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INVERSE LIMIT of GRASSMANNIANS Introduction INVERSE LIMIT OF GRASSMANNIANS AMELIA ALVAREZ¶ Y PEDRO SANCHO Introduction Grothendieck de¯ned functorially the Grassmannian of the vector subspaces of ¯nite codimension of a vector space, and proved that this functor was representable by a scheme. However, for the construction of the Hilbert scheme of the closed varieties of a projective variety (see [G]), and the moduli of formal curves and soliton theory, etc., it is convenient to consider Grassmannians in a more general situation. M. Sato and Y. Sato, [SS] and Segal and Wilson, [SW] considere, given a vector subspace E0 ½ E, the topology on E, for which the set of the vector subspaces F ½ E such that F + E0=F \ E0 are ¯nite dimensional vector spaces is a neighbourhood basis at 0. Then they prove that the subset of vector subspaces F of the completion of E, such that E=F + E0 and F \ E0 are ¯nite dimensional vector spaces is a scheme. Alvarez-Mu~noz-Plaza,[AMP]¶ de¯ne and prove these results functorially. In this paper we prove that, if E is the inverse limit of linear maps fEi ! Ejg, with ¯nite dimensional kernels and cokernels, and if we consider the Grassmannian of ¯nite codimensional vector spaces of Ei, G(Ei), then we get an inverse system of rational morphisms fG(Ei) 99K G(Ej)g, which is representable by a scheme, whose points represent the vector subspaces of F ½ E, such that the morphism F ! Ei is injective and the cokernel is a ¯nite dimensional vector space (1.10). We obtain the results of Alvarez-Mu~noz-Plaza[AMP],¶ as a consequence of 1.10. 1. Grassmannians of inverse limit of vector spaces 1. Teorema Grothendieck, [GD]: Let E be a k-vector space. The functor on the category of noetherian k-algebras ½ ¾ A-submodules M ½ E = E ­ A whose cokernel Gr(E)(Spec A) := A k is a projective ¯nitely generated module is representable by a scheme and is covered by the a±ne open subsets ¢ ¤ Homk(E1¡n;En) := Spec S (E1¡n ­ En) where En ⊆ E, dimk En < 1 and E1¡n is any vector subspace of E such that En © E1¡n = E (such open subsets are de¯ned as being the set of vector supplementary subspaces of En with respect to E). ` Moreover, Gr(E) = Grn(E), where n2N ½ ¾ A-submodules M ½ E = E ­ A whose cokernel Gr (E)(Spec A) := A k n is a locally free ¯nitely generated module of rank n Date: 24-1-2004. 1 2 AMELIA ALVAREZ¶ Y PEDRO SANCHO 2. De¯nici¶on: Given V and W submodules of a module E, we will say that V and W are supplementary vector spaces with respect to E if E = V © W . In particular, both V and W are direct summands of E. 3. Proposici¶on: Let L be a free A-module and L0 ½ L a ¯nitely generated submodule. Then L0 is a direct summand of L if and only if L=L0 is a projective module, if and only if L0 is a submodule on ¯bers for every closed point of Spec A, and if and only if L0 ! L is a universally injective morphism. Therefore, the fact that a ¯nitely generated submodule of L is a direct summand is a local question in the Zariski topology, and it is also a question on ¯bers for every closed point. Proof. We must only prove that if L0 is a submodule on ¯bers for every closed point of Spec A, then L=L0 is a projective module. Because L0 is ¯nitely generated, it injects into a free ¯nitely generated submodule of L that is a direct summand of it, therefore we can assume that L is also ¯nitely generated. Then, L=L0 is of ¯nite presentation and it will be projective if and only if so is locally for every closed point of Spec A. Let A be a local ring of maximal m. By assumption 1 0 0 TorA(L=L ; A=m) = 0, then L=L is projective. ¤ : : 4. Notaci¶on: We denote (Spec A) as the functor (Spec A) (X) = Homk¡sch(X; Spec A). 