Hypergraphs and Regularity of Square-free Monomial Ideals

Transcription

1 Hypergraphs and Regularity of Square-free Monomial Ideals Jason McCullough (Joint w/kuei-nuan Lin) MSRI Rider University January 9, 2013 Joint AMS-MAA Math Meetings Special Session in Commutative Algebra and Algebraic Geometry

2 Notation K any field (e.g. Q,C,Z/pZ)

3 Notation K any field (e.g. Q,C,Z/pZ) R = K[X 1,X 2,...,X n ], polynomial ring in n variables

4 Notation K any field (e.g. Q,C,Z/pZ) R = K[X 1,X 2,...,X n ], polynomial ring in n variables R = i=0 R i is a standard graded ring, where R i = K-vector space of homogeneous degree i polynomials

5 Notation K any field (e.g. Q,C,Z/pZ) R = K[X 1,X 2,...,X n ], polynomial ring in n variables R = i=0 R i is a standard graded ring, where R i = K-vector space of homogeneous degree i polynomials R( d) = rank one free module with generator in degree d (so R( d) i = R i d )

6 Notation K any field (e.g. Q,C,Z/pZ) R = K[X 1,X 2,...,X n ], polynomial ring in n variables R = i=0 R i is a standard graded ring, where R i = K-vector space of homogeneous degree i polynomials R( d) = rank one free module with generator in degree d (so R( d) i = R i d ) I = (f 1,...,f t ) R a homogeneous ideal (i.e. each f j is in some R i )

7 Graded Free Resolutions Minimal, graded free resolution of R/I: 0 R/I R j R( j) β ϕ 1j 1 R( j) β ϕ 2j 2 ϕ j j j R( j) β pj 0

8 Graded Free Resolutions Minimal, graded free resolution of R/I: 0 R/I R j R( j) β ϕ 1j 1 R( j) β ϕ 2j 2 ϕ j j j R( j) β pj 0 Regularity of R/I = reg(r/i) = max{j i β ij 0}

9 Betti Tables Record the Betti numbers β ij in a matrix called the Betti table:

10 Betti Tables Record the Betti numbers β ij in a matrix called the Betti table: i 0: β 0,0 β 1,1 β 2,2 β i,i 1: β 0,1 β 1,2 β 2,3 β i,i+1 2: β 0,2 β 1,3 β 2,4 β i,i j: β 0,j β 1,j+1 β 2,j+2 β i,i+j.....

11 Betti Tables Record the Betti numbers β ij in a matrix called the Betti table: i 0: β 0,0 β 1,1 β 2,2 β i,i 1: β 0,1 β 1,2 β 2,3 β i,i+1 2: β 0,2 β 1,3 β 2,4 β i,i j: β 0,j β 1,j+1 β 2,j+2 β i,i+j.. reg(r/i) = last nonzero row in Betti table....

12 Example: I = (efh,aefgij,bchij,dghij), R = K[a,...,j]

13 Example: I = (efh,aefgij,bchij,dghij), R = K[a,...,j] R R( 2) R( 5) 2 R( 6) R( 6) 4 R( 7) 0.

14 Example: I = (efh,aefgij,bchij,dghij), R = K[a,...,j] R R( 2) R( 5) 2 R( 6) R( 6) 4 R( 7) 0. Betti Table for R/I: : : : : : : :

15 Example: I = (efh,aefgij,bchij,dghij), R = K[a,...,j] R R( 2) R( 5) 2 R( 6) R( 6) 4 R( 7) 0. Betti Table for R/I: : : : : : : : reg(r/i) = 6

16 Definition V = finite set. Definition A labeled hypergraph (or just hypergraph) H on V with alphabet A is a tuple (V,X,E,E), where E : A P(V) is a function, X = {a A : E(a) } and E = Im(E).

17 Hypergraph of a Square-free Monomial Ideal I = (f 1,...,f µ ) R = K[A], Definition The labeled hypergraph of I is the labeled hypergraph H(I) = (V,X,E,E), where V = [µ] = {1,...,µ} and E : A P([µ]) is defined by E a = {j : a divides f j }.

18 Hypergraph of a Square-free Monomial Ideal I = (f 1,...,f µ ) R = K[A], Definition The labeled hypergraph of I is the labeled hypergraph H(I) = (V,X,E,E), where V = [µ] = {1,...,µ} and E : A P([µ]) is defined by E a = {j : a divides f j }. Note: A similar unlabeled version was used by Kimura-Terai-Yoshida to study arithmetical rank of square-free monomial ideals.

