On some local cohomology ltrations

Transcription

1 On some local cohomology ltrations Alberto F. Boix Universitat Pompeu Fabra Combinatorial Structures in Geometry, Osnabrück 2016

2 Motivation Graded pieces of local cohomology Betti numbers of arrangements Highlights of our work Filtrations of local cohomology modules Local cohomology decompositions The Frobenius structure of local cohomology

3 Based on joint work in progress with: Josep Àlvarez Montaner (Universitat Politècnica de Catalunya). Santiago Zarzuela (Universitat de Barcelona).

4 Local cohomology A commutative Noetherian ring, I A ideal. I = (a 1,..., a t ) M A-module H i I (M) := H i (0 M ) t M ai... M a1 a t 0. i=1

5 MOTIVATION

6 SHEAF COHOMOLOGY AND LOCAL COHOMOLOGY

7 Deligne Correspondence A Z n -graded, I A homogeneous. M graded A-module. U := Spec(A) \ V(I ), v Z n. Γ(U, M(v) 0 ) = D I (M) v, H i (U, M(v) 0 ) = H i+1 I (M) v, i 1.

8 GRADED PIECES OF LOCAL COHOMOLOGY

9 THE CASE OF STANLEY-REISNER RINGS

10 Stanley-Reisner rings simplicial complex (n vertices). K[ ] Stanley-Reisner ring. F, link(f ) := {G \ F : G, F G}. v Z n, v := {i {1,..., n} : v i < 0}.

11 Hochster formula (1996): statement H i m(k[ ]) v = H i v 1 (link(v ); K). Moreover, H i m(k[ ]) = F H m F (K[F ]) m i,f as Z n -graded K-vector spaces, where m i,f := dim K H i F 1 (link(f ); K).

12 Hochster formula: prime characteristic remark Enescu-Hochster (2008) If K Z/pZ, the decomposition is preserved by the natural Frobenius action. Consequence H i m(k[ ]) has nitely many F-compatible submodules.

13 Terai formula (1999): statement and remark K[ ] = K[x 1,..., x n ]/I. H i I (K[x 1,..., x n ]) v = Hn i v 1(link(v ); K). (Miller, 2000) Hochster and Terai formulas are equivalent.

14 THE CASE OF TORIC FACE RINGS

15 Toric face rings: denition Σ R n rational pointed fan. M := {M C C Z n : C Σ} monoidal complex. where K[M] := x v x v := { v Z n a v x v : a v 0 for nite v s { x v+v, if v, v M C for some C, 0, otherwise. },

16 Toric face rings: general examples (Stanley, 1987) If M C = C Z n C, K[M] = K[Σ]. If Σ = Fan(C), K[M] = K[M C ]. If C has exactly dim(c) generators C, then K[Σ] = K[ ].

17 Toric face rings: specic example O = (0, 0, 0), A 1 = (2, 0, 0), A 2 = (0, 2, 0), A 3 = (0, 0, 2), A 4 = (1, 1, 0). Σ := Fan (OA 1 A 2, OA 1 A 3, OA 2 A 3 ). M := { A 1, A 2, A 4, A 1, A 3, A 2, A 3 } K[M] = K[x 1, x 2, x 3, x 4 ] (x 1 x 2 x 2 4, x 3x 4 )

18 Brun-Bruns-Römer formula (2007) Look at Σ as poset (faces ordered by inclusion). (C, 1 Σ) := {D Σ : C D}. As graded Z n -vector spaces H i m(k[σ]) = C Σ H dim(c) m (K[C]) m i,c, where m i,c := dim K H i dim(c) 1 ( (C, 1 Σ); K )

19 Brun-Bruns-Römer formula: remarks Original proof: very technical. (Ichim, Römer (2007)) Alternative proof in the spirit of Hochster's one. What about the natural Frobenius action?

20 TOPOLOGICAL BETTI NUMBERS OF ARRANGEMENTS

21 Topological Betti numbers: reminder X topological space. β i (X ) := dim Q H i (X, Q).

22 Betti numbers of complements of arrangements K eld. A := {A j } collection of linear varieties K n. I K[x 1,..., x n ] its dening ideal. I = I 1... I t primary decomposition. P: all dierent sums of I i 's ordered by reverse inclusion. p P is just I p, sum of certain I i 's.

