SS SS S E

Rukkkk55 kk kk

τSSS)) SS SS

S F

E

⊕

F ^pr2

55k

kk kk kk k

is commutative, where

τ = (

^IdE

, ϕ)

^{. We equipE}

⊕

Fwith the Hermitian inner product

h· , ·i

₀such that, for any

(α, β) ∈ (

^E

⊕

F

)

^{2, we have}

h α, β i

₀

=

S⁻¹

α,

^S−1

β

, where

h· , ·i

is the orthogonal direct sum of Hermitian inner products onEand onF. Then, for anyx

,

^y

∈

_{E, one has}

h τ(

^x

), τ(

^y

) i

₀

= h

_SRu

(

^x

),

^SRu

(

^y

) i

₀

= h

_Ru

(

^x

),

^Ru

(

^y

) i = h

_u

(

^x

),

^u

(

^y

) i = h

_x

,

^y

i .

Finally, the kernel of pr₂is stable by the action ofS, so the projections of

h· , ·i

₀and of

h· , ·i

by pr₂are the same.

Remark 3.4. Similarly, Euclidean norms form a geometric structure on the category of all finite dimensional vector spaces overR. The linear maps compatible with geometric structures are just those of norm61.

3.2. Ultranormed space

Letkbe a field equipped with a non-Archimedean absolute value

| · |

under whichkis complete. We denote byVec_kthe category of finite dimensional vector spaces overk, which is clearly an Abelian category. LetE be the class of short exact sequences inVec_k. For any finite dimensional vector spaceXoverk, we denote byA

(

^X

)

the set of all ultranorms (that is, a norm

k · k

which verifies

k

x

+

y

k

₆max

( k

x

k , k

y

k )

^{) on}X. Suppose thathis an ultranorm onX. Iff

:

Y

→

Zis a subspace of E,f^∗

(

^h

)

denotes the induced norm onY; ifg

:

X

→

Zis a quotient space ofX,g∗

(

^h

)

denotes the quotient norm onF. Then

(

^Veck

,

^E

,

^A

)

is a geometric exact category. In particular, axiom (A7) is justified by the following proposition.

Proposition 3.5. Let

ϕ :

E

→

F be a linear map of vector spaces over k. Suppose that E and F are equipped respectively with the ultranorms h_E and h_F such that

k ϕ k

₆ 1. If we equip E

⊕

F with the ultranorm h such that, for any

(

^x

,

^y

) ∈

E

⊕

F , h

(

^x

,

^y

) =

max

(

^hE

(

^x

),

^hF

(

^y

))

, then in the decomposition E ^(Id,ϕ)//_E

⊕

F ^pr2 //_F of

ϕ

^{, we have}

(

^Id

, ϕ)

^∗

(

^h

) =

h_E and pr_2∗

(

^h

) =

h_F.

Proof. In fact, for any elementx

∈

E,h

(

^x

, ϕ(

^x

)) =

max

(

^hE

(

^x

),

^hF

(ϕ(

^x

))) =

h_E

(

^x

)

^sincehF

(ϕ(

^x

))

6

k ϕ k

h_E

(

^x

)

6 h_E

(

^x

)

^. Furthermore, by definition, one hash_F

=

pr_2∗

(

^h

)

. Therefore the proposition is true.

3.3. Filtrations in an Abelian category

LetCbe an essentially small Abelian category andEbe the class of all short exact sequences inC. For any objectXin C, denote byA

(

^X

)

the set of isomorphism classes ofR-filtrations ofX. The maps in (2)–(3) ofDefinition 3.1are respectively chosen to be induced filtration map and quotient filtration map (see Section2.1).

We claim that

(

^C

,

^E

,

^A

)

is a geometric exact category. In fact, axioms (A1)–(A5) are clearly satisfied. We now verify axiom (A6). Consider the diagram(1)inDefinition 3.1, which is the right sagittal square of the following diagram(2). Suppose given anR-filtration

(

^Yλ

)

λ∈RofY. For any

λ ∈

_R, letb_λ

:

Y_λ

→

Ybe the canonical morphism.

(2) Letd_λ

:

W_λ

→

Wbe the image ofqb_λinWandq_λ

:

Y_λ

→

W_λbe the canonical epimorphism. Let

(

^Zλ

,

^cλ

, v

λ

)

be the fiber product of

v

^anddλ, and

(

^Xλ

,

^aλ

,

^uλ

)

be the fiber product ofuandb_λ. Therefore, in diagram(2), the two coronal squares and the right sagittal square are cartesian, the inferior square is commutative. As

v

^paλ

=

qua_λ

=

qb_λu_λ

=

d_λq_λu_λ, there exists a unique morphismp_λ

:

X_λ

→

Z_λsuch thatc_λp_λ

=

pa_λand

v

λp_λ

=

q_λu_λ. It is then not hard to verify that the left sagittal square is cartesian, thereforep_λis an epimorphism, soZ_λis the image ofpa_λ. Axiom (A6) is therefore verified. Finally, the verification of the axiom (A7) follows from the following proposition.

