msjmeeting-2017sep-02i002 . Dehn Sommerville, . Gorenstein., ( ) 2 8, f 0 ( ) = 6, f 1 ( ) = 12, f 2 ( ) = 8 3 ( : )

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1 07 : msjmeeting-07sep-0i00 () Dehn Sommerville. Gorenstein.,... V ( ), V (i),(ii) : (i) F, G F G, (ii) v V {v}., F dim F = F, dim = max{dim F : F }, X X. f i ( ) i,,. d, f i ( ) = {F : F = i } f( ) = (f ( ), f 0 ( ),..., f d ( )) f-.,,.,,., 0 6,, 8, f 0 ( ) = 6, f ( ) =, f ( ) = 8 f( ) = (, 6,, 8) ,,, f Dehn Sommerville. Dehn Sommerville, 3., d S d d. (:50003) ( ) f ( ) =. f.

2 . f( ) = (f ( ), f 0 ( ), f ( ), f ( )) f 0 ( )., S f 0 ( ) f ( ) + f ( ) = f ( ) = 3f ( )., () (), 3., f ( ) = 3f 0 ( ) 6, f ( ) = f 0 ( ) f( ) = (, f 0 ( ), 3f 0 ( ) 6, f 0 ( ) ) f( ) = (f ( ), f 0 ( ), f ( ), f ( ), f 3 ( )) f 0 ( ) f ( )., S 3 f 0 ( ) f ( ) + f ( ) f 3 ( ) = 0 f ( ) = f 3 ( )., () 3 (), 3., f ( ) = f ( ) f 0 ( ), f 3 ( ) = f ( ) f 0 ( ) f( ) = (, f 0 ( ), f ( ), f ( ) f 0 ( ), f ( ) f 0 ( )). Dehn Sommerville. d d d i=0 ( )i f i ( ) = ( ) d+ f d ( ) = (d + )f d ( )., Dehn Sommerville. (Dehn Sommerville ) (d ) d ( ) j + ( ) d f k ( ) = ( ) j f j ( ) (k =, 0,,..., d ). () k + j=k [Gr]. Dehn Sommerville d S d f f( ) = (f ( ), f 0 ( ),..., f d ( )) f 0 ( ),..., f d ( ) ( x x ). f h Dehn Sommerville. h., f( ) d+ f 0( ),..., f d ( ),. [Gr, 9.].

3 h. (d ), h 0 ( ), h ( ),..., h d ( ) h i ( ) = i j=0 ( ) d j ( ) i j f j ( ) i j. h( ) = (h 0 ( ), h ( ),..., h d ( )) h. f h., f( ) h( ), t, d h i ( )t i = i=0 d f i ( )(t ) d i i=0, t t + f i ( ) = i j=0 ( ) d j h j ( ) i j. f ( )t 3 + f 0 ( )t + f ( )t + f ( ) = t 3 + 6t + t + 8 = (t + ) 3 + 3(t + ) + 3(t + ) h( ) = (, 3, 3, ). h Dehn Sommerville 3. (Dehn Sommerville (h )) (d ) h i ( ) = h d i ( ) (i = 0,,..., d).. Dehn SommervilleGoreinstein, Dehn Sommerville Gorenstein. Cohen Macaulay Gorenstein. K S = K[x,..., x n ], S. R = S/I S I S, R Krulld. dθ = θ,..., θ d S, dim K (R/ΘR) Θ M (homogeneous system of parameters)., Θ, Θ 3 () h i = h d i.

4 (linear system of parameters). K. f,..., f k, i =,,..., k f i R/(f,..., f i )R R. R Θ = θ,..., θ d R, R = S/I Cohen Macaulay. R i R i. R = s i=0 R i Krull 0( dim K R < ), R s = K, R i R s i R s = K i = 0,,..., s R ( s ) (Poincaré duality algebra). R R i = Rs i. R = S/I Cohen Macaulay, R Θ R/ΘR, R Gorenstein.. [n] = {,,..., n}., I S I = (x i x i x ik : {i,..., i k } [n], {i,..., i k } ) S. K[ ] = S/I ( K ). K[ ]., (d ), K[ ] Krull d ([St, II ] )., h,. S- N, H N (t) = (dim K N k )t k k Z N. ([St] ). (d ) H K[ ] (t) = d i=0 h i( )t i ( t) d. K[ ] h., K[ ] Cohen Macaulay, ([St] ). (d ). K[ ] Cohen Macaulay Θ K[ ] H K[ ]/ΘK[ ] (t) = d i=0 h i( )t i. dim K (K[ ]/ΘK[ ]) i = h i ( ) (i = 0,,,..., d). K[ ] Cohen Macaulay h( )., K[ ] Gorenstein K[ ]/ΘK[ ] h. Cohen Macaulay,., f R i fr s i, g R s i gr i.

5 , Hi ( ; K) K i, β i ( ; K) = dim K Hi ( ; K) i. F, lk (F ) = {G \ F : F G } F link (F =, lk (F ) = ). (Reisner ) d. () K[ ] Cohen Macaulay. () F (F = ), βi (lk (F ); K) i d F. Gorenstein ( [St, II 5] ).. K[ ] Cohen Macaulay, K Cohen Macaulay. d K Cohen Macaulay, F (F = ) H d F H(lk (F ); K) = K, K Gorenstein* 5. ([St, II 5] ). (d ). KGorenstein*K[ ]Gorenstein, K[ ]ΘK[ ]/ΘK[ ]d. R = d i=0 R i d R i = Rd i. (d ) Gorenstein* i = 0,,..., d h i ( ) = h d i ( ). Gorenstein*. Dehn Sommerville, (d ), K[ ]/ΘK[ ]., K[ ] = K[x, x,..., x 6 ]/(x x, x 3 x, x 5 x 6 ).. K[ ] Gorenstein, x x, x 3 x, x 5 x 6 K[ ], K[ ]/((x x, x 3 x, x 5 x 6 )K[ ]) = K[x, x 3, x 5 ]/(x, x 3, x 5). R, R = R 0 R R R 3, dim K R 0 =, dim K R = 3, dim K R = 3, dim K R 3 =. 5 Gorenstein*..

6 3. M M. Dehn Sommerville.. M, f 0 ( ) f ( ) + f ( ) = χ(m) 3f ( ) = f ( ). χ(m) M. f, f f 0, f( ) = (, f 0 ( ), 3(f 0 ( ) χ(m)), (f 0 ( ) χ(m)))., M f( ) M. Dehn Sommerville Klee [Kl]. (()Dehn Sommerville [Kl]) (d ) M ( ) d h i ( ) = h d i ( ) + ( ) d i (χ(m) χ(s d )) (i = 0,,..., d). i S S 7 3 f( ) = (, 7,, ), h( ) = (,, 0, ). χ(s S ) χ(s ) =, h 3 ( ) = h 0 ( ), h ( ) = h ( ) 3 ( ). 3 Klee Dehn Sommerville, d f f( ) f 0 ( ),..., f d ( ) M., χ(m) χ(s d ), h, Dehn Sommerville., Dehn Sommerville.,. (),. F lk (F ) K Gorenstein* K. K, Hdim ( ; K) = K, K. 7 K, char(k) char(k) =. 7

7 h, h. (d ), h - h ( ) = (h 0( ), h ( ),..., h 8 9 d ( )) { h h i ( ) ( d i ( ) = i) ( i j= β j ( ; K)), (i d ) h n ( ) ( ) d i ( d j= β () j ( ; K)), (i = d )., h Dehn Sommerville χ( ) χ(s d ), Dehn Sommerville Novik [No]. (Dehn Sommerville (h )) (d ) K. h i ( ) = h d i( ) (i = 0,,..., d). Dehn Sommerville.,, (Z/Z), K. S S 7 h( ) = (,, 0, ) 3. β (S S ) = h ( ) = h( ) (0, 0, 3, ) = (,,, ), Cohen Macaulay Reisner i < dim β i ( ) = 0, Cohen Macaulay, Cohen Macaulay., Buchsbaum. I, R = S/I Krull d. m = (x,..., x n ) S. R Θ = θ,..., θ d i =,,..., d (θ,..., θ i )R : R (θ i ) = (θ,..., θ i R : R m, R Buchsbaum., S N N I N : N I = {g N : f I fg N }. K[ ] Buchsbaum, K Buchsbaum., Buchsbaum link Cohen-Macaulay 0. 8 h. h i = ) h βi (i d), h d = h i + ( d i d. 9 h ( ),. 0 link.

