Cremona group functor and its lie algebra

JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0027 Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 3-13 This paper is available online at CREMONA GROUP FUNCTOR AND ITS LIE ALGEBRA Nguyen Dat Dang Faculty of Mathematics, Hanoi National University of Education Abstract. Let Crn(k) = Bir(Pnk ) denote the set of all birational maps of the projective space Pn k over a field k. It is clear that Crn(k) is a group under composition of dominant rational maps; called the Cremona group of order n. This group is not an algebraic group. It was studied for the first time by Luigi Cremona (1830 - 1903), an Italian mathematician. Although it has been studied since the 19th Century by many famous mathematicians, it is still not well understood. For example, we still don’t know if it has the structure of an algebraic group of infinite dimension. In this paper, we will construct the Cremona group functor, calculate its Lie algebra and show that its Lie algebra is simple. Keywords: Birational map, cremona group, group functor, Lie algebra. 1. Lie algebra of a group functor 1.1. Extension of categories When we study the Cremona group Crn(k), we have the following question: Is it an algebraic group, a variety, a scheme . . .? If none of the above, in what category does it belong? In this paper, we will present a category, which is larger than the category of group schemes and contains Crn(k) as its object. In classic algebraic geometry, we study the category of algebraic varieties, which is the sets of solutions to systems of polynomial equations in An ou Pn. However, because this category is not large enough, we extend it in the category of schemes, which is really larger. We can resume this extension stating the following theorem (cf. [1], page 78 and page 104). Theorem 1.1. Let k be an algebraically closed field. There is a natural fully faithful functor t : Var(k) −→ Sch(k) from the category Var(k) of varieties over k to Sch(k) schemes over k. For any variety V , its topological space is homeomorphic to the set of closed points of sp(t(V )), and its sheaf of regular functions is obtained by restricting the structural sheaf of t(V ) via this homeomorphism. The image of the functor t is exactly the set of quasi-projective integral schemes over k. The image of the set of projective varieties is the set of projective integral schemes. In particular, for any variety V , t(V ) is an integral, separated scheme of finite type over k. Even if, the theory of schemes is very beautiful and rich, it is not general enough, there are still geometric objects which are not in this category. A. Grothendieck extended this category Received September 16, 2016. Accepted October 15, 2016. Contact Nguyen Dat Dang, e-mail address: Dangnd@hnue.edu.vn 3 Nguyen Dat Dang in a more general direction: the category Fun ( Ring,Set ) of functors from the category Ring of commutative and unitary rings to the category Set of sets. We know that the study of algebraic varieties and schemes arises from the need to solve the systems of polynomial equations: Pi(x1, . . . , xn) = 0; i = 1, . . . ,m in a fixed field k. The principal idea of A. Grothendieck is to study the solutions of these systems, but instead of fixing the field k, he solves it in the different k-algebras and he has a functor from the category Alg(k) of k-algebras to the category Set. Alg(k) −→ Set R 7−→ {the solutions in R of the system : P1 = . . . = Pm = 0} ≃ HomAlg(R) ( R [x1, . . . , xn] (P1, . . . , Pm) ;R ) . We explain this idea below. We introduce the following concepts: If C is some category, we denote by Fun ( C op,Set ) the category whose the objects are the contravariant functors F from C to Set and whose the morphisms from object F to object G are the functorial morphisms φ : F → G, that is, the natural transformations. Hence, we have the natural following functor: h : C −→ Fun(Cop,Set) X 7−→ hX where hX : C op −→ Set Y 7−→ hX(Y ) = MorC(Y,X). Lemma 1.1. (Yoneda) (cf. [2], pages 252-253). LetX,X ′ be two objects of any category C. 1. If F ∈ Fun(Cop,Set), the natural transformations from hX to F are in one to one correspondence with the elements of F (X). In particular MorFun(Cop,Set) ( hX , hX′ ) ≃ MorC(X,X ′). 