., S/I (Castelnuovo Mumford)

Transkript

1 REGULARITY AND a-invariant OF CAMERON WALKER GRAPHS ( ), ( ), ( ), ( ) [7]. 1. S = K[x 1,...,x n ] K n degx i = 1. I S dims/i = d, S/I Hilber H (S/I, λ) H (S/I, λ) = h 0 + h 1 λ + h 2 λ h s λ s (1 λ) d. h i Z ([2, Proosiion 4.4.1]). H (S/I, λ) h(s/i, λ) = h 0 + h 1 λ + h 2 λ h s λ s ( h s 0), S/I h., degh(s/i, λ) dims/i S/I a-, a(s/i)., I, a(s/i) 0, F S/I : 0 S( ( + j)) β,+ j(s/i) S( (1 + j)) β 1,1+ j(s/i) S S/I 0 j 1 j 1 S/I S β i,i+ j (S/I) S/I (i, j), S/I (Caselnuovo Mumford) reg(s/i) = max{ j : β i,i+ j (S/I) 0}., Theorem 1.1. (cf. [11, Corollary B.4.1]) I S degh(s/i, λ) reg(s/i) dims/i deh(s/i), S/I Cohen-Macaulay, ( ) degh(s/i, λ) reg(s/i) = dims/i deh(s/i), S/I Cohen-Macaulay degh(s/i, λ) = reg(s/i). JSPS JP17K

2 Theorem 1.1, [2, Lemma ] H (S/I, λ) = i=0 ( 1) i β i,i+ j (S/I)λ i+ j j Z (1 λ) n = h(s/i,λ) (1 λ)n dims/i (1 λ) n, degh(s/i,λ) +reg(s/i) n+dims/i. n = deh(s/i),, ( ), β,+reg(s/i) (S/I) 0 (, S/I exremal ([5, Definiion ]) 1 ) [1, Lemma 3], S/I Cohen-Macaulay β,+reg(s/i) (S/I) 0, Theorem 1.1. Theorem 1.1 Remark 1.2. (1) I ure resoluion ([2,. 153]), S/I exremal 1. S/I ( )., I = (x 1 x 2 x 3,x 3 x 4,x 4 x 5 x 6 ), S/I Cohen-Macaulay ( ), I ure resoluion (2) Theorem 1.1, I = (x 1 x 2,x 2 x 3 ) ure resoluion S/I ( ), S/I Cohen-Macaulay. (3), degh(s/i,λ) reg(s/i) ([8, Theorem 1.2])., G (,, ), V (G) = {x 1,...,x n } G, E(G) G S = K[V (G)]., G I(G) I(G) = ( x i x j : {x i,x j } E(G) ) S I(G) a(s/i(g)) 0., I(G) I(G) = (x i : i V (G) \ T ) T V (G):., T V (G) G, i, j T {i, j} E(G) dims/i(g) = max{ T : T G }., a(s/i(g)) = 0 G I = I(G) ( ) G degh(s/i(g),λ) = reg(s/i(g)) G, Cameron Walker, 2

3 2. CAMERON-WALKER, Cameron Walker G, V (G), E(G) M E(G) G, e,e M e e = /0. G M, e,e M, e f /0 e f /0 f E(G) m(g) = max{ M : M G } im(g) = max{ M : M G }, G, Examle 2.1. (1) n K n n im(k n ) = 1, m(k n ) = 2 (2) K n1,n 2 (n 1 n 2 ) im(k n1,n 2 ) = 1, m(k n1,n 2 ) = n 1 (3) n ah P n (,E(P n ) = {{x 1,x 2 },{x 2,x 3 }...,{x n 1,x n }} ) n + 1 n im(p n ) =, m(p n ) = 3 2 (4) n C n im(c n ) = n n, m(c n ) = 3 2 im(g),reg(s/i(g)) m(g),. Theorem 2.2. ([4, Theorem 6.7], [9, Lemma 2.2]) G im(g) reg(s/i(g)) m(g) Theorem 2.2, im(g) = m(g) G im(g) = reg(s/i(g)) = m(g). Cameron Walker [3, Theorem 1], im(g) = m(g) G (, [6, Remark 0.1] ). 3