23 BETTI NUMBERS OF ARRANGEMENTS AND LOCAL COHOMOLOGY

24 Àlvarez Montaner, García López, Zarzuela Armengou (2003) A arrangement K n. I K[x 1,..., x n ] dening ideal of A There is a ltration F := {F r k } of H r I (K[x 1,..., x n ]) s.t. F r k /F r k 1 = ht(i p)=k H ht(ip) I p (K[x 1,..., x n ]) mr,p, where m r,p := dim K Hht(Ip) r 1((p, 1 P); K).

25 Àlvarez Montaner, García López, Zarzuela Armengou (2003) β r (R n \ j A j ) = p P m r+1,p (r 1). β r (C n \ j A j ) = p P m r+1 ht(ip),p (r 1). Both are consequences of Goresky-Macpherson formula.

26 HIGHLIGHTS OF OUR WORK

27 Highlights of our work General way to produce certain ltrations of local cohomology modules. Producing a decomposition for local cohomology of central arrangements of linear varieties. Alternative (and new) proof of Brun-Bruns-Römer formula. Brun-Bruns-Römer formula is preserved by Frobenius.

28 END OF INTRODUCTION

29 General setting I A ideal, I = I 1... I t decomposition. P: all dierent sums of I i 's ordered by reverse inclusion. p P is just I p, sum of certain I i 's. P := P {0 P, 1 P}.

30 A picture I I 1 I 2 I 1 + I 2 A

31 Inverse limit rings (A/I p A/I q ) p q inverse system of A-algebras A P := lim p P A/I p.

32 Inverse limit rings: examples simplicial complex. Σ R n rational pointed fan. M monoidal complex. K[ ] = lim F K[F ], K[M] = lim C Σ K[M C ].

33 Order complex of a poset and homology P nite poset, P := P {0 P, 1 P}. Q P subposet, (Q) order complex of Q. (p, 1 P) := {q P : q > p}. H r ((p, 1 P); K) := H r ( ((p, 1 P)); K)

34 FIRST FILTRATION

35 First ltration: setting A K, K eld. p P, H j I p (A) = 0 j h p := ht(i p ). p < q with h p h q, Hom A (H hp I p (A), H hq I q (A)) = 0.

36 First ltration: statement Theorem (Àlvarez Montaner, B., Zarzuela) If 0 r cd(i ), an increasing, nite ltration {Gk r} of HI r (A) by A-modules such that G r k /G r k 1 = p P H hp I p (A) mr,p, where m r,p := dim K Hhp r 1((p, 1 P); K)

37 SKETCH OF PROOF

38 Sketch of proof Roos ( 1 T [ ] (E 1 ) ) Roos ( 1 T [ ] (E 0 ) ) Roos ( 1 T [ ] (A) ) 0... Roos ( 0 T [ ] (E 1 ) ) Roos ( 0 T [ ] (E 0 ) ) Roos ( 0 T [ ] (A) ) 0... T (E 1 ) T (E 0 ) T (A) T := H 0 I, T [ ] := (H 0 I p ) p P.

39 Sketch of proof: Mayer-Vietoris spectral sequence E i,j 2 = L i colim p P H j I p (A) i H j i I (A).

40 Sketch of proof A K, K eld. p P, H j I p (A) = 0 j h p := ht(i p ). p < q with h p h q, Hom A (H hp I p (A), H hq I q (A)) = 0. These assumptions guarantee degeneration at second page.

41 WHEN THESE ASSUMPTIONS ARE FULFILLED?

42 When these assumptions are fullled? I denes an arrangement of linear varieties, I denes a toric face ring K[M], where K Z/pZ, p prime. Each K[M C ] is Cohen-Macaulay C Σ.

43 First ltration: some remarks Same formalism works for other functors (satisfying certain technical conditions). You can replace A by an A-module M (satisfying the corresponding assumptions).