5. Lema : Given a vector subspace E0 ½ E of ¯nite dimension, it holds that the set of vector subspaces of E00 ½ E, of ¯nite codimension such that E0 \ E00 = 0, is an open subset of the Grassmannian of subspaces of ¯nite codimension of E. Proof. The subfunctor U ½ Gr(E) from the statement is de¯ned by ½ ¾ M 2 Gr(E)(Spec A) such that M,! E =E0 U(Spec A) := A A is injective with projective cokernel ¢ ¢ We must prove that U £Gr(E) (Spec A) is an open subfunctor of (Spec A) . Given Spec A ! Gr(E), that is, an A-submodule M ½ EA with ¯nite projective cokernel, ¢ then by Proposition 1.3, U £Gr(E) (Spec A) is the open subset of the points of 0 Spec A where the morphism EA ,! EA=M is injective on ¯bers. ¤ Every linear application between vector spaces L : E1 ! E2, whose kernels and cokernels are of ¯nite dimension, induces a rational morphism between their Grassmannians LG : Gr(E1) 99K Gr(E2), V 7! L(V ). This morphism is de¯ned on 0 the open subset Gr (E1) of Gr(E1) made up by the vector subspaces V of E1 that do not intersect with ker L, more precisely, by the submodules V such that any of their supplementary submodules contain ker L, that is equivalent to saying that on 0 0 ¯bers ker L and V intersect in zero. Moreover, Grn(E1) = Grn(E1)\Gr (E1) is the maximal open subset of Grn(E1) where LG is de¯ned (for n > dimk ker L): Let us assume , for simplicity, that E1 is vector space of ¯nite dimension, V ½ E1 satis¯es 0 V \ker L = hwi and LG is de¯ned on V . Let be r+1 vectors of E1, v1; v1; v2; : : : ; vr, such that are linearly independents in E1= ker L and that V =< w; v2; : : : ; vr >. We would have that ¸!0 LG(< w; v2; : : : ; vr >) = LG(< w + ¸v1; v2; : : : ; vr >) = LG(< v1; : : : ; vr >) and ¸!0 0 0 LG(< w; v2; : : : ; vr >) = LG(< w + ¸v1; v2; : : : ; vr >) = LG(< v1; : : : ; vr >) INVERSE LIMIT OF GRASSMANNIANS 3 0 L that is, LG(< v1; : : : ; vr >) = LG(< v1; : : : ; vr >). But the morphism E1= ker L ,! 0 E2 is injective, so that L(< v¹1;:::; v¹r >) 6= L(< v¹ 1;:::; v¹r >), which leads to contradiction. 0 LG : Grn(E1) ! Gr(E2) is an a±ne morphism (for n > dimk ker L), because if U is the a±ne open subset of Gr(E2) of vector subspaces of E2 that are supplementary 0 ¡1 vector spaces of V , a subspace of E of ¯nite dimension, then LG (U) is the a±ne open subset of Gr(E1) of vector subspaces of E1 that are supplementary vector spaces of L¡1(V 0). For n · dimk ker L, LG is constant over Grn(E1) and its image is Im L 2 Gr(E2). 6. De¯nici¶on: Let f : X 99K Y be a rational morphism between schemes, being Y a separated one, and let g : T ! X, h : T ! Y be two morphisms of schemes. We shall say that f ± g = h if g values in the de¯nition domain of f and it holds, now with sense, that f ± g = h. We shall say that a scheme X = lim Xi is the inverse limit of the family of à i2I rational morphisms fXi 99K Xjgi>j if it holds that " # Homk¡sch(Y; X) = lim lim Homk¡sch(Y; Xi) ! à j2I i>j That is, giving a morphism Y ! X is equivalent to giving a family of morphisms Y ! Xi for all i greater than some j, satisfying the corresponding commutative diagrams, and we say two of these families are equal if the morphisms that value in the same Xi are equal. 7. Teorema : Let E = lim Ei be an inverse limit of vector spaces with kernels à i2I and cokernels of ¯nite dimension. Let us consider the projective system of rational morphisms fGr(Ei) 99K Gr(Ej)gi¸j. The scheme lim Gr(Ei) exists. à i2I " # Proof. The functor X à F (X) = lim lim Homk¡sch(X; Gr(Ei)) is a sheaf for ! à j2I i>j the Zariski topology, then to prove that it is representable it is su±cient to prove it locally. 0 Fixed an index j let us consider a subspace Vj of ¯nite dimension of Ej, and for 0 every morphism Ei ! Ej let us consider the vector subspace Vi of Ei inverse image 0 of Vj , and the a±ne schemes Ui ½ Gr(Ei), made up by the supplementary vector 0 spaces of V with respect to Ei. The inverse limit Gj = lim Ui, that is an a±ne i à i¸j scheme because lim Spec Ai = Spec lim Ai, turns out to be an open subfunctor of F : à ! i i ¢ Given a morphism (Spec A) ! F , that is, a family of morphisms Spec A ! Gr(Ei) for all i ¸ j0, where j0 is an index we can assume greater than j, we have to prove ¢ ¢ that Gj £F (Spec A) is an open subset of (Spec A) . The morphism Spec A ! ¢ Gr(Ej0 ) is de¯ned by a direct summand W of Ej0 A, and Gj £F (Spec A) identi¯es 0 with the open subset of the points of Spec A where the morphism V 0 ! Ej0 =W jA A is an isomorphism.
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