4 Theorem 2.3. ([3, Theorem 1], [6, Remark 0.1]) G G im(g) = m(g), G : (i) G sar(v) s : K 1,s (Figure 1 ) (ii) G (w) : w (Figure 1 ) (iii) {v 1,...,v m } {w 1,...,w }, v i G sar(v) s i (s i 1), w j G (w) j ( j 0) (Figure 2 ) x 1 x x s v 2 G sar(v) s w y 1, y, y 1,2 y,1 G (w) FIGURE 1. G sar(v) s G (w) x (1) 1 x s (1) 1 x (2) 1 x s (2) 2 x (m) 1 x s (m) m v 1 v 2 v m {v 1,...,v m } {w 1,...,w } w 1 w 2 w y (1) 1,1 y (1) y (1) 1,1 y () y (2) y (2) 1,1 1,1 y (1) 2,2 1,2 1,2 y (2) 1,2 y (2) 2,1 y () 1,2 y (),1 y (),2 FIGURE 2. Cameron Walker 4

5 , Theorem 2.3(iii) Cameron Walker. Definiion 2.4. ( cf. [6,. 259]) G Cameron Walker, im(g) = m(g), Remark 2.5. [10], im(g) = m(g) G Cameron Walker Cameron Walker Theorem 2.6. G Figure 2 Cameron Walker, S = K[V (G)]. (1) ([6, Theorem 1.3]) 5 : (a) S/I(G) unmixed. (b) S/I(G) Cohen Macaulay. (c) S/I(G) unmixed shellable. (d) S/I(G) unmixed verex decomosable. (e) s 1 = = s m = 1 1 = = = 1. (2) ([6, Theorem 2.1]) S/I(G) Gorensein (3) ([6, Theorem 3.1]) S/I(G) sequenially Cohen Macaulay. (4) im(g) = reg(s/i(g)) = m(g) = j + m. j=1 3.,, a(s/i(g)) = 0 Cameron Walker G I = I(G) ( ) Cameron Walker G degh(s/i(g),λ) = reg(s/i(g)) Cameron Walker G. Cameron Walker G, a(s/i(g)) = 0 Theorem 3.1. ([7, Theorem 1.1, Proosiion 1.3]) G Figure 2 Cameron Walker, S = K[V (G)]. degh(s/i(g),λ) = dims/i(g) = a(s/i(g)) = 0. 5 m i=1 s i + max { j,1 } j=1

6 Remark 3.2. a(s/i(g)) = 0 G, Cameron Walker. (1) n K n H (S/I(K n ),λ) = a(s/i(k n )) = 0. (2) K n1,n 2 (n 1 n 2 ) 1 + (n 1)λ 1 λ H (S/I(K n1,n 2 ),λ) = 1 + (1 λ)n 2 n 1 (1 λ) n 2 (1 λ) n 2 a(s/i(k n1,n 2 )) = 0. (3) G sar(v) s ( H S/I(G sar(v) s ) 1 + λ(1 λ)s 1 ),λ = (1 λ) s ( ) a S/I(G sar(v) s ) = 0. (4) G (w) ( H. S/I(G (w) ) ),λ = (1 + λ) + λ(1 λ) 1 (1 λ) { ( ) a S/I(G (w) 0 ( : odd) ) = 1 ( : even) I = I(G) ( ) degh(s/i, λ) reg(s/i) = dims/i deh(s/i) Cameron Walker G Theorem 3.3. ([7, Theorem 2.2]) G Figure 2 Cameron Walker, 1 j N Gbi (w j ) = { v i : {v i,w j } E(G) }. S = K[V (G)]., I = I(G) ( ), V {v 1,...,v m } s i + v i V { } j : N Gbi (w j ) V j + V N Gbi (w j ) V 6

7 Remark 3.4. (1) Theorem 3.3, N Gbi (w j ) V 1 j V {v 1,...,v m } Theorem 3.3, 1 j N Gbi (w j ) V s i V V, Theorem 3.3 v i V, s i 1 (2) V = {v 1,...,v m } Theorem 3.3, Cameron Walker G, I = I(G) ( ) m i=1 s i + n j + m = reg(s/i(g)) j=1 Examle 3.5. G Cameron Walker (Figure 2, m = 3, = 4, s 1 = 3,s 2 = 1,s 3 = 2, 1 = 0, 2 = 1, 3 = 4 = 2 ). G = 8 v 1 v 2 v V {v 1,...,v m } f (V ) = s i + v i V { } j : N Gbi (w j ) V N Gbi (w j ) V j + V. Theorem 3.3 Remark 3.4(1), I = I(G) ( ), V = {v 3 },{v 1,v 3 },{v 2,v 3 },{v 1,v 2,v 3 } f (V ) 0 f ({v 3 }) = (2 + 1) = 0, f ({v 1,v 3 }) = (2 + 2) = 3, f ({v 2,v 3 }) = (5 + 2) = 1, f ({v 1,v 2,v 3 }) = (5 + 3) = 2., I = I(G) ( ) 7

8 Theorem 3.3 Corollary 3.6. ([7, Corollary 2.4]) G Figure 2 Cameron Walker., 1 j j 1, I = I(G) ( ). Corollary 3.7. ([7, Corollary 2.6]) G Figure 2 Cameron Walker, G, I = I(G) m ( ), s i + n j + m i=1 j=1. Remark 3.8. I = I(G) ( ) G, Cameron Walker, reg(s/i(g)) = 1, I(G) linear resoluion, ure resoluion, Remark 1.2(1) I = I(G) ( )., G, I = I(G) ( ) ([7, Proosiion 2.10]). (1) n ah P n. (2) n C n. degh(s/i(g),λ) = reg(s/i(g)) Cameron Walker G Theorem 3.9. ([7, Theorem 3.1]) G Figure 2 Cameron Walker, S = K[V (G)]. degh(s/i(g),λ) reg(s/i(g)), degh(s/i(g),λ) = reg(s/i(g)), 1 i m s i = 1 1 j j 1. Acknowledgmen. 32, (IPMU), ( ) REFERENCES [1] M. Bigdeli and J. Herzog, Bei diagrams wih secial shae, Homological and Comuaional Mehods in Commuaive Algebra, Sringer INdAM Series 20 (2017), [2] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised ED., Cambridge Sud. Adv. Mah., vol. 39, Cambridge Universiy Press, Cambridge, [3] K. Cameron and T. Walker, The grahs wih maximum induced maching and maximum maching he same size, Discree Mah. 299 (2005), [4] H. T. Hà and A. Van Tuyl, Monomial ideals, edge ideals of hyergrahs, and heir graded Bei numbers, J. Algebraic Combin. 27 (2008), [5] J. Herzog and T. Hibi, Monomial ideals, Graduae Texs in Mahemaics 260, Sringer, London, [6] T. Hibi, A. Higashiani, K. Kimura and A. B. O Keefe, Algebraic sudy on Cameron Walker grahs, J. Algebra 422 (2015),

9 [7] T. Hibi, K. Kimura, K. Masuda and A. Tsuchiya, Regulariy and a-invarian of Cameron Walker grahs, arxiv: [8] T. Hibi and K. Masuda, Regulariy and h-olynomials of monomial ideals, Mah. Nachr. 291 (2018), [9] M. Kazman, Characerisic-indeendence of Bei numbers of grah ideals, J. Combin. Theory Ser. A 113 (2006), [10] T. N. Trung, Regulariy, machings and Cameron Walker grahs, arxiv: [11] W. V. Vasconcelos, Comuaional Mehods in Commuaive Algebra and Algebraic Geomery, Sringer-Verlag, KITAMI INSTITUTE OF TECHNOLOGY, KITAMI, HOKKAIDO , JAPAN address: kaz-masuda@mail.kiami-i.ac.j 9