sequentially Cohen Macaulay Herzog Cohen Macaulay 5 unmixed semi-unmixed 2 Semi-unmixed Semi-unmixed G V V (G) V G V G e (G) G F(G) (G) F(G) G dim G G

Size: px

Start display at page:

Download "sequentially Cohen Macaulay Herzog Cohen Macaulay 5 unmixed semi-unmixed 2 Semi-unmixed Semi-unmixed G V V (G) V G V G e (G) G F(G) (G) F(G) G dim G G"

けいしょうおおふさ
5 years ago
Views:

1 Semi-unmixed 1 K S K n K[X 1,..., X n ] G G G 2 G V (G) E(G) S G V (G) = {1,..., n} I(G) G S square-free I(G) = (X i X j {i, j} E(G)) I(G) G (edge ideal) 1990 Villarreal [11] S/I(G) Cohen Macaulay G 2005 Herzog H S/I(H) Cohen Macaulay ( 4.9 ) 2007 Villarreal H S/I(G) unmixed ( 2.5 ) semi-unmixed unmixed Cohen Macaulay 2 Herzog Cohen Macaulay Villarreal unmixed semi-unmixed 3 semi-unmixed regularity 2 semi-unmixed Kummini unmixed semi-unmixed regularity 4 semi-unmixed sequentially Cohen Macaulay semi-unmixed hirotaka.higashidaira.eq.sun@gmail.com

2 sequentially Cohen Macaulay Herzog Cohen Macaulay 5 unmixed semi-unmixed 2 Semi-unmixed Semi-unmixed G V V (G) V G V G e (G) G F(G) (G) F(G) G dim G G unmixed G V dim G = V 2.1 G semi-unmixed G Z Z G V dim G = V G Z G G Z semi-unmixed semi-unmixed V V (G) V G G V G V V (G V ) = V E(G V ) = {{i, j} E(G) i, j V } G \ V G V (G)\V V G N G (V ) = {w V (G) v V ; {v, w} E(G)} V = {v} N G (V ) = N G (v) v G N G [V ] = N G (V ) V, N G [v] = N G (v) {v} deg G (v) = N G (v) v G deg G (v) = 0 v G G iso(g) G semi-unmixed G \ iso(g) semi-unmixed Stanley Reisner square-free (G) V (G) I(G) (G) Stanley Reisner S/I(G) I(G) Min(G) (G) F(G) P C = (X i i C) Stanley Reisner Min(G) = {P C V (G) \ C F(G)} S/I(G) (G) 1 dim S/I(G) = dim (G)+1

3 V 1,..., V c V (G) G = (G; V 1,..., V c ) V (G) = V 1 V c G V 1,..., V c G G = (G; V 1,..., V c ) c = 2 G = (G; V 1, V 2 ) (G; V 1, V 2 ) v 1 V v 2 V 2 {v 1, v 2 } G (G; V 1,..., V c ) i, j (G Vi V j ; V i, V j ) H H = (H; X, Y ) r = X, s = Y Semi-unmixed 2.2 Z H (H; X, Y ) (1) H Z semi-unmixed (2) X Z r = s H X = {x 1,..., x r }, Y = {y 1,..., y s } (CM1) i (1 i r) {x i, y i } H (CM3) Z {x i, y j } {x j, y k } H {x i, y k } Z {x i, y k } H (CP) Z j (1 j s) {x j, y j } Z = H (NH (y j ) Z) (Y \Z) H (X\Z) (NH (x j ) Z) (3) P I(H) P P V (H)\Z dim S/P = dim S/I(H) (4) F (H) Z dim F = dim (H) ( ) (1) (3) (3) (4) 2.1 Stanley Reisner (1) (2) : H X Y H X Z H semi-unmixed r = X = dim H X Y = s r = s X = Z H semi-unmixed s = Y = dim H dim H = Y X = {x 1,..., x r }, Y = {y 1,..., y s } (CM1) (CM3) Z (CP) Z (CM3) Z H {x i, y j } {x j, y k } {x i, y k } Z {x i, y k } / E(H) {x i, y k } F H F F Z H semi-unmixed dim H = F

4 {x j, y j } F = F < Y = dim H (CM3) Z 1 j s {x j, y j } F = (x j 1 {y j } ) x i N H (y j ) Z, y k Y \ Z {x i, y k } H (Z Y ) {x i, y k } F F F(H) y k F \ Z F Z H Z semi-unmixed F = dim H x i F y j / F r + 1 j s x j / V (H) F < Y = dim H 1 j r x j F y l N H (x j ) Z {x j, y l } H y l Y Z F {x j, y l } F H (NH (y j ) Z) (Y \Z) H (X\Z) (NH (x j ) Z) (CP) Z (2) (1) : F F(H) F Z, F < Y (A) j (1 j r) {x j, y j } F = X Z (2) r = s F < Y X = Z y k F (CP) Z F {y r+1,..., y s } F < Y (A) (A) F {x i, y k } {x i, y j } {x j, y k } H {x i, y k } Z (CM3) Z {x i, y k } Z (CP) Z Z = X semi-unmixed 2.3 (H; X, Y ) (1) H X semi-unmixed (2) H X = {x 1,..., x r }, Y = {y 1,..., y s } (CM1) i (1 i r) {x i, y i } H (CM3) {x i, y j } {x j, y k } H {x i, y k } H (CP) j (r + 1 j s) H NH (y j ) Y G Semi-unmixed G unmixed G V G V semiunmixed (H; X, Y )

5 2.4 (H; X, Y ) (1) H unmixed (2) H Z H Z semi-unmixed (3) X = Y H X semi-unmixed ([12] ) (H; X, Y ) (1) H unmixed (2) X = Y H X = {x 1,..., x r }, Y = {y 1,..., y r } (CM1) i (1 i r) {x i, y i } H (CM3) {x i, y j } {x j, y k } H {x i, y k } H 3 Regularity regularity G G regularity S/I(G) Castelnuovo Mumford regularity reg(g) = reg(s/i(g)) = max{i + j Hm(S/I(G)) i j {0}} = max {j i, Tor i S(K, S/I(G)) i+j 0} m = (X 1,..., X n ) Eisenbud [1] regularity M E(G) M G M = 1 M 2 e f e N G [f] = G G im(g) im(g) = max{ M M G } Katzman im(g) reg(g) [7] Kummini 3.1 ([8] ) H H unmixed reg(h) = im(h) Kummini semi-unmixed regularity X semi-unmixed

6 3.2 (H; X, Y ) H X semi-unmixed reg(h) = im(h) ( ) Unmixed X < Y X V (H) X =, X = Y X H V (H ) = (X X ) Y, E(H ) = E(H) {{x, y} x X, y Y } H = (H ; X X, Y ) H H reg(h) reg(h ) ([6] ) im(h) = im(h ) F(H ) = (F(H) \ X) {X X } H unmixed im(h) reg(h) reg(h ) = im(h ) = im(h) im(h) = reg(h) Z X semi-unmixed ( 3.4) regularity G G reg(g) = 1 ( [2, Theorem 3.4] ) G V (G) = V (G) E(G) = {{v, w} {v, w} / E(G), v w} G 5 im(g) = 1 reg(g) = 1 ( ) reg(g) > 1 G 4 C C 4 C 2 G im(g) > 1 C 5 C 5 G reg(g) = (H; X, Y ) Z H Z X H Z semi-unmixed reg(h) = im(h) ( ) H im(h) reg(h) X Y, iso(h) = r = X, s = Y 2.2 H r = s X = {x 1,..., x r }, Y = {y 1,..., y r } (CM1) (CM3) Z (CP) Z H (H ; X, Y ) E(H ) = E(H) {{x j, y k } i; {x i, y i } Z =, x j N H (y i ) Z, y k N H (x i ) Z} S V (H) S = K[v v V (H)]

7 I(H ) = I(H) + (x i y k {x i, y k } E(H ) \ E(H)) (CP) Z I(H ) (w w / Z) = I(H) I(H ) I(H) H unmixed H I(H ) + (w w / Z) = P Q S 0 S/I(H) S/I(H ) S/(w w / Z) S/P Q 0 regularity reg(h) max{reg(s/p Q) + 1, reg(s/i(h ))} P Q 3.2 reg(s/p Q) + 1 = reg(s/i(h )) = im(h ) reg(h) im(h ) H im(h ) im(h) reg(h) = im(h) H H semi-unmixed reg(h) = im(h) 4 Sequentially Cohen Macaulay Cohen Macaulay unmixed semi-unmixed sequentially Cohen Macaulay ( 4.8) ( 4.4 ) S = K[X 1,..., X n ] K K Sequentially Cohen Macaulay Stanley 4.1 ([9]) M Z- S- M sequentially Cohen Macaulay M M i {0} = M 0 M 1 M r = M (1) i (1 i r) M i /M i 1 Cohen Macaulay (2) dim(m 1 /M 0 ) < dim(m 2 /M 1 ) < < dim(m r /M r 1 ) G S/I(G) sequentially Cohen Macaulay G

8 sequentially Cohen Macaulay G G Cohen Macaulay S/I(G) Cohen Macaulay 4.2 (H; X, Y ) 1. H Cohen Macaulay 2. X = Y H semi-unmixed sequentially Cohen Macaulay ( ) 4.1 S/I(H) Cohen Macaulay S/I(H) {0} M 1 = S/I(H) sequentially Cohen Macaulay S/I(H) P dim S/P = dim S/I(H) X = Y H semi-unmixed (P) V (G) \ iso(g) = {v 1,..., v t } (P) (P 1 ),..., (P l ) G (P 1 ),..., (P l ) V (G)\iso(G) = {v 1,..., v t } (P 1 ),..., (P l ) (CM1) (CP) H = (H; X, Y ) X semi-unmixed (H; X, Y ) (CM1) (CM3) (CP) sequentially Cohen Macaulay Tuyl Villarreal ([10, Lemma 3.9] ) (H; X, Y ) X Y E(H) H sequentially Cohen Macaulay H 1 y Y 4.4 ([10, Corollary 3.11]) H H v w N H (v) = {w} H sequentially Cohen Macaulay H \ N H [v] H \ N H [w] sequentially Cohen Macaulay H = (H; X, Y ) r = X, s = Y H

9 4.5 (H; X, Y ) Z H H 1 y Y H (CM1) (CM3) Z X = {x 1,..., x r}, Y = {y 1,..., y s} (CM1) (CM3) y 1 = y ( ) X = {x 1,..., x r }, Y = {y 1,..., y s } (CM1) (CM3) Z y = y α α > r x β N H (y α ) β r y β y α deg H (y α ) = 1 (CM1) (CM3) Z α r φ {1,..., r} i (1 i r) x φ(i) = x i, y φ(i) = y i j (r + 1 j s) y j = y j X = {x 1,..., x r}, Y = {y 1,..., y s} (CM1) (CM3) Z φ φ(α) = 1 y = y α = y Z H H X = {x 1,..., x r }, Y = {y 1,..., y s } (CM1) (CM3) Z Z r i=1 {x i, y i } H sequentially Cohen Macaulay (H; X, Y ) (CM1) (CM2) (CM3) Z () i (1 i r) (CM2) i : {x i, y j } H i j H sequentially Cohen Macaulay y Y y = y α (CM3) Z Z r i=1 {x i, y i } α r α 4.5 deg H (y 1 ) = 1 (H; X, Y ) (CM1) (CM2) 1 (CM3) Z Z r i=1 {x i, y i } H 2 = H \ N H [y 1 ] (H 2 ; X \ {x 1 }, X \ {y 1 }) (CM1) (CM3) Z Z r i=2 {x i, y i } 4.4 H 2 sequentially Cohen Macaulay 4.5 deg H2 (y 2 ) = 1 (CM2) 1 (H; X, Y ) (CM1) (CM2) 1 (CM2) 2 (CM3) Z H 3 = H 2 \ N H [y 2 ] 4.7 (H; X, Y ) (CM1) (CM2) (CM3) H sequentially Cohen Macaulay () X = r r = 1 H sequentially