Proposition 3.6. Let X and Y be two objects inCand letF

= (

^Xλ

)

λ∈_R(resp.G

= (

^Yλ

)

λ∈_R) be anR-filtration of X (resp. Y ). If f

:

X

→

Y is a morphism which is compatible with the filtrations

(

^F

,

^G

)

, then the filtrationH

= (

^Xλ

⊕

Y_λ

)

λ∈Rof X

⊕

Y verifies Γf^∗H

=

_F andpr_2∗H

=

_G, whereΓf

= (

^Id

,

^f

) :

X

→

X

⊕

Y is the graph of f andpr₂

:

X

⊕

Y

→

Y is the projection onto the second factor.

Proof. By definition, one has pr_2∗H

=

_G. For any

λ ∈

_R, consider the square X_λ ^φλ //

(Id,fλ)

X (Id,f)

X_λ

⊕

Y_λ

Φλ //_X

⊕

Y

(3)

where

φ

λ

:

X_λ

→

Xand

ψ

λ

:

Y_λ

→

Yare canonical morphisms,Φλ

= φ

λ

⊕ ψ

λ, andf_λ

:

X_λ

→

Y_λis the morphism through which the morphismf

φ

λfactorizes. The square(3)is commutative. Suppose that

α :

Z

→

Xand

β = (β

1

, β

2

) :

Z

→

X_λ

⊕

Y_λ are two morphisms such that

(

^Id

,

^f

)α =

_Φλ

β

^.

Z β1

##

α

""

β

##

X_λ ^φλ //

(Id,fλ)

X (Id,f)

X_λ

⊕

Y_λ

Φλ //X

⊕

Y

(4)

Then we have

α = φ

λ

β

1andf

α = ψ

λ

β

2. So

ψ

λ

β

2

=

f

α =

f

φ

λ

β

1

= ψ

λf_λ

β

1. As

ψ

λis a monomorphism, we obtain f_λ

β

1

= β

2. So

β

1

:

Z

→

X_λis the only morphism such that the diagram(4)commutes. Hence we getF

= (

^Id

,

^f

)

^∗H. 4. Harder–Narasimhan categories

4.1. Degree function and rank function on a geometric exact category

Let

(

^C

,

^E

,

^A

)

be a geometric exact category. In the following, a geometrical object

(

^X

,

^h

)

ⁱⁿ

(

^C

,

^E

,

^A

)

will be simply denoted byX if this notation does not lead to any ambiguity; and we use the expressionh_X to denote the underlying geometric structure ofX.

We say that a geometric objectXisnon-zeroif its underlying object inCis non-zero. SinceCis essentially small, the isomorphism classes of objects inCAform a set.

We denote byEAthe class of diagrams of the form

0 //_X^{0 i} //_X ^p //_X⁰⁰ //₀

,

⁽⁵⁾

where the underlyingC-diagram lies inE,X⁰is a geometric subobject ofXandX⁰⁰is a geometric quotient ofX.

LetK₀

(

^C

,

^E

,

^A

)

be the free Abelian group generated by isomorphism classes inCA, modulo the subgroup generated by elements of the form

[

X

] − [

X⁰

] − [

X⁰⁰

]

, where

0 //_X^{0 i} //_X ^p //_X⁰⁰ //₀

is a diagram inEA. The groupK₀

(

^C

,

^E

,

^A

)

is called theGrothendieck groupof the geometric exact category

(

^C

,

^E

,

^A

)

^{. We have} a ‘‘forgetful’’ homomorphism fromK₀

(

^C

,

^E

,

^A

)

^toK0

(

^C

,

^E

)

, the Grothendieck group of the exact category

(

^C

,

^E

)

, which sends

[

X

]

to the class of the underlyingC-object ofX.

In order to define the semistability of geometric objects and to establish the Harder–Narasimhan formalism, we need two auxiliary homomorphisms of groups. The first one, fromK₀

(

^C

,

^E

,

^A

)

^toR, is called adegree functionon

(

^C

,

^E

,

^A

)

^{; and} the second one, fromK₀

(

^C

,

^E

)

^toZ, which takes strictly positive values on classes represented by non-zero objects inC, is called arank functionon

(

^C

,

^E

)

^.