2. Two functors hX and hX′ are isomorphic in Fun ( C op,Set ) if and only if X and X ′ are isomorphic in C. Hence, the functor h : C → Fun(Cop,Set) defines an equivalence of categories between C with a full subcategory of Fun ( C op,Set ) . Definition 1.1. A functor F ∈ Fun(Cop,Set) is called representable if there exists an objectX ∈ C such that F ≃ hX in Fun ( C op,Set ) . It is clear that object X is unique up to an isomorphism of C. We say that F is the associate functor with X. Consequently, the image of h is the full subcategory of representable functors of Fun ( C op,Set ) . Now, we apply the Yoneda lemma to the case where C = Sch(R) is the category of schemes over some ring R and we have an extension of categories: h : Sch(R) −→ Fun(Sch(R)op,Set). 4 Cremona group functor and its lie algebra If R = Z is the ring of integer numbers, we have an extension of categories: h : Sch −→ Fun(Schop,Set). Consequently, a scheme is well defined by its associate functor. Furthermore, by the definition of scheme, we know that a scheme is defined by gluing affine subschemes X = ∪i∈ISpecAi, so a R-scheme X is defined uniquely by the restriction of its associate functor to the category of affine R-schemes, that is equivalent to the category of R-algebras. Hence, we have an extension of categories: h : Sch(R) −→ Fun(Alg(R),Set) X 7−→ hX . For R = Z h : Sch −→ Fun(Alg,Set) X 7−→ hX . Definition 1.2. (cf. [2], page 252) Given X,Y ∈ Sch. The set hX(Y ) = MorSch(Y,X) is called the set of Y -value points of X. In particular, if Y = SpecR, these points are simply called the R-value points of X. If X is a scheme over a field k, the k-value points of X are exactly the k-rational closed points of X, that is, the closed points of X with the residue field κ(p) = k. In fact, there exists functors F ∈ Fun(Alg,Set), which are not representable using any scheme. 1.2. Tangent space of a functor Let F : Alg(k)→ Set be a k-functor and p ∈ F (k) a k-value point. We define the tangent space of F at p as the fiber TpF := F (π)−1(p) where π : k(ε) ։ k, ε 7→ 0 and k(ε) is the ring of dual numbers on k. We recall that the ring of dual numbers k(ε) is defined as the quotient ring k[X]/(X2) of the polynomial ring k[X] on the principal ideal (X2) and ε = X + (X2) is the class of the variable X. In any case, we can always define the multiplication of a scalar λ ∈ k with a vector x of TpF : k× TpF −→ TpF (λ, x) 7−→ λ.x := F (∗λ)(x) where ∗λ : k(ε) → k(ε) which sends a in λa. However, we can only define an addition in TpF such that it becomes a vector space in the case where F preserves the fiber product, in particular, the case where F is representable by a scheme. For detail, you see it in the reference [2], pages 257-258. 1.3. Group schemes and group functors In classical algebraic geometry, we define the concept of algebraic group, that is, simultaneously an algebraic variety and a group such that the maps G → G,x 7→ x−1 and G×G→ G, (x, y) 7→ xy are both regular. In the theory of schemes, by generalizing this concept, we obtain the concept of group scheme (cf. [1], page 324). 