HILBERT FUNCTIONS. 1. Introduction
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1 HILBERT FUCTIOS JORDA SCHETTLER 1. Introduction A Hilbert function (so far as we will discuss) is a map from the nonnegative integers to themselves which records the lengths of composition series of each layer in a graded module. In many situations of interest, the Hilbert function attached to a module agrees for sufficiently large inputs with a polynomial, called a Hilbert polynomial. Thus an infinite amount of information is encoded in a finite object. Indeed, the degree and coefficients of a Hilbert polynomial represent important invariants from which one can potentially read off properties of the module. Of interest to us here is that the coordinate ring of a projective variety is a graded module for which there is an associated Hilbert polynomial that tells us information about the space such as its dimension and degree. Also, there are Hilbert polynomials which capture the dimension and multiplicity at a point in a quasi-projective variety. First, we ll build up the needed machinery with a review of graded modules. Then a crucial result and corollary about Poincaré series is proved which will allow us to conclude that a wide range of Hilbert functions have the property of eventually behaving like polynomials mentioned above. ext, we will explore the appropriate notion of a Hilbert function for a projective variety. Lastly, Hilbert functions of local rings (in particular, stalks at points in a quasi-projective variety) are introduced and studied. An effort was made to include proofs in cases where either the result was fundamental or the source was unclear. On the other hand, I ve intentionally omitted standard or otherwise unenlightening details which can be found in my references. 2. Graded Modules Definition 2.1. We define CommRing to be the category whose class of objects CommRing consists of all nonzero commutative rings with identity and whose morphisms are ring homomorphisms which preserve 1. Recall that R CommRing is said to be (nicely) graded (by 0 ) when R as abelian groups with R 0 {0} and R n R i R j R i+j 1
2 2 JORDA SCHETTLER for all i, j 0 ; an R-module M (assumed to be unitary) is graded when M M n as abelian groups and R i M j M i+j for all i, j 0. Example 2.2. The polynomial ring S R[x 0,..., x ] is graded in a natural way whenever R CommRing by taking S n equal to the set of homogeneous polynomials in S of degree n along with 0. Here S is said to be graded by degree. ote that S 0 R. Remark 2.3. Let R CommRing be graded. Then we may write 1 where r n R n for all n 0 and all but finitely many summands are 0. Thus if ρ 0 R 0, then ρ 0 r n ρ 0 1 ρ 0 R 0, so ρ 0 ρ 0 r 0 since each ρ 0 r n R n. Hence r 0 is a multiplicative identity in R 0, so we have R 0 CommRing because R 0 R 0 R 0 {0}; in particular, R is an R 0 -algebra. Also, it s clear that if M is a graded R-module, then M n is a R 0 -module for all n 0. r n Lemma 2.4. Let R CommRing be graded. Then R is oetherian (R 0 is oetherian and R is finitely-generated as an R 0 -algebra). Sketch. ( ) Suppose R is oetherian. Then the irrelevant ideal R + : is finitely-generated, wlog, by r i R k(i) for i 1,..., s, and satisfies R 0 R/R+ (thus R 0 is oetherian). One can use induction to show that R n R 0 [r 1,..., r s ] for all n 0, so R R 0 [r 1,..., r s ]. ( ) Conversely, suppose R 0 is oetherian and R is finitely-generated as an R 0 -algebra. Then R is oetherian by the Hilbert basis theorem. n1 R n Lemma 2.5. Let R CommRing be graded and oetherian, and suppose M is a finitelygenerated graded R-module. Then M n is a finitely-generated R 0 -module for all n 0.
3 HILBERT FUCTIOS 3 Proof. We have R R 0 [r 1,..., r s ] by lemma 2.4 and M Rm Rm t where wlog r i R k(i) for all i 1,..., s and m i M l(i) for all i 1,..., t. Thus fixing n 0 and m M n, we have t m ρ i m i i1 for some ρ 1,..., ρ t R. Since m M n we may assume ρ i R n l(i) for all i 1,..., t where we use the convention that R k {0} whenever 0 > k Z. Thus each ρ i is of the form J(i) ρ i f j (r 1,..., r s ) j1 where every f j R 0 [x 1,..., x s ] is a sum of monomials of bounded degree (with a bound that may be taken independent of i). Therefore M n is a finitely-generated R 0 -module. 3. Poincaré Series Definition 3.1. Let R CommRing. A function λ from a subclass S of the class all R-modules to Z is said to be additive on S if whenever is a short exact sequence of R-modules in S. λ(b) λ(a) + λ(c) 0 A B C 0 Remark 3.2. Let R CommRing be graded and oetherian, and suppose λ is an additive function with values in Z on the class S all finitely-generated R 0 -modules. Then by lemma 2.5 we may define the Poincaré series of a finitely-generated graded R-module M as P M (x) λ(m n )x n Z[[x]]. Theorem 3.3 (Hilbert-Serre). Let R, λ, M be as in remark 3.2. Then f(x) P M (x) s (1 x k(i) ) i1 for some f(x) Z[x] and some k(1),..., k(s) 0.
