Extensions of Stanley-Reisner theory: Cell complexes and be

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1 : Cell complexes and beyond February 1, 2012

2 Polyhedral cell complexes Γ a bounded polyhedral complex (or more generally: a finite regular cell complex with the intersection property).

3 Polyhedral cell complexes Γ a bounded polyhedral complex (or more generally: a finite regular cell complex with the intersection property). k a field. Form the k-vector space with basis the cells of dimension p, C p (Γ) = k F. F cell of dimension p

4 Homology and cohomology 1. Augmented chain complex: C (Γ) : C i (Γ) with differential d(f) = d C i 1 (Γ) C 0 (Γ) k, sgn(g, F) G. G is p 1 face of F

5 Homology and cohomology 1. Augmented chain complex: C (Γ) : C i (Γ) with differential d(f) = d C i 1 (Γ) C 0 (Γ) k, sgn(g, F) G. G is p 1 face of F Reduced homology H i (Γ) = H i (C (Γ)).

6 Homology and cohomology 1. Augmented chain complex: C (Γ) : C i (Γ) with differential d(f) = d C i 1 (Γ) C 0 (Γ) k, sgn(g, F) G. G is p 1 face of F Reduced homology H i (Γ) = H i (C (Γ)). 2. Augmented cochain complex: C (Γ) = Hom(C (Γ), k). Reduced cohomology H i (Γ) = H i (C (Γ)).

7 Squarefree modules Let ǫ by i th unit vector in N n. An N n -graded S-module M is squarefree if for b N n the multiplication x M i b Mb+ǫi is an isomorphism whenever b i 1.

8 Squarefree modules Let ǫ by i th unit vector in N n. An N n -graded S-module M is squarefree if for b N n the multiplication x M i b Mb+ǫi is an isomorphism whenever b i 1. Identify subset, f.ex. R = {1, 2, 5} [6], with its characteristic vector r = (1, 1, 0, 0, 1, 0) in N 6. Let M R = M r.

9 Squarefree modules Let ǫ by i th unit vector in N n. An N n -graded S-module M is squarefree if for b N n the multiplication x M i b Mb+ǫi is an isomorphism whenever b i 1. Identify subset, f.ex. R = {1, 2, 5} [6], with its characteristic vector r = (1, 1, 0, 0, 1, 0) in N 6. Let M R = M r. The module M is determined by M R where R [n], and by the multiplications between them.

10 Alexander dual Cellular complexes The Alexander dual module M is defined by: (M ) R is the vector space dual Hom k (M R c, k).

11 Alexander dual Cellular complexes The Alexander dual module M is defined by: (M ) R is the vector space dual Hom k (M R c, k). The multiplication (M x ) i R (M ) R {i} is the dual map of the multiplication M R c \{i} x i MR c.

12 Free squarefree modules Free squarefree module: R [n] S β R R. Example S.{1, 2, 4} = S.(1, 1, 0, 1, 0) is a free squarefree module. S.(1, 2, 0, 1, 0) is a free module but not squarefree.

13 Standard duality Cellular complexes F sq is the category of complexes of free squarefree modules. Example The enriched chain and cochain complexes E(Γ) and E (Γ) are in F sq. Standard duality D : F sq F sq defined by D(P ) = Hom S (P,ω S ).

14 Alexander duality Cellular complexes Alexander duality A : F sq F sq. First form Alexander dual (P ). Then take a minimal squarefree resolution Q (P ). Define A(P ) = Q.

15 Alexander duality Cellular complexes Alexander duality A : F sq F sq. First form Alexander dual (P ). Then take a minimal squarefree resolution Q (P ). Define A(P ) = Q. Example P = S.(1, 1, 0, 1, 0). Alexander dual (P ) = S/(x 1, x 2, x 4 ). Minimal squarefree resolution of S/(x 1, x 2, x 4 ) is: S S 3 S 3 S. Then Alexander dual Q = A(P ) is this complex.

16 Cellular complexes Yanagawa Q D Q A R A D P D P A[ n] R

17 Cellular complexes For simplicial complexes E is the enriched chain complex, F the resolution of the Stanley-Reisner ring. E[ 1] D A E [ 1] G A[ n] D F D F A G

18 Stanley-Reisner complex instead of Stanley-Reisner resolution For cell complexes Γ one has enriched chain complex E(Γ). May turn the wheel back and get a complex F : F p F 1 F 0.

