Inverse limits of projective systems

Transcription

1 Inverse limits of projective systems November 6, 2015

2 Part 1, Splines as inverse limits of projective systems E p,q 2 = lim (p) Γ lim (q) F lim ( ) Λ F κ(γ) Inverse limits of projective systems November 6, / 33

3 λ d Λ = 3 λ d = 2 λ d = 1 m Λ = 0 Inverse limits of projective systems November 6, / 33

4 Definition (Abstract cell complex) An abstract cell complex is a ranked poset satisfying the following condition: For any λ 1 2 λ 2, λ 2 λ 1 = {τ 1, τ 2 }, i.e. there exists exactly two elements in λ 2, τ 1 and τ 2, such that λ 1 < τ i < λ 2, i = 1, 2. OK Not OK Inverse limits of projective systems November 6, / 33

5 Definition (Local orientation) A local orientation of an abstract cell complex Λ is a map ɛ : Cov(Λ) {±1} such that for λ 1 2 λ 2, with λ 2 λ 1 = {τ 1, τ 2 } we have ɛ λ1 <τ 1 ɛ τ1 <λ 2 + ɛ λ1 <τ 2 ɛ τ2 <λ 2 = 0 where we use the notation ɛ λ<τ = ɛ(λ <τ). Inverse limits of projective systems November 6, / 33

6 (Dual poset) Dual poset Λ : Same objects as Λ, but with reversed ordering d(λ ) = d Λ d(λ) q-skeleton of Λ corresponds to the (d Λ q)-skeleton of Λ λ = ( λ) σ τ 1 τ 2 τ 2 τ 1 σ Inverse limits of projective systems November 6, / 33

7 (Order complex) Order complex Λ (1) : Ranked poset of geometric dimension d Λ (1) = r(λ) and minimal rank m Λ (1) = 0 Order dimension : r(λ) = max{p λ 0 <λ 1 < <λ p Λ} The p-skeleton of Λ (1) consists of sequences λ 0 <λ 1 < <λ p and the ordering is by inclusion. Inverse limits of projective systems November 6, / 33

8 Λ Λ (1) Inverse limits of projective systems November 6, / 33

9 Definition (Projective system) Let Λ be a poset, and let A be the category of abelian groups (or any abelian category). A projective system with values in A on Λ is a contravariant functor F : Λ A, where Λ is considered as a category, i.e. for any λ Λ we associate an object F(λ), and for a relation λ <σ, a morphism F λ<σ : F(σ) F(λ) in A. σ F(σ) τ 2 τ 1 F(τ 1 ) F(τ 2 ) γ F(γ) Inverse limits of projective systems November 6, / 33

10 Definition (Cellular homology groups) H p (Λ, F) = H p (C (Λ, F), δ), p 0 Let Λ be a locally oriented ranked poset of geometric dimension d Λ, with local orientation ɛ, and let F be a projective system on Λ. C p (Λ, F) = λ Λ p F(λ)[λ] p 0 with differential δ : C p (Λ, F) C p 1 (Λ, F) given by δ(f σ [σ]) = λ σ ɛ λ<σ F λ<σ (f σ )[λ], f σ F(σ) Inverse limits of projective systems November 6, / 33

11 Definition (Cellular cohomology groups) H p (Λ, F ) = H p (C (Λ, F ), ), p 0 Let F be a projective system of abelian groups on Λ. C p (Λ, F ) = λ Λ p F (λ ) Differential : C p (Λ, F ) C p+1 (Λ, F ) given by ξ(σ) = λ σ ɛ λ<σ F σ <λ ξ(λ) Inverse limits of projective systems November 6, / 33

12 H q (Λ, F ) H q (Λ, F), q 0 A projective system F on Λ induces a projective system F = Hom k (F, k) on Λ by F (λ ) = Hom k (F(λ), k) = F(λ) and F σ <λ : F (λ ) F (σ ) given by F σ <λ (φ(λ)) = φ(λ) F λ<σ Inverse limits of projective systems November 6, / 33