4 4 JORDA SCHETTLER Proof. By lemma 2.4 we may write R R 0 [r 1,..., r s ] where wlog r i R k(i) for all i 1,..., s. We ll use induction on s. If s 0, then R R 0, so M is a finitely generated R 0 -module, say M R 0 m R 0 m t where wlog m i M l(i) for all i 1,..., t, but then M n {0} for n > max{l(1),..., l(t)}, whence P M (x) is a polynomial because λ(m n ) 0 for all sufficiently large n by additivity (apply λ to the SES consisting entirely of 0 s). ow assume s > 0 and that the statement holds for s 1. For every n 0, consider the multiplication map ϕ n : M n M n+k(s) : m r s m, which gives rise to short exact sequences of R 0 -modules ker(ϕ n ) M n r s M n r s M n M n+k(s) M n+k(s) /r s M n, so additivity gives λ(m n+k(s) ) λ(m n ) λ(m n+k(s) /r s M n ) λ(ker(ϕ n )). ote that (assuming wlog k(s) > 0) R : R 0 [r 1,..., r s 1 ] R/r s R (R n + r s R)/r s R is a graded element of CommRing with the last equality on the right as abelian groups. In fact, K : ker(ϕ n ) ker(ϕ 0 ϕ 1 ) and L : M n /(M n r s M) (M n + r s M)/r s M M/r s M
5 HILBERT FUCTIOS 5 are finitely-generated graded R -modules. Thus the induction hypothesis implies that there are f L (x), f K (x) Z[x] such that (1 x k(s) )P M (x) λ(m n )x n λ(m n )x n+k(s) where g(x) + g(x) + (λ(m n+k(s) ) λ(m n ))x n+k(s) (λ(m n+k(s) /r s M n ) λ(ker(ϕ n )))x n+k(s) P L (x) x k(s) P K (x) f L(x) x k(s) f K (x) s 1 (1 x k(i) ) i1 f(x) s 1 (1 x k(i) ) i1 g(x) k(s) 1 λ(m n )x n Z[x] f(x) f L (x) x k(s) f K (x) Z[x]. Corollary 3.4. Let R, λ, M be as in 3.2. If R R 0 [r 1,..., r s ] for some r 1,..., r s R 1, then there exists an h M (x) Q[x] called the Hilbert polynomial of M such that for all sufficiently large n 0. Proof. By theorem 3.3 we have P M (x) λ(m n ) h M (n) f(x) (1 x) g(x) s (1 x) w where g(x) Z[x] and g(1) 0. Thus writing g(x) a n x n and noting that (with the convention ( ) ( n ) for n 0 ) ( ) w + n 1 (1 x) w x n, w 1
6 6 JORDA SCHETTLER we find n 0 implies λ(m n ) i0 which is a polynomial in n with leading term ( ) w + n i 1 a i w 1 g(1) (w 1)! nw Hilbert Functions Remark 4.1. Recall that if R CommRing is Artinian and M is a finitely-generated R- module, then every chain of submodules of length n M M 0 > M 1 >... > M n {0} can be extended to a composition series (i.e., a maximal chain) and that every such composition series has the same length. We write l R (M) for the common length of all composition series. ote that l R is additive on the class of all finitely-generated R-modules. To see this, suppose A f B g C is a short exact sequence of finitely-generated R-modules. Let be a composition series for A and be a composition series for C. Then A 0 >... > A a C 0 >... > C c B g 1 (C 0 ) >... > g 1 (C c ) f(a 0 ) >... > f(a a ) {0} is a composition series for B of length a + c. Definition 4.2. Let R CommRing be graded and finitely-generated as an R 0 -algebra with R 0 Artinian, and suppose M is a finitely-generated graded R-module. Then R 0 is oetherian by Hopkin s theorem, so R is oetherian by 2.4, whence 2.5 and 4.1 allow us to define the Hilbert function of M by for all n 0. H M (n) : l R0 (M n )