19 Stanley-Reisner complex instead of Stanley-Reisner resolution For cell complexes Γ one has enriched chain complex E(Γ). May turn the wheel back and get a complex F : F p F 1 F 0. The homology at F p is given by: Its squarefree parts of degree d are the (d 1 p)-dimensional faces of Γ with p vertices.

20 Stanley-Reisner complex instead of Stanley-Reisner resolution For cell complexes Γ one has enriched chain complex E(Γ). May turn the wheel back and get a complex F : F p F 1 F 0. The homology at F p is given by: Its squarefree parts of degree d are the (d 1 p)-dimensional faces of Γ with p vertices. Point: Stanley-Reisner theory can be done equally well for polyhedral complexes as for simplicial complexes!

21 Numerical invariants Symmetry breakdown. A complex P of free graded S-modules P : S( j) bi j comes with three sets of numerical invariants: The graded Betti numbers bj i.

22 Numerical invariants Symmetry breakdown. A complex P of free graded S-modules P : S( j) bi j comes with three sets of numerical invariants: The graded Betti numbers b i j. The homology modules and their Hilbert functions h i j = dim k H i (P ) j.

23 Numerical invariants Symmetry breakdown. A complex P of free graded S-modules P : S( j) bi j comes with three sets of numerical invariants: The graded Betti numbers b i j. The homology modules and their Hilbert functions h i j = dim k H i (P ) j. The cohomology modules and their Hilbert functions c i j = dim k H i (Hom(P,ω S )) j.

24 Squarefree invariants A complex P of free squarefree S-modules P : R (S k BR i ). Three sets of invariants for subsets R [n]. Betti spaces BR i. Homology spaces HR i = Hi (P ) R. Cohomology spaces CR i = Hi Hom(P,ω S ) R.

25 Cellular complexes Rotations of invariants Perfect symmetry! The functor A D cyclically rotates the homological invariants. B i R (A D(P )) = H i+r R c (P ). H i R (A D(P )) = C i R (P ). C i R (A D(P )) = B i+r R c (P ). H B A D 7 A D C A D

26 From pure resolutions to pure complexes. A resolution of a graded S-module of the form S( d 0 ) β 0 S( d 1 ) β 1 S( d r ) βr is called a pure resolution.

27 From pure resolutions to pure complexes. A resolution of a graded S-module of the form S( d 0 ) β 0 S( d 1 ) β 1 S( d r ) βr is called a pure resolution. A complex P of free squarefree module is pure if P i = R [n] S k BR i where all R have the same cardinality d i.

28 Cellular complexes Resolutions of Cohen-Macaulay modules Q A D A D 6 A D P R

29 Cellular complexes Resolutions of Cohen-Macaulay modules Q A D A D 6 A D P R Fact P is a resolution of a Cohen-Macaulay module if and only if Q and R are linear complexes.

30 Cellular complexes Resolutions of Cohen-Macaulay modules Q A D A D 6 A D P R Fact P is a resolution of a Cohen-Macaulay module if and only if Q and R are linear complexes. In particular P is a pure resolution of a Cohen-Macaulay module if and only if: i.p is pure, ii.q is linear, iii.r is linear.

31 Problem Construct complexes P,Q, and R which are all pure.

32 Problem Construct complexes P,Q, and R which are all pure. Example S = k[x 1, x 2, x 3 ]. P Q R : S [x 1x 2,x 1 x 3,x 2 x 3 ] S( 2) 3 [ x1 x 2 x 3 : S 2 S( 2) 3 S( 3) : S( 1) 3 S( 2) 6 S( 3) 2 ]

33 Resolutions but not necessarily of ideals Only for few integer sequences 0 = d 0 < d 1 < < d r can we hope to get a squarefree free resolution S( d 0 ) β 0 S( d 1 ) β 1 S( d r ) βr of a quotient ring S/I, i.e. with β 0 = 1.

34 Resolutions but not necessarily of ideals Only for few integer sequences 0 = d 0 < d 1 < < d r can we hope to get a squarefree free resolution S( d 0 ) β 0 S( d 1 ) β 1 S( d r ) βr of a quotient ring S/I, i.e. with β 0 = 1. Are there natural classes of squarefree modules which give as interesting resolution theory as what one has for ideals? And which have the potential of giving all (or many) pure resolutions?

35 Resolutions but not necessarily of ideals If simplicial complex with SR-ideal I = (x F 1, x F 2,...,x Fm ), then SR-resolution starts with S i S( F i ). Choose integers a and b and consider resolutions which start with S a where φ is a general map. φ i S( F i ) b