10 Cohen Macaulay r > 1 X = {x 1,..., x r }, Y = {y 1,..., y s } (CM1) (CM2) (CM3) H 1 = H \ N H [y 1 ], H 2 = H \ N H [x 1 ] N H [y 1 ] = {x 1, y 1 }, iso(h 1 ) {y r+1,..., y s } H H 1 (CM1) (CM2) (CM3) H 1 sequentially Cohen Macaulay H 2 (CM3) iso(h 2 ) = {x i y i N H (x 1 ), 2 i r} N H (x 1 ) iso(h 2 ) = {x i, y i y i N H (x 1 ), 2 i r} {y j y j N H (x 1 ), r < j} H 2 H H H 2 (CM1) (CM2) (CM3) H 2 sequentially Cohen Macaulay 4.4 H sequentially Cohen Macaulay (CM3) Z Z = X 4.7 Z X 4.8 Z X (CP) X Z : i (1 i r) H N H (y i ) Z (Y \Z) (CP) X Z semi-unmixed ( 2.2 ) 4.8 Z H Z < X H X = {x 1,..., x r }, Y = {y 1,..., y s } (CP) X Z Z r i=1 {x i, y i } (H; X, Y ) (CM1) (CM2) (CM3) Z H sequentially Cohen Macaulay (H; X, Y ) Z semi-unmixed (1) H sequentially Cohen Macaulay (2) (H; X, Y ) (CM1) (CM2) (CM3) Z ( ) (1) (2) : 4.6 (2) (1) : (H; X, Y ) (CM1) (CM2) (CM3) Z Z = X 4.7 H sequentially Cohen Macaulay Z X (CP) Z 2.2 (1) (2)

11 4.8 H sequentially Cohen Macaulay ([4, Corollary ] [3] ) (H; X, Y ) (1) H Cohen Macaulay (2) X = Y (H; X, Y ) (CM1) (CM2) (CM3) 5 Unmixed semi-unmixed unmixed unmixed 5.1 ([5]) c G = (G; V 0, V 1,..., V c ) (c+1) G G \ V 0 = (G; V 1,..., V c ) c (G; V 0, V 1,..., V c ) i (1 i c) G i := G V0 V i (G i ; V 0, V i ) 5.2 (G; V 0, V 1,..., V c ) G unmixed i (1 i c) G i semi-unmixed 5.1 G unmixed G G regulariy 5.3 ([5, Lemma 5.2] ) (G; V 0, V 1,..., V c ) (1) reg(g) = max{reg(g 1 ),..., reg(g c ), 1} (2) im(g) = max{im(g 1 ),..., im(g c ), 1} (3) i (1 i c) reg(g i ) = im(g i ) reg(g) = im(g) [5] semi-unmixed

12 5.4 ([5, Theorem 1.4] ) G unmixed reg(g) = im(g) ( ) G = (G; V 0, V 1,..., V c ) i (1 i c) G i := G V0 V i G i 5.1 i (1 i c) G i semi-unmixed 3.5 i (1 i c) reg(g i ) = im(g i ) 5.2 reg(g) = im(g) [1] W. Bruns and J. Herzog, Cohen Macaulay rings, revised edition, Cambridge studies in advanced mathematics 39, Cambridge University Press, [2] A. Dochtermann and A. Engström, Algebraic properties of edge ideals via combinatorial topology, Electron. J. Combin. 16(2) (2009), #R2. [3] J. Herzog and T. Hibi, Distributive lattices, bipartitegraphs and Alexander duality, J. Alg. Combin. 22 (2005), [4] J. Herzog and T. Hibi, Monomial ideals, Graduate Texts in Mathematics 260, Springer-Verlag London Limited, [5] H. Higashidaira, On the sequentially Cohen Macaulay properties of almost complete multipartite graphs, Commun. Algebra 45(6) (2017), [6] S. Jacques, Betti Numbers of Graph Ideals, Ph.D. thesis, The University of Sheffield, (2004). [7] M. Katzman, Characteristic-independence of Betti numbers of graph ideals, J. Combin. Theory Ser. A 113 (2006), [8] M. Kummini, Regularity, depth and arithmetic rank of bipartite edge ideals, J. Alg. Combin. 30 (2009), [9] R. P. Stanley, Combinatorics and Commutative Algebra, Second Edition., Progr.Math., vol.41, Birkhäuser Boston, Inc., [10] A. Van Tuyl and R. H. Villarreal, Shellable graphs and sequentially Cohen- Macaulay bipartite graphs, J. Combin. Theory Ser. A 115 (2008), [11] R. H. Villarreal, Cohen-Macaulay graphs, Manuscripta Math. 66 (1990), [12] R. H. Villarreal, Unmixed bipartite graphs, Rev. Colombiana Mat. 41 (2007),

i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,.

R-space ( ) Version 1.1 (2012/02/29) i Version 1.1, (2012/02/22 24),.,..,.,,. R-space,, ( R- space),, Kahler (Kähler C-space)., R-space,., R-space, Hermite,. ii 1 Lie 1 1.1 Killing................................