7 HILBERT FUCTIOS 7 Remark 4.3. Suppose R CommRing is Artinian and let S : R[x 0,..., x ] be graded by degree. Then since x 0,..., x S 1 remark 4.1 and corollary 3.4 imply that for every finitely-generated graded S-module M there is a Hilbert polynomial h M (x) Q[x] such that h M (n) H M (n) for all sufficiently large n 0. In fact, if w is the order of the pole at x 1 in P M (x), then by 3.4 we have deg(h M ) w 1. Definition 4.4. Let X P be a projective variety. Recall that k[x] S/I(X) where S k[x 0,..., x ] for an algebraically closed field k and I(X) is the homogeneous radical ideal in S consisting of those polynomials which vanish on all of X. Then k[x] is a finitelygenerated graded S-module (S graded by degree) with k[x] n (S n + I(X))/I(X) and S 0 k Artinian, so we may define the Hilbert function of X by H X (n) : H k[x] (n) l k (k[x] n ) dim k ((S n + I(X))/I(X)) for all n 0, which by 4.3 agrees for sufficiently large n with the Hilbert polynomial of X given by h X (x) : h k[x] (x) Q[x]. Example 4.5. Consider X P. Then k[x] S, so ( ) + n H P (n) dim k (S n ) #monic monomials of degree n, giving h P (x) 1! (x + )(x + 1) (x + 1) 1! x ( 1)! x As we ll see below, this implies the degree of P is 1. Theorem 4.6. Let X P be a projective variety with dim(x) D. Then deg(h X ) D (with the convention deg(0) 1), and if X, then the leading term of h X (x) is dx D D! for some d called the degree of X. Moreover, if X V (F ) is a hypersurface, then d deg(f ). Proof. To show h X has degree D, we use induction on D. If D 1, then X, so I(X) S and k[x] n (S n + S)/S {0} for all n 0, giving l k (k[x] n ) 0, whence h X 0 as needed. ow suppose D 0 and that deg(h Y ) dim(y ) whenever Y P has dim(y ) < D. We may assume X is irreducible (see [3], pg.s 50-51, for this reduction). Then I(X) p is a homogeneous prime ideal. ote that x i / p for some i {0,..., } since otherwise V (p) V (I(X)) X. Hence Y : X V (x i ) V (p + x i S) has dim(y ) D 1. Then x i S 1 is not a zero divisor of k[x] since S/p is an integral domain
8 8 JORDA SCHETTLER and x i / p, so using the notation in the proof of theorem 3.3 we have maps (which are now injective) ϕ n k[x] n k[x] n+1 : m x i m for all n 0. Injectivity implies K {0}, so f K (x) 0 and whence the proof of corollary 3.4 implies (1 x)p k[x] (x) P L (x), leading term of h X (x) d (w 1)! xw 1 where d : g(1) (since g(x) Z[x] and h X (n) l k (k[x] n ) 0 for sufficiently large n 0 ). Thus On the other hand, L deg(h X ) + 1 w k[x] n + x i k[x] x i k[x] so the induction hypothesis gives order of pole at 1 in P k[x] (x) 1 + order of pole at 1 in P L (x) 2 + deg(h L ). (S n + x i S + p)/p (x i S + p)/p k[y ] n k[y ], deg(h X ) 1 + deg(h L ) 1 + dim(y ) D. ow assume X V (F ) is a hypersurface. Then if n δ : deg(f ), we have so Thus S n + (F ) (F ) H V (F ) (n) dim k (S n /F S n δ ) S n S n (F ) S n/f S n δ, dim k (S n dim k (S n δ ) ( ) ( ) + n + n δ 1! n ( 1)! n 1 1! n 2δ n 1. 2( 1)! leading term of h V (F ) (x) giving d δ deg(f ) (and dim(x) 1, as expected). ( + 1 2( 1)! 2δ ) x 1 2( 1)! δ ( 1)! x 1,
9 HILBERT FUCTIOS 9 5. Local Rings and Multiplicity Definition 5.1. Given R CommRing and a proper ideal I of R, we define the associated graded ring of R with respect to I by gr I (R) : I n /I n+1 as abelian groups with I 0 R where for a I i, b I j we have a well-defined product given by (a + I i+1 )(b + I j+1 ) : ab + I i+j+1 I i+j /I i+j+1. If M is an R-module, then an I-filtration F of M is a decreasing chain of submodules M M 0 M 1 M 2... such that IM n M n+1 for all n 0 ; if, in addition, IM n M n+1 for all sufficiently large n 0, we say F is stable. We define the associated graded module of M with respect to F by gr F (M) : M n /M n+1 as abelian groups. This becomes a gr I (R)-module by taking whenever r I i, m M j. (r + I i+1 )(m + M j+1 ) : rm + M i+j+1 Lemma 5.2. Let R, I, M, F be as in definition 5.1 with R oetherian. Suppose M is