5 Nguyen Dat Dang Definition 1.3. A scheme X with a morphism to another scheme S is a group scheme over S if there is a section e : S → X (the identity) and a morphism ρ : X → X over S (the inverse) and a morphism µ : X ×X → X over S (the group operation) such that 1. the composition µ ◦ (id × ρ)◦ △X : X → X is equal to the structural morphism X → S followed by e, where △X is the diagonal morphism of X. 2. the two morphisms µ ◦ (id× µ) and µ ◦ (µ × id) from X ×X ×X → X are the same. It is clear that ifX is a group k-scheme, then its closed points form an algebraic k-group in the usual sense. In generalizing, we have the following concept: Definition 1.4. A group S-functor is a contravariant functor from the category of S-schemes to the category gr of groups: F : Sch(S)op → gr. Remark 1.1. If X is a group S-scheme, then its associate functor F = hX is a group S-functor and representable. Reciprocally, if F is a group S-functor and representable by a S-scheme X, then X is also a group S-scheme. Example 1.1. We meet the group k-schemes: Ga,Gm,GL(n), . . . Suppose that H is a k-algebra. The functor hSpecH associate to the affine scheme SpecH is an affine group k-scheme, if and only if, H is a Hopf k-algebra. There is an equivalence of categories between the category of affine group k-schemes and the category of Hopf k-algebras. 1.4. Lie algebra of a group functor LetG be a group k-scheme. The tangent space of G at the identity element e of G will be a Lie k-algebra, denoted by Lie(G) = TeG. If ϕ : H → G is a morphism of group k-schemes, we also have a corresponding homomorphism of Lie k-algebras: Lie(ϕ) = Teϕ : Lie(H) = TeH −→ TeG = Lie(G). In taking the derivative of the inner automorphisms φg : G → G,h 7→ ghg−1, we have a morphism of group k-schemes Ad : G −→ GL(Lie(G)) g 7−→ Ad(g) := Lie(φg) = Teφg which induces a corresponding homomorphism of Lie k-algebras ad = Lie(Ad) : Lie(G) −→ Lie(GL(Lie(G))) = glk(Lie(G)) where we denote by glk(V ) the Lie algebra of the general linear group GLk(V ) of the vector space V . As a vector space, glk(V ) = Endk(V ) is the k-vector space of endomorphisms of V . We know that ad is exactly the adjoint representation of the Lie k-algebra Lie(G), and the Lie bracket [x, y] := ad(x)(y). 6 Cremona group functor and its lie algebra The structure of Lie k-algebra over Lie(G) is completely defined if we can calculate the morphisms Ad and ad. You can see these arguments in the reference [3]. If G is only a group k-functor, non necessarily representable, we can still define Lie(G) = TeG as the tangent space of G at the identity element e of G(k), that is, the fiber of the homomorphism G(π) : G(k(ε)) → G(k) at e ∈ G(k). As we said in the section 1.2., Lie(G) is a k-vector space. Furthermore, in this case, we can still define a Lie bracket over Lie(G) such that it becomes a Lie k-algebra as follows: Consider the following diagram where k(ε),k(δ) are rings of dual numbers over k. Indeed, let X,Y ∈ Lie(G) be two tangent vectors, that is, 1+ εX ∈ G(k(ε)) and 1+ δY ∈ G(k(δ)). Then, the Lie bracket [X,Y ] is defined as the coefficient of εδ in the decomposition of (1+ εX)(1+ δY )− (1+ δY )(1+ εX) ∈ G(k(ε, δ)). 2. Cremona group functor and its Lie algebra 2.1. Recalls on the birational maps In classical algebraic geometry, we define the concept of rational map between two algebraic varieties. Now, in order to introduce the Cremona group functor, we need generalize this concept for any schemes. In reality, there are many variants of this concept, the reader can see it in E.G.A. I, [4], §7, pages 155-161 and in E.G.A. IV, [4], §20, pages 231-251. Now, we suppose always that S is a scheme and X,Y are two smooth, separated S-schemes of finite type. Definition 2.1. We say that a open subset U of X is S-dense (cf. [5]) if for each s ∈ S, the open subset Us := U ×S Specκ(s) of the fiber Xs of X at s is dense. We call a rational S-map (S-pseudo-morphism ) (cf. E.G.A. IV [4], §20, pages 231-251 or (cf. [5]) from X to Y , denoted by X 99K Y , an equivalence class of S-morphisms from S-dense open subsets of X to Y . Here, if f : U → Y and g : V → Y are two such S-morphisms, we say that f and g are equivalent if there exists an open subset W ⊂ U ∩ V , S-dense in X in which f and g coincide. It is clear that the relation defined above is well an equivalence relation since the intersection of two S-dense open subsets is still a S-dense open subset. If ϕ : X 99K Y is a rational S-map, the reunion dom(ϕ) of S-dense open subsets U for all the couples (U, f) representing ϕ is still a S-dense open subset of X, called the domain of definition of ϕ. Remark 2.1. We know that if the fibers Us are irreducible for all s ∈ S, then, U is S-dense in X if and only if Us is non-empty for all s ∈ S, that is, the structural morphism U ։ S is 7 Nguyen Dat Dang surjective. In particular, if S = Speck is the prime spectrum of some field k and X is irreducible, then an open subset U of X is k-dense if and only if it is dense in X. Therefore, the concept of rational k-map of varieties (k-pseudo-morphism) coincide with the usual concept already defined in classical algebraic geometry. Definition 2.2. Let ϕ : X 99K Y and ψ : Y 99K Z be two rational S-maps. Consider the representing morphisms f : dom(ϕ) → Y of ϕ and g : dom(ψ) → Z of ψ. If there exists a W ⊂ f(dom(ϕ)) ∩ dom(ψ) such that f−1(W ) is a S-dense open subset of X, then the couple( f−1(W ), g ◦ (f |f−1(W )) ) defines a rational S-map from X to Z , called the composite rational S-map of ψ and of ϕ, denoted by ψ ◦ ϕ. Definition 2.3. (cf. [5]). A rational S-map ϕ : X 99K Y is called birational S-map if there exists a rational S-map ψ : Y 99K X such that the two composite rational maps ψ ◦ϕ,ϕ◦ψ are defined and equivalent respectively idX , idY . It is clear that a such rational map ψ is unique and called the inverse of ϕ, denoted by ϕ−1 = ψ. Denote by BirS(X,Y ), the set of birational S-maps from X onto Y . When X = Y , we write simply BirS(X) and when S is understood, we write also simply Bir(X,Y ),Bir(X). If there exists a birational S-map ϕ from X onto Y , we say that X is birational S-isomorphic to Y or S-birational to Y , and denote by ϕ : X ∼ 99K Y. In general, we can not compose two rational maps. However, there are classes of rational maps in which the composition is always defined. Now, we will present a such class. Definition 2.4. (cf. [5]). A rational S-map ϕ : X 99K Y is called S-dominant if for all s ∈ S, the rational maps induced on the fibers ϕs : Xs 99K Ys are dominant in the usual sense, that is, ϕs ( dom(ϕs) ) contains a dense open subset of Ys. It is easy to show that we can always compose ψ ◦ ϕ if ϕ is S-dominant. Obviously, all birational S-maps are S-dominant. Consequently, the set BirS(X) of birational S-maps of X is a group with the composition of dominant rational S-maps. Proposition 2.1. (cf. [5]) A rational S-map ϕ is S-birational, it is necessary and sufficient that its domain of definition contains a S-dense open subset of X such that ϕ induces a S-isomorphism from U onto a S-dense open subset of Y . Proposition 2.2. (cf. [5]) Let k be a field andX,Y two smooth, separated, irreducible k-schemes, of finite type with the field of rational functions k(X),k(Y ) respectively. The dominant rational k-maps from X to Y are in bijective correspondence with the k-homomorphisms of fields from k(Y ) to k(X). Consequently, the birational k-maps from X onto Y identify the k-isomorphisms of fields from k(Y ) onto k(X): Birk(X,Y ) ≃ Isok(k(Y ),k(X)). In particular, the group Birk(X) is aussi the Galois group of k-automorphisms of the field of rational functions: Birk(X) ≃ Autk(k(X)) = Gal ( k(X)upslopek ) . 8 Cremona group functor and its lie algebra 2.2. Cremona group functor Let S be a scheme and X a smooth, separated S-scheme of finite type. We have the following group S-functor birS(X) : Sch(S) op −→ gr S′ 7−→ birS(X)(S′) := BirS′(X ×S S′) ↑ ↓ S′′ 7−→ birS(X)(S′′) := BirS′′(X ×S S′′) which associated with each S-scheme S′, the group BirS′(X ×S S′) and each morphism of S-schemes ϕ : S′′ → S′, the group homomorphism birS(X)(ϕ) : BirS′(X ×S S′)→ BirS′′(X ×S S′′) defined as follows: each birational S′-map f : X ×S S′ ∼99K X ×S S′ is transformed to the birational S′′-map: f ×S′ idS′′ : X ×S S′′ ≃ ( X ×S S′ )×S′ S′′ ∼99K (X ×S S′)×S′ S′′ ≃ X ×S S′′. Proposition 2.3. The group S-functor birS(X) is a birational invariant of the S-scheme X, that is, if there exists a birational S-map ϕ : X ∼ 99K Y , then we have also an isomorphism of group S-functors: birS(X) ≃ birS(Y ). Indeed, for all S-scheme S′, we have an isomorphism of groups BirS′(X ×S S′) ∼→ BirS′(Y ×S S′) which associates to each f ∈ BirS′(X ×S S′), (ϕ× idS′) ◦ f ◦ ( ϕ−1 × idS′ ) . Definition 2.5. (cf. [5]) When X is a rational k-scheme of dimension n, according to the proposition 2.3, we can suppose that X = Pnk the projective space of dimension n over k. The group Birk(Pnk) ≃ Gal(k(x1, . . . , xn)upslopek) of birational k-maps (birational k-automorphisms) of Pnk is called the Cremona group of order n, denoted by Crn(k). The group k-functor birk(P n k) is called the Cremona group k-functor, denoted by crn(k). It is clear that the k-value points of the Cremona group k-functor crn(k) are exactly the elements of the Cremona group Crn(k). 2.3. Lie algebra of the Cremona group functor The Cremona group k-functor crn(k) is well defined by its restriction to Alg(k) F = crn(k) : Alg(k) −→ gr R 7−→ crn(k)(R) = BirR(PnR). The Lie algebra k-functor of F is defined as follows lie(F ) : Alg(k) −→ gr R 7−→ lie(F )(R) = Ker (F (π) : F (R(ε))→ F (R)) 9 Nguyen Dat Dang where R(ε) := R[X]upslope(X2) is the ring of dual numbers over R and π : R(ε)→ R, ε 7→ 0, so lie(F )(R) = Ker(F (π)) = Ker ( BirR(ε)(P n R(ε)) −→ BirR(PnR) ) . The Lie algebra of the Cremona group is defined to be the Lie algebra of k-value points of the functor lie(F ) Lie(Crn(k)) := lie(crn(k))(k) = Ker ( Birk(ε)(P n k(ε)) −→ Birk(Pnk) ) . In the convenient affine coordinate system, each rational map σ : Pnk(ε) 99K P n k(ε) is expressed in the form σ = (f1 + εg1, . . . , fn + εgn) where the fi, gi are rational maps on Pnk . The homomorphism F (π) is associated with σ, the n-tuple (f1, . . . , fn), its kernel is Lie(Crn(k)) = { σ = (x1 + εg1, . . . , xn + εgn) ∈ Birk(ε)(Pnk(ε)) : gi ∈ k(x1, . . . , xn) } . We will show that for any g = (g1, . . . , gn) ∈ k(x1, . . . , xn)n, σ = id + εg is an element of Birk(ε)(Pnk(ε)). Indeed, it suffices to choose τ = id− εg, we have, for all x: (σ ◦ τ)(x) = (id + εg)(x − εg(x)) = x− εg(x) + εg(x− εg(x)) = x, (τ ◦ σ)(x) = (id − εg)(x + εg(x)) = x+ εg(x) − εg(x+ εg(x)) = x. Hence Lie(Crn(k)) = {id+ εg : g = (g1, . . . , gn) ∈ k(x1, . . . , xn)n} ≃ {g = (g1, . . . , gn) : gi ∈ k(x1, . . . , xn)} . As a vector space, Lie(Crn(k)) ≃ k(x1, . . . , xn)n is a k(x1, . . . , xn)-vector space of dimension n. Now, we calculate the Lie bracket: Let X = (f1, . . . , fn) ∈ Lie(Crn(k)) and Y = (g1, . . . , gn) ∈ Lie(Crn(k)) be two tangent vectors of F at the origin, that is, id+ ε1X ∈ F (k(ε1)) and id+ ε2Y ∈ F (k(ε2)). Consider (id+ ε1X) ◦ (id+ ε2Y )− (id+ ε2Y ) ◦ (id+ ε1X) ∈ F (k(ε1, ε2)). Calculating minutely each term, we have (id+ ε1X) ◦ (id+ ε2Y ) = id+ ε1X + ε2Y + ε1ε2X ′.Y (2.1) where X ′ is the matrix of first partial derivatives. Indeed, for all x, we have ((id+ ε1X) ◦ (id+ ε2Y )) (x) = (id+ ε1X)(x + ε2Y (x)) = x+ ε2Y (x) + ε1X(x+ ε2Y (x)) = x+ ε2Y (x) + ε1 [ X(x) + ε2X ′.Y (x) ] = x+ ε1X(x) + ε2Y (x) + ε1ε2X ′.Y (x). Similarly, we also have (id+ ε2Y ) ◦ (id + ε1X) = id+ ε1X + ε2Y + ε1ε2Y ′.X. (2.2) 10 Cremona group functor and its lie algebra According to (2.1) and (2.2), we have (id+ ε1X) ◦ (id+ ε2Y )− (id+ ε2Y ) ◦ (id+ ε1X) = ε1ε2 ( X ′.Y − Y ′.X) . The coefficient of ε1ε2 is also the Lie bracket: [X,Y ] = X ′.Y −Y ′.X. Now, if we identify X with the operator: X = f1 ∂∂x1 + · · ·+ fn ∂∂xn and Y = g1 ∂∂x1 + · · ·+ gn ∂∂xn , then X ′.Y − Y ′.X =   n∑ j=1 ( ∂g1 ∂xj fj − ∂f1 ∂xj gj ) , . . . , n∑ j=1 ( ∂gn ∂xj fj − ∂fn ∂xj gj ) = X ◦ Y − Y ◦X. Consequently, the Lie bracket in Lie(Crn(k)) coincide with the Lie bracket in the Lie algebra of k-derivations Derk(k(x1, . . . , xn)) (in order to know Derk(k(x1, . . . , xn)), the reader can see it in [6]). We fin