19 Example: H((efh, aefgij, bchij, dghij))

20 Example: H((efh, aefgij, bchij, dghij)) 1 a 2 3 4

21 Example: H((efh, aefgij, bchij, dghij)) 1 a b,c

22 Example: H((efh, aefgij, bchij, dghij)) 1 a b,c d

23 Example: H((efh, aefgij, bchij, dghij)) 1 e,f a b,c d

24 Example: H((efh, aefgij, bchij, dghij)) 1 e,f a 2 g 3 4 b,c d

25 Example: H((efh, aefgij, bchij, dghij)) 1 e,f a 2 h g 3 4 b,c d

26 Example: H((efh, aefgij, bchij, dghij)) 1 e,f a 2 h i,j g 3 4 b,c d

27 Separated Hypergraphs Definition A hypergraph H = (V,X,E,E) is separated if for every pair of vertices v,w V, there exist edges F,G E such that v F G and w G F.

28 An Equivalence Proposition There is a one-to-one correspondence { } { } separated labeledhypergraphs up to vertex per- Square-free monomials ideals mutation I H(I) I H H

29 Taylor Resolution for I = (f 1,...,f µ ) T 1 = R e 1,...,e µ. T i = i T 1 = R e F = e j1 e ji : F = {j 1 < j 2 < < j i } i (e F ) = i ( 1) k lcm(f F ) lcm(f F {jk }) e F {j k }, k=1

30 Taylor Resolution for I = (f 1,...,f µ ) T 1 = R e 1,...,e µ. T i = i T 1 = R e F = e j1 e ji : F = {j 1 < j 2 < < j i } i (e F ) = i ( 1) k lcm(f F ) lcm(f F {jk }) e F {j k }, k=1 Note: The Taylor Resolution forms a resolution of R/I but is rarely minimal.

31 Taylor Resolution for I = (f 1,...,f µ ) T 1 = R e 1,...,e µ. T i = i T 1 = R e F = e j1 e ji : F = {j 1 < j 2 < < j i } i (e F ) = i ( 1) k lcm(f F ) lcm(f F {jk }) e F {j k }, k=1 Note: The Taylor Resolution forms a resolution of R/I but is rarely minimal. However, it shows: reg(r/i) max{deg(lcm(f j1,...,f ji )) i : 1 i µ}.

32 When is the Taylor Resolution Minimal? Definition A labeled hypergraph H = (V,X,E,E) is saturated if for all v V, {v} E.

33 When is the Taylor Resolution Minimal? Definition A labeled hypergraph H = (V,X,E,E) is saturated if for all v V, {v} E. H(I) is saturated every minimal generator contains at least one variable not dividing any other generator.

34 When is the Taylor Resolution Minimal? Definition A labeled hypergraph H = (V,X,E,E) is saturated if for all v V, {v} E. H(I) is saturated every minimal generator contains at least one variable not dividing any other generator. A saturated example: k e,f a h i,j g b,c d

35 When is the Taylor Resolution Minimal? Proposition Let I = (f 1,...,f µ ) R be a square-free monomial ideal and let H = H(I) = (V,X,E,E). Then the following are equivalent: (1) H is saturated. (2) The Taylor resolution of R/I is minimal. In this case, we have reg(r/i) = X V = #variables #generators.

36 Example: I = (efhk, aefgij, bchij, dghij) k e,f a h i,j g b,c d

37 Example: I = (efhk, aefgij, bchij, dghij) k e,f a h i,j g b,c d Saturated reg(r/i) = X V = 11 4 = 7

38 Example: I = (efhk, aefgij, bchij, dghij) k e,f a h i,j g b,c d Saturated reg(r/i) = X V = 11 4 = 7 Question What if H is not saturated? How far does the regularity stray from this formula?

39 Isolated Open Vertices v V is open if {v} / E; otherwise v is closed

40 Isolated Open Vertices v V is open if {v} / E; otherwise v is closed For any v V, we define the neighbors of v in H to be the set N H (v) = {w V : w v and F E such that v,w F}.

41 Isolated Open Vertices v V is open if {v} / E; otherwise v is closed For any v V, we define the neighbors of v in H to be the set N H (v) = {w V : w v and F E such that v,w F}. Definition We say that H has isolated open vertices if for every open vertex v and every w N H (v), w is closed.