8 ([Sc, Mi]) () K[ ] Buchsbaum. (), F lk (F ) K Cohen Macaulay. K, linkcohen MacaulayGorenstein*, K K Buchsbaum. 980 [Go]. I S, R = S/I Krull d. R Θ = θ,..., θ d, R Σ(Θ; R) Σ(Θ; R) = ΘR + d (θ,..., ˆθ i,..., θ d )R : R θ i. i=, Cohen Macaulay K[ ]/(ΘK[ ]), K[ ]/Σ(Θ; K[ ]),. ([Go, NS, MNY]) (d ), Θ = θ,..., θ d K[ ]. K[ ] Buchsbaum H K[ ]/Σ(Θ;K[ ]) (t) = d i=0 h i ( )t i,, i = 0,,,..., d dim K (K[ ]/Σ(Θ; K[ ])) i = h i ( ). ( [NS, MNY]) (d ) K. K[ ]/Σ(Θ; K[ ]) d.,, Dehn Sommerville K[ ]/Σ(Θ; K[ ]).. K = Z/Z, RP 6. ( ) x x x 3, x x x 5, x x 3 x 6, x x x 5, x x x 6 I = x x 3 x, x x x 6, x x 5 x 6, x 3 x x 5, x 3 x 5 x 6, Θ = (x + x 3 + x 5, x + x 3 + x 5, x + x 5 + x 6 ) K[ ]. Σ(Θ, R) I Θ, x x 3 + x x 5 + x 3 x 5 + x x 6 + x 3 x 6 + x 5 x 6, X = x x + x 3 x 5 + x x 6 + x 3 x 6 + x 5 x 6,. x x + x x 5 + x 3 x 5 + x x 6 + x 3 x 6 + x x 6 + x 5 x 6 ΘR+ d

9 K[ ]/(Σ(Θ; K[ ]) K[x,..., x 6 ]/(I + (Θ) + (X)) = K[x, y, z]/(x + xy + yz, y + xz + yz, z + xy + xz)., [MNY], [Go, NS, NS]. ) [Go]., [NS] Σ(Θ; K[ ]) h K[ ], [NS].. d (i),(ii),(iii), () K : (i) K Buchsbaum, (ii) F β d F (lk (F ); K) 0, (iii) = {F : βd F (lk (F ); K) = 0} (d ) K. M K, M. K, H dim (, ) = K,. Γ, f i (, Γ) Γ i, β i (, Γ; K) = dim K Hi (, Γ; K)., h(, Γ), h (, Γ). Dehn Sommerville, h [MN] [MN] ). (()Dehn Sommerville ) (d ) K. h i ( ) = h d i(, ) (i = 0,,..., d)., Γ K[, Γ]. K[, Γ] = I Γ /I ([MNY]) (d ) K, R = K[ ], C = K[, ], Θ = θ,..., θ n R.. () S (R/Σ(Θ;R)) (t) = d k=0 h k ( )tk. () S (C/Σ(Θ;C)) (t) = d k=0 h k (, )tk. Σ(Θ; C) Σ(Θ; C) = ΘC + d i= (θ,..., ˆθ i,..., θ d )C : C θ i.

10 (3), (C/Σ(Θ; C)) d = K, 3 (a, b) ab,. (R/Σ(Θ; R)) i (C/Σ(Θ; C)) d i (C/Σ(Θ; C)) d C Rcanonical module, Buchsbaum K R canonical module C, R Θ, R/Σ(Θ; R) C/Σ(Θ; C) Matlis dual. [Go] S. Goto, On the associated graded rings of parameter ideals in Buchsbaum rings, J. Alg. 85 (983), [Gr] B. Grünbaum, Convex Polytopes, nd edn, Springer, New York, 003. [Kl] V. Klee, A combinatorial analogue of Poincaré s duality theorem, Canad. J. Math. 6 (96), [Mi] M. Miyazaki, Characterizations of Buchsbaum complexes, Manuscr. Math. 63 (989), 5 5. [MN] S. Murai and I. Novik, Face numbers of manifolds with boundary, Int. Math. Res. Not., to appear, arxiv: [MNY] S. Murai, I. Novik and K. Yoshida, A duality in Buchsbaum rings and triangulated manifolds, Algebra and Number Theory (07), [No] I. Novik, Upper bound theorems for homology manifolds, Israel J. Math. 08 (998), 5 8. [NS] I. Novik and E. Swartz, Socles of Buchsbaum modules, complexes and posets, Adv. Math. (009), [NS] I. Novik and E. Swartz, Gorenstein rings through face rings of manifolds, Compos. Math. 5 (009), [Sc] P. Schenzel, On the number of faces of simplicial complexes and the purity of Frobenius, Math. Z. 78 (98), 5. [St] R.P. Stanley, Combinatorics and commutative algebra, Second edition, Progr. Math., vol., Birkhäuser, Boston, (C/Σ(Θ; C)) R/Σ(Θ; R)-.

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