13 Proposition H p (Λ, F) H d p (Λ, F), p 0 For finite Λ and a projective system F on Λ we have an isomorphism C p (Λ, F) = λ (Λ ) p F(λ) β λ Λ d p F(λ)[λ] = C d p (Λ, F) given by β(ξ) = λ Λ d p ξ(λ )[λ], β (ξ) = δβ(ξ) Inverse limits of projective systems November 6, / 33

14 Definition (Inverse limit functor) lim (p) F = H p (D (Λ, F)), p 0 Λ Define complex D p (Λ, F) = F(λ 0 ) λ 0 <λ 1 < <λ p Λ with differential δ : D p (Λ, F) D p+1 (Λ, F) given by δξ(λ 0 < <λ p+1 ) = F λ0 <λ 1 ξ(λ 1 < <λ p+1 ) p+1 + i=1 ( 1) i ξ(λ 0 < ˆλ i <λ p+1 ) Inverse limits of projective systems November 6, / 33

15 D p (Λ, F) = C p (Λ (1), F) A projective system F on Λ induces a projective system F on the dual of the order complex, (Λ (1) ), given by and F(λ 0 < <λ p ) = F(λ 0 ) F(λ 0 < ˆλ i <λ p λ 0 < <λ p ) = { Fλ0 <λ 1 if i = 0 Id F (λ0 ) if i 0 Inverse limits of projective systems November 6, / 33

16 D (Λ, F) = lim Λ D (F) Denote by D p (F), for p 0, the projective system on Λ given by D p (F)(λ) = C p (ˆλ (1), F) = λ 0 <λ 1 < <λ p λ F(λ 0 ) where, for λ <σ, D p (F)(σ) D p (F)(λ) is the projection. Inverse limits of projective systems November 6, / 33

17 and E p,q 2 = H p (Λ, lim (q) F) ˆλ E p,q 2 = H q (H p (Λ, D (F))) both converge to the cohomology of the total complex. First-quadrant double complex C p,q = C p (Λ, D q (F )) δ : C p,q C p,q+1 is the differential of the complex D (ˆλ, F) : C p,q C p+1,q is the differential of the cell complex. Inverse limits of projective systems November 6, / 33

18 Proposition E p,q 2 = H p (Λ, lim ˆλ (q) F ) = { H p (Λ, F) for q = 0 0 for q 0 For any λ Λ we have { (q) F(λ) if q = 0 lim F = 0 if q 0 ˆλ Inverse limits of projective systems November 6, / 33

19 Theorem a) There is a first quadrant spectral sequence E p,q 2 = lim Λ (q) H p ( ˆλ, F(λ 0 )) converging to cellular cohomology H n (Λ, F). b) If Λ has acyclic closed cells, then H p (Λ, F) = lim Λ (d p) F, p 0 Λ has acyclic closed cells means H p ( ˆλ { F(λ0 ) for p = 0, F(λ 0 )) = 0 for p 0 Inverse limits of projective systems November 6, / 33

20 Theorem Let κ : Γ Λ (1) be a order-preserving map such that κ(γ) Λ (1) is closed for all γ Γ with im κ = Λ, and such that if γ 1, γ 2 Γ, and λ κ(γ 1 ) κ(γ 2 ), then there exixts γ < γ 1, γ 2 such that λ κ(γ). Then there is a spectral sequence E p,q 2 = lim Γ (p) lim (q) F κ(γ) converging to lim ( ) F Λ Inverse limits of projective systems November 6, / 33

21 Part 2, Morgan-Scott triangulation of a triangle Inverse limits of projective systems November 6, / 33

22 Definition (Spline space) Let r be a positive integer and a triangulation of a plane area S. The vector space S r n( ) consists of all piecewise polynomial functions f of total degree less than n which has r 1 continuous derivatives on, and the r-th derivative D r f is continuous everywhere except for a finite number of points. Inverse limits of projective systems November 6, / 33

23 τ 6 τ 2 σ 2 τ 7 Σ γ 1 Σ 1 3 τ 4 γ 3 σ 0 τ τ 3 9 τ σ 1 γ 2 σ 3 1 τ 5 Σ 2 τ 8 Figure: A Morgan-Scott triangulation MS Inverse limits of projective systems November 6, / 33

24 Lemma Let f S r n( ) be a piecewise polynomial function over a triangulation of a plane region S. Let σ 1 and σ 2 be two adjacent 2-cells, and τ σ 1 σ 2 a 1-cell defined by a linear functional L τ. Then f has continuous r-th derivatives almost everywhere if and only if f (σ 1 ) f (σ 2 ) (L τ ) r+1. Inverse limits of projective systems November 6, / 33

25 Sn( r MS ) = limf 2-cell σ; F(σ) = R n = k[x, y] n 1-cell τ Λ MS corresponds to the zero set of a linear functional L τ : F(τ) = (R/I τ ) n, where I τ = (L r τ). A vertex γ: I γ = τ γ I τ, F(γ) = (R/I γ ) n. As a vector space over k we have dim(r n ) = ( ) n+2 2. Inverse limits of projective systems November 6, / 33

26 dim(lim F) = dim(lim F 0 ) + ( ) n + 2 F 0 is obtained from F by annihilating the center triangle, i.e. F 0 = F except for F 0 (σ 0 ) = 0, Consider F 0 as a sub projective system of F with cokernel Q. The projective system Q is given by Q(σ 0 ) = R and 0 elsewhere. Obviously we have lim Q = R and lim 2 (1) Q = 0 for i 0. The map F Q has a section, inducing a short-exact sequence 0 lim F 0 lim F Q 0 Inverse limits of projective systems November 6, / 33

27 We introduce another system F 1 which agree with F 0 on all cells except for the boundary of the middle triangle, i.e. F 1 (τ i ) = R, i = 1, 2, 3, F 1 (γ j ) = R/I {τj+1,τ 2j+1 }, j = 1, 2, 3 Induces short-exact sequence Kernel K given by 0 K F 1 F 0 0 K (τ i ) = I i, i = 1, 2, 3, K (γ j ) = I γj /I γj \{τ j+1,τ 2j+1 }, j = 1, 2, 3 and K = 0 for the rest of the cells. Inverse limits of projective systems November 6, / 33

28 We compute cohomology of F 1 limf 1 (I 4 I 7 ) (I 5 I 8 ) (I 6 I 9 ), Λ MS lim Λ MS (j) F 1 = 0, j 1 Since K = 0 on all maximal cells, we have lim K = 0. By a dimension argument we have lim (j) K = 0 for j 3. Inverse limits of projective systems November 6, / 33

29 (Conclusion so far) Long-exact sequence 0 limf 1 limf 0 lim (1) K 0 and an isomorphism lim (1) F 0 lim Λ MS (2) K dim(lim F) = dim(lim F 1 ) + ( ) n dim(lim (1) K ) Inverse limits of projective systems November 6, / 33

30 Can compute and dim(lim F 1 ) = 3 ( ) n 2r 2 0 lim (1) K I 1 I 2 I 3 (I 1 + I 2 /J 1 ) (I 2 + I 3 /J 2 ) (I 3 + I 1 /J 3 ) lim (2) K 0 Inverse limits of projective systems November 6, / 33

31 ( ) n r + 1 dim(i) n =, n d r 2d 2r r <d < 4 dim (I 1 + I 2 /J 1 ) d = 3 r r d r d <2r d 2r + 1 Inverse limits of projective systems November 6, / 33

32 Theorem (Morgan-Scott formula) The dimension of the truncated ring of r times continous differentiable piecewise polynomial functions of degree n over MS is given by dim(sn(λ r MS ) = 1 2 n(n 18r + 3) r(13r 1) + 1 ɛ + dim(lim (2) K ) where ɛ = 1 for r 2 (mod 3) and ɛ = 0 for r 2 (mod 3). The term lim (2) K is independent of n forn 0. Λ MS Inverse limits of projective systems November 6, / 33

33 This gives for r = 1, For r = 2 we get dim(limf) = 1 2 n(7n 15) dim(lim (2) K ) dim(limf) = 1 2 n(7n 33) dim(lim (2) K ) Inverse limits of projective systems November 6, / 33

34 For r = 1 the condition for non-vanishing of lim (2) K is the existence of a non-trivial relation a 4 (L 4 ) a 9 (L 9 ) 2 = 0 This is equivalent to the existence of a conic section, tangent to the six (2) K is 1. lines L j, j = 4,..., 9. In this case the dimension of lim Inverse limits of projective systems November 6, / 33