42 Isolated Open Vertices v V is open if {v} / E; otherwise v is closed For any v V, we define the neighbors of v in H to be the set N H (v) = {w V : w v and F E such that v,w F}. Definition We say that H has isolated open vertices if for every open vertex v and every w N H (v), w is closed. isolated opens not isolated opens

43 Isolated Open Vertices Theorem Let I R be a square-free monomial ideal and suppose that H = H(I) = (V,X,E,E) has only isolated open vertices. Then reg(r/i) X V.

44 Example: I = (efh, cefgij, abhij, dghij) 1 e,f a 2 h i,j g 3 4 b,c d

45 Example: I = (efh, cefgij, abhij, dghij) 1 e,f a 2 h i,j g 3 4 b,c d Isolated open vertices reg(r/i) X V = 10 4 = 6. actually, reg(r/i) = 6(firstexample)

46 Fill-in Theorem Q: What if you don t have isolated open vertices?

47 Fill-in Theorem Q: What if you don t have isolated open vertices? A: Fill vertices in until you do.

48 Fill-in Theorem Q: What if you don t have isolated open vertices? A: Fill vertices in until you do. Proposition Suppose that if we can add edges of size 1, say {v 1 },...,{v t } to E, where v 1,...,v t V, we obtain a new hypergraph with isolated open vertices. Then reg(r/i) X V +t.

49 Example: I = (di,ade,bij,fgij,efg,jh,ch) b b i j i j d f,g h d f,g h a e c a e c

50 Example: I = (di,ade,bij,fgij,efg,jh,ch) b b i j i j d f,g h d f,g h a e c a e c reg(r/i) X V +1 = = 4.

51 Simple Edges Definition An edge F E of H is called simple if F 2 and F has no proper subedges besides.

52 Simple Edges Definition An edge F E of H is called simple if F 2 and F has no proper subedges besides. simple edge not a simple edge

53 Isolated Simple Edges

54 Isolated Simple Edges Definition We say that H has isolated simple edges if every open vertex is contained in exactly one simple edge.

55 Isolated Simple Edges Definition We say that H has isolated simple edges if every open vertex is contained in exactly one simple edge. one isolated simple edge

56 Isolated Simple Edges Definition We say that H has isolated simple edges if every open vertex is contained in exactly one simple edge. one isolated simple edge not isolated simple edges

57 Isolated Simple Edges Theorem Suppose that H(I) = (V,X,E,E) has isolated simple edges. Then reg(r/i) = X V + F E F simple ( F 1).

58 Example H((ab,bcdef,ac,eg,fg,gh,hi)) a b c d e f g h i

59 Example H((ab,bcdef,ac,eg,fg,gh,hi)) a b c d e f g h i H has isolated simple edges E a and E g

60 Example H((ab,bcdef,ac,eg,fg,gh,hi)) a b c d e f g h i H has isolated simple edges E a and E g reg(r/i) = X V + F E ( F 1) = F simple 9 7+(2 1)+(3 1) = 5.

61 One-Dimensional Hypergraphs Corollary Suppose that dim(h) = 1 and that there exist closed vertices c 1,...,c t such that Then For all 1 i < j t, c i N H (c j ). For every open vertex v, there exists 1 i t such that c i N H (v). For all 1 i t, there is a unique a X with E a = {c i }. All open vertices are isolated. reg(r/i) = X V.

62 Example: H((aef,bgh,ei,hk,cgij,dfjk)) e a f g b h i k c j d

63 Example: H((aef,bgh,ei,hk,cgij,dfjk)) e a f g b h i k c j d

64 Example: H((aef,bgh,ei,hk,cgij,dfjk)) e a f g b h i k c j d reg(r/i) = X V = 11 6 = 5

65 Closing Remarks 1. All of our results are independent of char(k), even though reg(r/i) in general depends on char(k).

66 Closing Remarks 1. All of our results are independent of char(k), even though reg(r/i) in general depends on char(k). 2. No restriction on the degrees of the generators or geometry of the hypergraph. (One can at least always apply the Fill-In Theorem.)

67 Closing Remarks 1. All of our results are independent of char(k), even though reg(r/i) in general depends on char(k). 2. No restriction on the degrees of the generators or geometry of the hypergraph. (One can at least always apply the Fill-In Theorem.) 3. Gives a quick eyeball upper bound on reg(r/i) that is easy to compute by hand (small cases) or by computer.

68 Reference: Thank you! K. Lin and J. McCullough. Hypergraphs and Square-free Monomial Ideals. submitted. arxiv: