THE ARITHMETIC COHEN-MACAULAY CHARACTER OF SCHUBERT SCHEMES
B Y
DAN LAKSOV
Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.
1. I n this p a p e r we prove the following theorem. (For the notation, see Section 2.) Trrv, oR~l~ 1. Let R be a Cohen-Macaulay ring. Then then homogenous coordinate ring ol the Schubert scheme ~(al... as) is Cohen-Macaulay o/relative dimension
The theorem was also proved b y M. Hochster. For an announcement of his results, see [5].
I t was proved in a weaker local case (see Theorem 12 below) b y J. A. Eagon and M. Hochster in "Cohen-Macaulay rings, invariant theory and the generic perfection of determinental loci" (to appear, cf. [3]).
The proof below owes m a n y of its ideas to Eagon and Hochster a n d to G. Kempf.
F r e q u e n t discussions with S. K l e i m a n and T. Svanes have also been helpful. I a m espe- cially grateful to K l e i m a n for his p a t i e n t help preparing this material.
The proof goes as follows. We assume b y induction t h a t the homogeneous coordinate rings of small Schubert schemes are Cohen-Macaulay (the smallest being empty). Given a Schubert scheme we intersect it properly with a given hyperplane. The intersection t h e n breaks up into the union of smaller Schubert schemes in the w a y described b y a clas- sical formula of M. Pieri. (We derive this formula from a result of W. V. D. Hodge. H e d g e ' s result is the central p a r t of the p a p e r and we prove it following a m e t h o d of J.-I. Igusa.) A result similar to lemmas of Eagon and Hochster shows t h a t since the smaller Schubert schemes are Cohen-Macaulay, their union is also. As the equation of the hyperplane is not a zero-divisor in the homogeneous coordinate ring of the bigger Schubert scheme, this ring is t h e n itself Cohen-Macaulay.
1 -- 722901 Acta mathematica 129. I m p r i r n ~ le 1 J u i n 1972
DAN LAKSOV
2. L e t R be a n y ring and E a free R-module with basis e 1 .. . . . e~. Consider the grass- m a n n i a n Gr(E) which parametrizes submodules of E which are direct summands of r a n k d where d = n - r . Equivalently Gr (E) parametrizes projective quotient modules of E of r a n k r, which we call r-quotients for short. (See [8] for more precise definitions.)
An r-quotient K of E gives rise to a 1-qu0tient A r K of A rE. Writing
_P(t) = el~ A . . . A e~ r,
we find (by Laplace expansion, see [2] Chap. I I I , w 6, no. 4, p. 84) t h a t the images of the To) in A r K satisfy the following quadratic relations
sign (a) p~...~_l,q... ~fr | P~J ... J~J~ + 1-.. Jr = 0 (*) where the sum ranges over all permutations a of (i~... ir~l... ~) such t h a t ai~ < ... < air and all < . . . < a]~ (see [7], p. 310). On the other hand, it is known t h a t a 1-quotient L of A rE arises from some r-quotient K of E if the images of the Po) in L satisfy the above relations (see [6], Vol. 1, Ch. V I I , w 6, Theorem I I , p. 312, the discussion at this point is independent of the characteristic zero assumption). Therefore G r (E) is isomorphic to the scheme of zeros of the ideal Q generated b y the quadratic relations in the polynomial ring P over R in the variables p(~) for all (k). We thus obtain an embedding called the Pliicker embedding of Gr(E) in P( A rE), which is projective ~V-space over R where
F i x 0 < a~ < a 2 < ... < aa ~< n. F o r I ~< i ~< d, let A~ be the free submodule of E genera- ted b y e~ .. . . . e~. The subseheme of G~(E) parametrizing the r-quotients K of E such t h a t the canonical m a p A ('a*-~+l)A~ ~ A (~-~+I)K is zero for all i is denoted ~ ( a ~ . . . aa) and called the Schubert subscheme of Gr(E) corresponding to the conditions A l e ... = Aa (see [8]). I n t u i t i v e l y ~ ( a l . . . aa) parametrizes those direct s u m m a n d s of E of r a n k d which intersect A~ in a module of r a n k at least i for 1 ~< i ~< d.