Now let deg

:

K₀

(

^C

,

^E

,

^A

) →

_Rbe a degree function on

(

^C

,

^E

,

^A

)

^{and rk}

:

K₀

(

^C

,

^E

) →

_Zbe a rank function on

(

^C

,

^E

)

^. For any geometric objectXin

(

C

,

E

,

^A

)

, we denote by deg

(

^X

)

the value deg

( [

X

] )

, called thedegreeofX. Denote by rk

(

^X

)

^the function rk evaluated on the class of the underlyingC-object ofX, called therankofX. Note that rk

(

^X

)

does not depend on the geometric structure ofX.

IfXis non-zero, the quotient

µ(

^X

) =

_deg

(

^X

)/

^rk

(

^X

)

is called theslopeofX. We say that a non-zero geometric objectXis semistableif for any non-zero geometric subobjectX⁰ofX, we have

µ(

^X0

)

⁶

µ(

^X

)

^.

The following proposition provides some basic properties of degrees and slopes.

Proposition 4.1. Let us keep the notation above.

(1) If 0 //_X⁰ //_X //_X⁰⁰ //₀ is a diagram inEA, then deg

(

^X

) =

deg

(

^X0

) +

deg

(

^X00

).

(2) If X is a geometric object of rank 1, then it is semistable.

(3) Any non-zero geometric object X is semistable if and only if for any non-trivial geometric quotient X⁰⁰(i.e., X⁰⁰does not reduce to zero and is not canonically isomorphic to X ), we have

µ(

^X

)

⁶

µ(

^X00

)

^.

Proof. Since deg is a homomorphism fromK₀

(

^C

,

^E

,

^A

)

^toR, (1) is clear.

(2) Assume thatX⁰is a non-zero geometric subobject ofX. It fits into a diagram 0 //_X^{0 f} //_X //_X⁰⁰ //₀

inCA. SinceX⁰is non-zero, rk

(

^X0

)

^>1. Therefore rk

(

^X00

) =

0 and henceX⁰⁰

=

0. In other words,fis an isomorphism. So we have

µ(

^X0

) = µ(

^X

)

^.

(3) For any diagram 0 //_X⁰ //_X //_X⁰⁰ //_{0 inEA,X}⁰⁰is non-trivial if and only ifX⁰is non-trivial. IfX⁰ andX⁰⁰are both non-trivial, we have the following equality

µ(

^X

) =

^rk

(

^X0

)

rk

(

^X

) µ(

^X0

) +

^rk

(

^X00

)

rk

(

^X

) µ(

^X00

).

Therefore

µ(

^X0

)

⁶

µ(

^X

) ⇐⇒ µ(

^X00

)

^>

µ(

^X

)

^.

4.2. Harder–Narasimhan category and Harder–Narasimhan sequence

We are now able to introduce a condition ensuring the existence of the Harder–Narasimhan ‘‘flag’’.

Definition 4.2. Let

(

C

,

E

,

^A

)

be a geometric exact category, deg

:

_K0

(

C

,

E

,

^A

) →

_R be a degree function and rk

:

K₀

(

^C

,

^E

) →

_Zbe a rank function. We say that

(

^C

,

^E

,

^A

,

^deg

,

^rk

)

^{is a}Harder–Narasimhan categoryif the following axiom is verified:

(HN) For any non-zero geometric objectX, there exists a geometric subobjectX_desofXsuch that

µ(

^Xdes

) =

sup

{ µ(

^Y

) |

Yis a non-zero geometric subobject ofX

} .

Furthermore, any non-zero geometric subobjectZofXsuch that

µ(

^Z

) = µ(

^Xdes

)

is a geometric subobject ofX_des. Note that ifXis a non-zero geometric object, thenX_desis unique up to a unique isomorphism. Moreover, it is a semistable geometric object. IfXis not semistable, we say thatX_desdestabilizes X.

Theorem 4.3. Let

(

C

,

E

,

^A

,

^deg

,

^rk

)

be a Harder–Narasimhan category. If X is a non-zero geometric object, then there exists a sequence of admissible monomorphisms

0

=

X₀ //_X1 //

· · ·

//X_n−1 //_Xn

=

X

,

⁽⁶⁾

such that

(1) for any integer j

∈ {

1

, . . . ,

ⁿ

}

, the geometric quotient X_j

/

^Xj−1is semistable;

(2) the inequalities

µ(

^X1

/

^X0

) > µ(

^X2

/

^X1

) > · · · > µ(

^Xn

/

^Xn−1

)

^hold.

Proof. We prove the existence by induction on the rankrofX. The case whereXis semistable is trivial, anda fortiorithe assertion is true forr

=

1 (seeProposition 4.12). Now we consider the case whereXis not semistable. LetX₁

=

X_des. It is a semistable geometric object, andX⁰

=

X

/

^X1is non-zero. The rank ofX⁰being strictly smaller thanr, we can therefore apply the induction hypothesis onX⁰. We then obtain a sequence of admissible monomorphisms

0

=

X₁^{0 f}

0

1 //X₂⁰ //

· · ·

//X_n⁰₋₁ ^f

0

n−1 //X_n⁰

=

X⁰ verifying the desired conditions.

Since the canonical morphism fromXtoX⁰is an admissible epimorphism, for anyi

∈ {

1

, . . . ,

ⁿ

}

, if we noteX_i

=

X

×

_X0X_i⁰, then by (Ex6), the projection

π

i

:

_Xi

→

_Xi⁰is an admissible epimorphism. For any integeri, 16i

<

n, we have a canonical morphismf_ifromX_itoX_i+1and the square

X_i ^fi //

πi

X_i+1 πi+1

X_i⁰

f0 i

//X⁰_i+1

(7)

is cartesian. Sincef_i⁰ is an monomorphism, also isf_i (cf. [11] V. 7). Moreover, since the square(7)is cartesian,f_i is the kernel of the composed morphism X_i+1

πi+1////X_i⁰₊₁ ^pi ////X_i⁰₊₁

/

^Xi⁰, wherep_iis the canonical morphism. Since

π

i+1and p_iare admissible epimorphisms, also isp_i

π

i+1(by (Ex4)). Thereforef_iis an admissible monomorphism. Hence we obtain a commutative diagram

0

=

X₀ //X₁ ^f1 //

π1

X₂ //

π2

· · ·

//X_n−1

fn−1 //

πn−1

X_n

=

X πn

0

=

X₁⁰

f0 1

//X⁰₂ //

· · ·

//X⁰_n−1

f0 n−1

//X⁰_n

=

X⁰

where the horizontal morphisms are admissible monomorphisms and the vertical morphisms are admissible epimorphisms.

Furthermore, for any integeri

∈ {

1

, . . . ,

ⁿ

−

1

}

, we have a natural isomorphism

ϕ

ifromX_i+1

/

^XitoX_i⁰₊₁

/

^Xi⁰. We denote by g_i(resp.g_i⁰) the canonical morphism fromX_i(resp.X_i⁰) toX(resp.X⁰). Leth_i

=

g_i^∗

(

^hX

)

^(resp.h0i

=

g_i^0∗

(

^hX0

)

) be the induced geometric structure onX_i(resp.X_i⁰). By (A6), one has

π

i∗

(

^hi

) = π

i∗f_i^∗

(

^hX

) =

f_i^0∗

π

^∗

(

^hX

) =

h⁰_i. Therefore

ϕ

i∗sends the quotient geometric structure onX_i+1

/

^Xito that onX_i⁰₊₁

/

^Xi⁰. Hence the geometric objectX_i+1

/

^Xiis semistable and we have the equality

µ(

^Xi+1

/

^Xi

) = µ(

^Xi⁰+1

/

^Xi⁰

)

. Finally, sinceX₁

=

X_des, one has

µ(

^X2

/

^X1

) =

^rk

(

^X2

)µ(

^X2

) −

rk

(

^X1

)µ(

^X1

)

rk

(

^X2

) −

rk

(

^X1

) < µ(

^X1

).

Therefore the sequence 0

=

X₀ //X₁ //

· · ·

//X_n−1 //X_n

=

X satisfies the desired conditions.

Definition 4.4. In the proof ofTheorem 4.3, we have actually constructed by induction a sequence 0

=

X₀ //_X1 //

· · ·

//X_n−1 //_Xn

=

X

of geometric subobjects ofXsuch thatX_i

/

^Xi−1

= (

^X

/

^Xi−1

)

desfor anyi

∈ {

1

, . . . ,

ⁿ

−

1

}

. This sequence satisfies the two conditions inTheorem 4.3. We call it theHarder–Narasimhan sequenceofX. The real numbers

µ(

^X1

)

^and

µ(

^X

/

^Xn−1

)

^are respectively called themaximal slopeand theminimal slopeofX, denoted by

µ

max

(

^X

)

^and

µ

min

(

^X

)

^.

The following proposition shows that in the Abelian category case, condition

(

^HN

)

is automatically satisfied.

Proposition 4.5. Let

(

^C

,

^E

,

^A

)

be a geometric exact category,degandrkare respectively a degree function and a rank function on it. If

(

^C

,

^E

)

is an Abelian category, then

(

^C

,

^E

,

^A

,

^deg

,

^rk

)

is a Harder–Narasimhan category.