finitely generated over R and that F is stable. Then gr F (M) is a finitely-generated gr I (R)-module. Proof. See Proposition in [1]. Remark 5.3. Let R CommRing be local and oetherian with maximal ideal m, and suppose M is a finitely generated R-module. Then there s a natural stable m-filtration F of M given by M m 0 M mm m 2 M.... Hence gr F (M) is a finitely generated gr m (R)-module by 5.2. Also gr m (R) 0 R/m is a field (so Artinian) and if m 1,..., m s generate m as an R-module, then gr m (R) (R/m)[m 1 + m 2,..., m s + m 2 ]. In particular, gr m (R) is finitely generated as an R/m-algebra. Thus in the context of definition 4.2 we may define the Hilbert function of M by H M (n) : Hgr F (M)(n) dim R/m (m n M/m n+1 M)
10 10 JORDA SCHETTLER for all n 0. As in remark 4.3, we have m i + m 2 gr m (R) 1 for all i 1,..., s, so again by corollary 3.4 there is a Hilbert polynomial h M of M with for all sufficiently large n 0. h M (n) H M (n) Definition 5.4. Let X P be a quasi-projective variety and fix p X. Then the stalk O : O X,p of X at p is a local oetherian ring with, say, maximal ideal m, so remark 5.3 allows us to define Hilbert function of X at p as n 1 n 1 H X,p (n) : dim k (O/m n ) dim k (m i O/m i+1 O) H O (i) for all n 0 where k O/m. i0 i0 Theorem 5.5. Let X, p, O, m be as in 5.4 with D p dim p (X). There there is an h X,p Q[x] called the Hilbert polynomial of X at p such that h X,p (n) H X,p (n) for all sufficiently large n 0. In fact, the leading term of h X,p is m D p! xdp for some m. Moreover, if X V (f) A is a hypersurface, then m is the multiplicity m p (X) of p on X. Proof. ote that H X,p (n + 1) H X,p (n) H O (n) h O (n) for all sufficiently large n 0. This implies H X,p is a polynomial function h X,p with rational coefficients for large n. In fact, if the leading term of h X,p (x) is αx β, then the above relation along with the proof of corollary 3.4 implies m (w 1)! xw 1 leading term of h O αβx β 1 where w is the order of pole at x 1 in Pgr F (O)(x) and m g(1) (again positivity follows since H X,p 0), so β w and α m/w!. The Krull dimension theorem implies that w least #generators of m dim(o) dim p (X) D p (see [1], pg.s ). ow assume X V (f) A is a hypersurface and denote µ m p (X). Then wlog p (0,..., 0), so setting I (x 1,..., x ) and R k[x 1,..., x ], we have O/m n R/(I n, f)
11 HILBERT FUCTIOS 11 for all n 0. Also, for n µ there s a short exact sequence of k-vector spaces R/I n µ R/I n R/(I n, f) r + I n µ fr + I n r + I n r + (I n, f) where injectivity of the first map can be seen as follows, while exactness elsewhere is clear. Let r R, and write f f µ + f µ+1 + and r r ν + r ν+1 + where f i, r i R i for all i and f µ 0 r ν. Hence if fr I n, then µ + ν n, so ν n µ, giving r I n µ. ote that R/I n is generated as a k-vector space by all monic monomials having degree less than n, so Therefore dim k (R/I n ) H X,p (n) dim k (O/m n ) dim k (R/(I n, f)) 1! n 1 #monic monomials of degree j j0 n 1 ( ) 1 + j 1 j0 +n 2 j 1 ( ) j 1 ). ( + n 1 dim k (R/I n ) dim k (R/I n µ ) ( ) ( ) + n 1 + n µ 1 (n + 1) n (n µ + 1) (n µ) ( ( )! ( n + n 1 n µ + 2 µ ( 1)! n 1 + for all sufficiently large n 0, giving whence m µ m p (X). leading term of h X,p (x) References µ ( 1)! x 1, ( )) ) n M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Westview Press, David Eisenbud, Commutative Algebra with a View Towards Algebraic Geomtry, Springer, Robin Hartshorne, Algebraic Geomtry, Springer-Verlag, 1977.
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