L ~ M ~ 2. The homogeneous ideal 1(al... aa) o / ~ ( a l . . , a~) (in the Pliiclcer embedding) is generated by those Pm such that ](a~-~+ l) <~ a, /or some i and by the quadratic relations (*).
Proo]. L e t K be an r-quotient of E and a: E - + K the canonical map. The m a p A (~-~+~)a: A (~-t+~)E -~ A (a*-~+~)K gives rise to a c o m m u t a t i v e diagram,
A(a~-~+I)A~@ A(r-(a~-~+l))K-~ A(a~-~+~)K| h(r-(a~-~+~))K :' _~ ArK
A(a~-~+~)A~| A(r-(a~-~+l))~__~ A ( a ~ - t + l ) . ~ @ A ( r - ( a ~ - t + l ) ) E ,.~ ~ A r E
THE ARITHMETIC COHEI~--MACAULAY CHARACTER OF SCHUBERT SCHEMES 3
where ~ a n d ~ are t h e canonical pairings. I t is easily checked t h a t A (a~-t+l)(~ ] A~) is zero if a n d only if t h e m a p A (~- i+ 1)A~ | A (r-(a~-i+ 1)) E _+ A r K is zero. E q u i v a l e n t l y A (a~-t+ 1)(~ [ A~) is zero if a n d o n l y if p(j) = ej~ A .../~ ej~ is m a p p e d to zero w h e n e v e r at least (a~ - i + 1) of t h e e/s are in A~ for some i, t h a t is w h e n e v e r j(a~-~+l)<~a~ for some i.
T h e ring P / I ( a 1 ... as) is t h e h o m o g e n e o u s coordinate ring of ~ ( a 1 ... a~) in t h e Plfieker embedding 9 W e will, for short, call it t h e h o m o g e n e o u s coordinate ring of
~(a~... a~).
3. A p o l y n o m i a l / = ~(~)(j)...(oa(~)(j)...(z)p(~)p(~)... P(o in P is said t o be in s t a n d a r d / o r m if, for all nonzero terms, we h a v e it ~ ~t <~ ... ~< It for 1 ~ t 4 r. W e easily check (see [6], Vol. I I , Chapter X I V , w 9 T h e o r e m I, p. 378) t h a t a n y p o l y n o m i a l can be w r i t t e n in s t a n d a r d f o r m m o d u l o t h e q u a d r a t i c relations.
Consider t h e m a t r i x M = (x~) of t h e f o r m
~11 X12 9 Xl, a l - 1 0 X22
Xal-l, al-1
X l , al ~ l , a l + l . . . X l , a , - I Xl,az . . . Xl,ad-1 Xl,a d . . . Xln
:
Xa~-:l,a~ Xa~-l,a~+l
0 Xa~,a~+ 1
Xaa-2,a~-I Xa~-2.az 0 ".
where t h e nonzero x,j are indeterminates.
Xae~-d,ad_ 1 Xad-d,a d 0
Xrn
L e t S be t h e localization of t h e p o l y n o m i a l ring over R in t h e variables x,j, for all i, j, at t h e r • r - d e t e r m i n a n t f o r m e d f r o m the last r columns of M . T h e n M defines a surjeetive h o m o m o r p h i s m f r o m E to a free S-module F of r a n k r. T h u s M defines a p o i n t m of Gr(E ) (with values in S). As the ( a , - i + 1) • ( a , - i + 1) s u b d e t e r m i n a n t s of t h e first at columns of M clearly vanish for all i, m lies on g2(al.., a~).
P R O P O S l T I O N 3. (Hodge). Let / = ~ a(o (j) ... (z) P(o P(J)... P(z) be a homogeneous poly.
n o m i a l i n s t a n d a r d / o r m with all a(o (~)... (I) E R . T h e n / v a n i s h e s at m i / and only i / w e have p(~) E I ( a 1 ... a~) /or all (i) such that a(o (j)... (z) 4 0 .
4 D A I ~ L A ~ S O V
Proo]. (Igusa). T h e s t a t e m e n t t h a t ] vanishes a t m m e a n s t h a t t h e i m a g e of / u n d e r t h e m o r p h i s m f r o m P = S y m s ( A rE) to S = S y m s ( A rF) induced b y h r M is zero, t h a t is t h e following relation holds in S
~ a ( 0 ( t ) . . . ( D
~ l f l '"" X l t t
:
X r t a 9 Z r t r
i
X l t ~ 9 9 9 Xli~. [ ~11~ " 9 9 ~ 1 l r
i i I"" ! i
I x r i ~ "" X r t r I I x r l ~ " ' " X r l r
= 0 .