44 First ltration: some remarks A K, K eld, A regular. If K Q, it is a ltration of D-modules. If K Z/pZ, it is a ltration of F -modules.

45 SECOND FILTRATION

46 Second ltration: setting A K, K eld. A/[ ] is asque. H j m(a/i p ) = 0 j d p := dim(a/i p ). p < q ( with d p d q, ) Hom A Hm dp (A/I p ), Hm dq (A/I q ) = 0.

47 Second ltration: statement A P := lim p P A/I p. Theorem (Àlvarez Montaner, B., Zarzuela) For 0 r dim(a P ) an increasing nite ltration {H r k } of H r m(a P ) by A-modules such that H r k/h r k 1 = p P H dp m (A/I p ) mr,p, where m r,p := dim K H r d p 1 (( p, 1 P ) ; K )

48 SKETCH OF PROOF

49 Sketch of proof... 0 Roos 1 (T (A/[ ])) Roos 1 (T (I 0 )) Roos 1 (T (I 1 ))... 0 Roos 0 (T (A/[ ])) Roos 0 (T (I 0 )) Roos 0 (T (I 1 ))... 0 lim p P (T (A/[ ])) lim p P (T (I 0 )) lim p P (T (I 1 )) T := H 0 m.

50 Sketch of proof: spectral sequence A P := lim p P A/I p. E i,j 2 = R i lim p P H j m (A/[ ]) i H i+j m (A P ).

51 Sketch of proof A K, K eld. A/[ ] is asque. H j m(a/i p ) = 0 j d p := dim(a/i p ). p < q ( with d p d q, ) Hom A Hm dp (A/I p ), Hm dq (A/I q ) = 0. These assumptions guarantee degeneration at second page.

52 WHEN THESE ASSUMPTIONS ARE FULFILLED?

53 When these assumptions are fullled? I denes a central arrangement of linear varieties. I denes a Stanley toric face ring K[Σ].

54 Second ltration: some remarks Same formalism works for other functors (satisfying certain technical conditions). You can replace A/[ ] by any asque inverse system (satisfying the corresponding assumptions).

55 LOCAL COHOMOLOGY DECOMPOSITIONS

56 Some short exact sequences H r 1 H r 2 H r 2 H r 3 H r 2/H r 1 H r 3/H r 2. H r c H r m(a P ) H r m(a P )/H r c 1. Question How far are these short exact sequences from being splitted?

57 In general, no splitting as A-modules I := (x, yz) K[x, y, z] (x,y,z) := A m := (x, y, z) A. 0 Hm(A/m) 0 Hm(A/I 1 ) Hm(A/(x, 1 y)) Hm(A/(x, 1 z)) 0. If was splitted {m} = Att A (Hm(A/m)) 0 Att A (Hm(A/I 1 )) = {(x, y), (x, z)}, contradiction!!

58 Hochster type formula Theorem (Brun, Bruns, Römer) There is a K-vector space isomorphism H j m ( lim p P A/I p ) = p P H dp m (A/I p ) m j,p, where m j,p = dim K H j d p 1 ( (p, 1 P); K ).

59 Hochster type formula: remark In case A P is Z n -graded, the isomorphism preserves this grading. Same decomposition works for other functors (satisfying similar assumptions).

60 Hochster type formula for central arrangements A central arrangement K n of linear varieties. I K[x 1,..., x n ] dening ideal of A H j m ( lim p P A/I p as Z n -graded K-vector spaces. ) = p P H dp m (A/I p ) m j,p

61 FROBENIUS STRUCTURE

62 Frobenius structure: assumptions p prime, K Z/pZ eld. Σ R n rational pointed fan.

63 Frobenius structure: statement Theorem (Àlvarez Montaner, B., Zarzuela) H j m (K[Σ]) = C Σ H dim(k[c Zn ]) m (K[C Z n ]) m j,c is preserved by the natural Frobenius action. In particular, H j m (K[Σ]) has nitely many F -stable submodules.

64 DANK FÜR IHRE AUFMERKSAMKEIT