Suppose P(o ~ I(a~... aa). T h e n m o r e t h a n ( a ~ - i + 1) columns in t h e d e t e r m i n a n t
X l h 9 . . ; V l i , :
~r~-x " " 9 X r i r
are t a k e n f r o m t h e first a, columns of M . T h u s L a p l a c e e x p a n s i o n along t h e first (a~ - i + 1) columns shows t h a t t h e d e t e r m i n a n t is zero. Therefore if p(oEI(at ... aa) for all (i) such t h a t a(0 (j)... (~) 4= 0, t h e n we h a v e t h a t / vanishes a t m.
Conversely, t a k e an i n d e t e r m i n a t e t a n d let x,r te~yi~ where ~ = ( n - i ) ( n - i + ?').
W e easily check t h a t t h e lowest p o w e r of t in
..
is ( ~ 1 ~ + ~ , + - . . + ~ m ) (see [6], Vol. 2, Chap. X I V , w p. 381). T h u s in t h e expression for t h e i m a g e of / in S t h e initial coefficient ( t h a t is t h e first n o n v a n i s h i n g coefficient of a p o w e r of t) is of t h e f o r m
~ a(o (~) . . . (l) X l t l 9 9 9 x r i r x 1 1 1 9 9 9 Z r j ' r 9 9 9 X l l x 9 9 9 g r i t (**) where the s u m is t a k e n o v e r all t e r m s where (~,~,+ ... + & ~ , + ~ l j , + . . . + ~ J , + . . . + Q r Z , ) is m i n i m a l a n d where (i) is such t h a t P(o ~ I ( a l . . . aa) a n d a t least one a,~ (j~... ~o 4= 0. Since the i m a g e of / is zero, (**) has to be zero. H o w e v e r as ] is expressed in s t a n d a r d form, we easily convince ourselves t h a t no two of the m o n o m i a l s Xl~... xa, xljl... Xrz, are equal.
Thus, all a(0(~...(l ~ a p p e a r i n g in (**) would h a v e to be zero. Therefore we h a v e P(o E I ( a l . . . an) for all (i) such t h a t a(~)(j)... (1) 4 = 0.
COROLLARY 4. We have / e I ( a l . . . a d ) q and only i] p,~EI(a 1 ...aa) ]or all (i) such that a(o (j)... (z) appearing in (3) is nonzero.
T H E A R I T H M E T I C C O H E N - - M A C A U L A Y C H A R A C T E R O F S C H U B E R T S C H E M E S 5
COROLLARY 5. The monomial8 in standard /orm are linearly independent over R modulo Q. I n particular, P / I ( a l . . . as) is torsion/ree over R.
Proo/. W i t h ai = ( r + i ) , we h a v e I ( a 1 . . . a s ) = Q . Thus, / is in Q if a n d o n l y if a(~) (t)... (z) is zero.
COROLLARY 6. I / / ~ I ( a l . . . as) and P(m~ r as), then p ( ~ ) / ~ I ( a l . . , as).
Proo/. Suppose P(m~"/6I(al ...as). T h e n proceeding as in t h e proof of Proposition 3, we see t h a t this relation m e a n s t h a t
X l m I . . . Xrrnr ~a(,) (f) ... (~) x~, ... X r i r X l / , . . . Xrj r . . . X l l 1 . . . X r l T = 0
where t h e sum ranges over (~lt~ + ... -~- ~rt, -~ ~111 -~- -o- ~- ~r) r'~- ... -~ ~rlr) minimal a n d (i) such t h a t P(i)(~I(al ... aa). However, as no two of t h e m o n o m i a l s x l ~ . . , x ~ x l ~ . . , cc~z, are equal (since / is expressed in s t a n d a r d form), all a(~) (j)... (1) are zero. Therefore, we h a v e / E I ( a l . . . as), a contradiction.
L e t H be t h e h y p e r p l a n e section of G~ (E) defined b y t h e vanishing of e = e l A ...
A~a,A
...A~a,A
...A~A
. . . A e n(where v means t h e s y m b o l has been deleted). T h a t is, t h e ideal I of H in t h e p o l y n o m i a l ring P is g e n e r a t e d b y e a n d Q. I n o t h e r words, H is t h e S c h u b e r t scheme of direct sum- m a n d s of E of r a n k d t h a t (intuitively) intersect t h e free submodule of E g e n e r a t e d b y e I . . . e a . . . e a . . . Cad . . . e n in a m o d u l e of r a n k ~> 1.
P R O P O S I T I O N 7. (Pieri). We have
that is
~ ( a l ... aa) N H = ~ ~ ( a l ... (a, - 1 ) . . . as);
a ~ - a l _ l > l
I (al ... ad) 4- I = N l" (al ... (a I - 1 ) . . . aa).
a t - a l _ l > l
Proo/. The generators of all t h e ideals being known, this proposition is a " c o m b i n a - torial" consequence of P r o p o s i t i o n 3.
Indeed, t h e relation ejl A ... A ej, E I ( a l . . . ad) clearly implies t h a t (as - i + 1) of t h e es's lie in Al for some i. Thus, certainly ( ( a i - i ) - i + l ) = ( a ~ - i ) of t h e ej's are in t h e set {el . . . e(a _1)}. Moreover, e has precisely (a i - i) of t h e ej's in this module w h e n (a~ - at-l) > 1.
T h u s t h e inclusion = holds.
F o r t h e opposite inclusion let ]E [7 a~-a~_l>lI(al... (at-- I) ... ad) be a h o m o g e n o u s
6 DAN LXKSOV
polynomial. W r i t e / = / ' + / " + q where f ' = ~ a ( t ) u ) . . . (z~ P(o P(J) ... Pu~ w i t h P(o ~ I ( a l . . . an) for all nonzero a ( o u ) . . , u) a n d / " = ~ b ( o u ) . . . ( o P ( o P u ) . . . P ( t ) w i t h p , ~ E I ( a 1 ...an) for all nonzero b(ou)...u~, a n d where [' a n d f " are in s t a n d a r d f o r m a n d q E Q . Since ( / ' + q ) E I ( a l ... aa) we h a v e / ' E n ~ _ , ~ _ 1 > 1 1 ( a 1 ... ( a t - 1) ... aa). T h u s b y P r o p o s i t i o n 3, p(~)E I ( a l . . . (at - 1 ) . . . an) for all (ai - at-i) > 1 a n d f o r all (i) a p p e a r i n g in t h e expression f o r / ' . H o w e v e r , since these p(~) are in I ( a l . . . (a t - 1 ) . . . aa) b u t n o t in I ( a l . . . aa) , e x a c t l y ( a y - ~) of t h e e / s are in t h e set {e 1 .. . . . eaj-1). Therefore P(o is equal to t h e g e n e r a t o r e of I . T h u s we h a v e / ' E I t h a t i s / E ( I ( a 1 ... a n ) + I ) .
4. L ~ M ~ t 8. L e t 11 . . . I~ be ideals in a local ring A a n d r an integer. S u p p o s e that /or a n y choice o / ( d i s t i n c t ) indices k 1 . . . k t a n d / o r a n y u < t where t = 1 . . . p, we have
a n d
d e p t h ( A / ( I k , + . . . + Ik,)) = (r -- t).
T h e n we have d e p t h ( A / n It) = (r - 1 ).
~ 1
P r o o / . P u t I = (Ik, + . . . + Ikt). T h e n we h a v e a c o m m u t a t i v e d i a g r a m o ~ I ~ . l ( z ~ n I ) ~ A I ( I ~ , n I ) ~ A l i a , -~ 0
0-+ (Ik, + I ) 1 I , A / I , A/(Ik~ + I ) -+ O.
F r o m t h e b o t t o m sequence, we conclude, looking a t t h e long e x a c t sequence of E x t ' s , t h a t we h a v e d e p t h ((Ik, + I ) / I ) = (r - t + 1) a n d then, f r o m t h e t o p sequence, t h a t we h a v e d e p t h ( A / ( I k , n I ) ) = (r -- t + 1). Consequently if we p u t J2 = (11 N 12) . . . J~ = (11 fl IT), we h a v e d e p t h (A/(Jk~ + ... + Jk,)) = (r + 1 -- t). Thus, t h e J k ' s satisfy the first condition of t h e l e m m a , a n d t h e d i s t r i b u t i v i t y follows i m m e d i a t e l y f r o m t h e d i s t r i b u t i v i t y of t h e Ik's.
T h e result n o w follows b y induction.
COROLLARY 9. I / A / ( I k , + ... + Ik,) is C o h e n - M a c a u l a y o/ d i m e n s i o n ( r - t ) , then A / f ' l "t=llt is C o h e n - M a c a u l a y o/ d i m e n s i o n (r - 1).
I n d e e d , d i m ( A / [7 ~=lIt) = m a x . d i m ( A / I t ) = (r - 1) = d e p t h ( A / ['] ~-1 It).
Lv,~IMA 10. L e t I t = I ( a l . . . (at - 1 ) . . . aa) /or (a, - a t - l ) > 1. T h e n these ideals s a t i s / y the d i s t r i b u t i v i t y condition o / L e m m a 8. M o r e o v e r
t
~ I k , = I ( a 1 ... ( a k ~ - 1 ) . . . ( % - - 1 ) . . . ( a ~ - 1 ) . . . a a ) .
T H E A R I T H M E T I C C O H E N - - M A C A U L A Y C H A R A C T E R OF S C H U B E R T S C H E M E S 7 Indeed, L e m m a 10 is an easy consequence of Proposition 3 using the method of proof of Proposition 7.
LEMMA 11. A finitely generated homogenous algebra A = k[x o . . . xn] over a field k is Cohen-Macaulay i / a n d only if it becomes Cohen-Macaulay after localization at the irrelevant maximal ideal (x o . . . x~).
Proof. Write A as a quotient of the polynomial ring B = k [ X o . . . X,] b y an ideal I. If dim ( A ) = m, then A is (equidimensional and) Cohen-Macaulay if and only if (ExtqB (A, B))N= 0 for q > n - - m and all maximal ideals N of B containing I (see [1]
Corollary (5.22), p. 66). However, the Ext's, being graded B-modules of finite type, are zero if and only if t h e y become zero when localized at (X 0 .. . . . Xn).
Proo/ o/ Theorem 1. Obviously Gr (E) is obtained b y change of ground ring from the grassmannian of r-quotients of the free Z-module with basis e 1 .. . . . en. Similarly the Schubert scheme ~ ( a l . . . aa) is obtained b y change of ground ring from a Schubert scheme defined over Z. B y Corollary 5 the homogeneous coordinate ring of the latter Schubert scheme is torsion free over Z and therefore faithfully flat over Z. Thus the homogenous coordinate ring of ~ ( a 1 ...aa) is faithfully flat over R. From [4] (Chap. IV, Corollary (6.1.2), p. 135, and Proposition (6.3.1), p. 138) (or from [1], Proposition (4.2), p. 143), we conclude t h a t we only need to prove Theorem 1 in the case when R is a field.
We will prove t h a t the homogenous coordinate ring of ~(al ... aa) is Cohen-Macaulay b y induction on ~=la~. So assume P / I ( b 1 ...ha) is Cohen-Macaulay of dimension (5~--~ b , - 89 d(d + 1) + 1) when ~f=l b, < X~=I a,.
B y Lemma 11, we m a y localize at the irrelevant maximal ideal of P and thus assume t h a t P is local.
From Corollary 9 and L e m m a 10 and the induction assumption, we conclude t h a t P / J is Cohen-Macaulay of dimension (3 = (~=1 a, - 89 d(d + 1)), where
J = f'l I ( a l . . . (a~ - 1 ) . . . aa).
at-a~_l>l
Moreover, b y (7) we have J = ( I ( a l . . . aa) + I ) where I is the ideal generated b y the ele- ment e and the quadratic relations Q. Thus, P / ( l ( a l . . . aa)+ (e)) is Cohen-Macaulay of dimension c3. However, by Corollary 6, e is not a zero-divisor in P[I(a~ ...aa). Thus P ] I ( a l . . . aa) is Cohen-Macaulay of dimension (~ + 1).
~5. T~EORWM 12. Let R be a Cohen-Macaulay ring and x~j indeterminates, where i = 1 . . . r and ] = 1 . . . d. Let J be the ideal in R[x~j] generated by the determinants o / t h e
8 DAN LAKSOV
k >< k-submatrices o/ the first s k columns of the matrix (x~j), where k = 1 . . . t and where the
sk are integers such that 0 < s I < ... < s t = d. T h e n R[x~j]/J is a Cohen-Macaulay R-algebra o/relative dimension [{(d + t) (d - t + 1) + (r + d) t - r - ~ = 1 s~].
Proo/. W e will s h o w t h a t t h e r e e x i s t s a n o p e n affine s u b s e t of ~ ( a 1 ... a~), for s o m e choice of a 1 .. . . . a~, whose a s s o c i a t e d r i n g is R[xu]/J. This r i n g is t h e n e a s i l y seen t o b e C o h e n - M a c a u l a y b e c a u s e t h e h o m o g e n e o u s c o o r d i n a t e r i n g of g s aa) is so, a n d i t s r e l a t i v e d i m e n s i o n is t h e r e l a t i v e d i m e n s i o n of ~ ( a l . . . ad).
L e t F (resp. G) b e t h e free s u b m o d u l e of E g e n e r a t e d b y ea+~ . . . e~ (resp. e I . . . ea).
(Recall: n = r + d . ) W e k n o w t h a t t h e r e is a n o p e n affine s u b s e t U of Gr(E) of t h e f o r m H o m n ( G , F ) ( s e e [8], p r o p . (1.2), p. 283.) ( H e r e we i d e n t i f y H o m n ( G , F ) w i t h t h e s u b s e t of h o m o r p h i s m s of H o m n (E, F ) t h a t l e a v e e~ f i x e d for i = d + 1 . . . n).
L e t r~ = (s k - k + 1) for k = 1 . . . t a n d l e t a 1 = 1, a 2 -- 2 . . . at, = sl a n d ar~+i = (S 1 + 2 ) , a~+~ = (s I + 3) . . . at, = s~ a n d a~,+l = (s2 + 2) . . . a~, = s a a n d . . . . a~t = s~ a n d art+l = (n -- (d - r~) + 1), a,,+2 = (n - (d - rt) + 2) . . . a a = (n - (d - n~) + ( d - rt)) = n. A p o i n t yE U, w i t h c o r r e s p o n d i n g m a t r i x M ( y ) e H o m n ( G , F ) , lies in ~(a~...aa) if a n d o n l y if A r is zero for i = 1 . . . d, or e q u i v a l e n t l y if a n d o n l y if t h e d e t e r m i n a n t s of t h e (a~ - i + 1) • (a, - i + 1) s u b m a t r i c e s of t h e first a~ c o l u m n s of t h e m a t r i x M ( y ) a r e zero for i = 1 . . . d. W i t h t h e a b o v e choice of a / s , we e a s i l y see t h a t t h e l a t t e r c o n d i t i o n m e a n s t h a t t h e d e t e r m i n a n t s of t h e k • k s u b m a t r i c e s of t h e f i r s t s k c o l u m n s of M ( y ) a r e zero for k ~- 1 . . . t. Or, i n o t h e r w o r d s , U N s 1 .. . . a~) looks like (Spec. of) t h e r i n g R[x,j]/J.
A s a n o t h e r a p p l i c a t i o n of t h e m a i n t h e o r e m , we will c o m p u t e t h e c o h o m o l o g y g r o u p s of t h e t w i s t e d (with r e s p e c t t o t h e P l f i c k e r i m b e d d i n g ) s t r u c t u r e sheaf O ~ ( k ) w h e r e
~ ~ ( a ~ . . . aa). F o r t h i s w e n e e d t h e following r e s u l t . ( F o r a p r o o f see [9], p r o p . 2.2.4.) P R O P O S I T I O N 13. ( G r o t h e n d i e c k . ) Let S o be a field, S = ~ ~=oSn a graded algebra o/
finite type generated by S 1. P u t X = P r o j (S) and let ~: S-> | 1 6 2 1 7 6 O x (k)) be the ca.
nonieal homorphism. P u t d = d e p t h (SM) where M = | n~ l Sn and assume d >~ 2. T h e n ~ is bi~ective and H I ( X , O x (lc) ) = 0 / o r all 1 ~ i <~ d - 2 and all k.
F r o m T h e o r e m 1 a n d P r o p o s i t i o n 13 we i m m e d i a t e l y c o n c l u d e t h a t if R is a field a n d d i m ( ~ ) >i 1, t h e n H t ( ~ , On(k)) = 0 for 0 < i < d i m ( ~ ) a n d all k a n d t h e n P / I ( a l . . . ad) a n d
| k = - ~ H0 (s O n (k)) are i s o m o r p h i c g r a d e d a l g e b r a s .
F o r a n y R, t h e g r a d e d pieces of P / I ( a l . . . ad) a r e free R - m o d u l e s g e n e r a t e d b y t h e r e s i d u e classes of t h e m o n o m i a l s of P in s t a n d a r d form, n o t c o n t a i n e d in I ( a l . . . aa), ( b y C o r o l l a r y 5). T h e r a n k of t h e k t h g r a d e d piece for k t> 0 is
THE ARITHMETIC COHEIr--MACAULAY CHARACTER OF SCHUBERT SCHEMES
')
o)(k)= \ m + l /
r e + d - 1
(
ad ~-m-- 2~ ... ~ a d - b m - d ] m - 1 / \ m - d + 1 /,)
(see [6], vol. 2, chap. X I V , w 9, Theorem I I I ) .
Assume now t h a t R is a field. Since the higher cohomology groups of 0 a (k) vanish for large k, the H i l b e r t polynomial of ~2 m u s t be equal to co. B y the above, h~ O~(k)) is equal to w(k) for k>~ 0 and zero for m < 0. Since co(k) = [h~ Oa(k)) - h~m(~)(~, 0~(k))], we m u s t thus h a v e Hal~(~)(~, Oa(k))= 0 for k~> 0.
The above results all hold for a n y noetherian ring R b y virtue of the " p r o p e r t y of exchange" ([4], Chap. I I I , 7.75) because, as we noted in the proof of Theorem 1 at the end of Section 4, ~ is flat over R. Therefore, we h a v e established the following theorem.
T H ]~ 0 R E M 14. Let R be any noetherian ring, put ~ = f l ( a l . . , aa), and assume dim (~)/> 1.
Then we have H * ( ~ , O a ( k ) ) = O /or: (i) 0 < i < d i m (~) and all k: (ii) i = O and k<O:
(iii) i = dim (~) and k >10. For i = 0 and k >~ 0 a n d / o r i = dim (~) and k <. 0, H ~ (~1, 0 ~ (k)) is a / t e e R-module whose rank is given by the polynomial ~(k) above.
R e f e r e n c e s
[I]. ALT~AN, A., & KLEIMAN, S. L., Introduction to Grothendieck duality theory. Lecture notes in mathematics no. 146, Springer Verlag, 1970.
[2]. BO~RBA]~I, 1~., Alg~bre, Chap. 3. Alg~bre Multilingaire. Act. Sci. Ind. 1044, Hermann, 1948.
[3]. EAGON, J. A., & HOeHSTER, M., A class of perfect determinental ideals. Bull. Amer. Math.
Soc., 76 (1970), 1026-1029.
[4]. GROT]~ENDIECK, A., (with J. Dieudonn6), Elemdnts de g6om~trie alg~brique. Chap. I I I . Publ. Math. I.H.E.S., 17 (1963) and Chap. IV, ibid., 24 (1965).
[5]. HOeHSTER, M., Cohen-Macaulay rings of invariants, rings generated by monomials, and polytypes. Notices Amer. Math. Soc., 18 (1971), 509.
[6]. ttODG]~, W. V. D. & PEDOE, D., Methods o] algebraic geometry, vol. I and I I . Cambridge Univ. Press, 1968.
[7]. IGUSA, J.-I., On the arithmetic normality of the Grassmann variety. Proc. 1Vat. Acad.
Sci. U.S.A., 40 (1954), 309-313.
[8], KLEII~A!V, S. L., Geometry of Grassmannians and applications to splitting bundles and smoothing cycles. Publ. Math. I.H.E.S., 36 (1969), 281-298.
[9]. KLEIMAN, S. L., & LANDOLFI, J., Geometry and deformations of special Schubert vari- eties. (To appear in Compositio Math.)
Received July 1, 1971.
