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On finite generation of symbolic algebras ofmonomial primesHema Srinivasan aa Departamento de Mathematics , University of Missouri , Columbia, MO, 65211Published online: 27 Jun 2007.

To cite this article: Hema Srinivasan (1991) On finite generation of symbolic algebras of monomial primes,Communications in Algebra, 19:9, 2557-2564

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COMMUNICATIONS I N ALGEBRA, 1 9 ( 9 ) , 2557-2564 ( 1 9 9 1 )

O n Finite Generation of Symbolic Algebras

of Monomial Primes

HEMA SRIXIVASAS~

Department of Mathematics

University of Missouri

Columbia. MO 65211

The symbolic algebra of an ideal I is the graded ring en,, - I (" ) of the n~:h symbolic

powers of I. .4n element f is in the symbolic power I ( n ) if there exists z $ fi sluch that x f

is in I n . Symbolic algebras, especially symbolic algebras of monomial primes. have been

of interest for a long time. By a monomial prime, we mean the prime P = P ( a , b, c) of the

space curve parametrized by ( t o , tb , t C ) . Thus P is generated by the 2x2 minors of a 2x3

matrix M ( a , b, c) over a polynomial ring k [ x , y , z].

The question of when these algebras are finitely generated has been studied in [El,

[HI], [H2], [H-U], [Hb], [Sc], [V] and [C2]. In [HI] Huneke proved that the symbtAic algebras

S ( P ( a , b, c)) of monomial primes P(a , b, c ) are finitely generated if a 5 3 and 1, and c are

arbitrary, and in [H2], he showed that finite generation holds true if a = 4, for. any b and

c. In this paper we prove that S ( P ( a , b, c ) ) is finitely generated if a = 6 for arbitrary b

and c. In [C2], Cutkosky showed that the symbolic algebra is finitely generated if the

anti-canonical dimension & - ' ( X ) , of an associated singular surface X obtained by blowing

up the nonsingular closed point P(a, b, c ) of the weighted projective space pro.i(k[x, y , z]),

is positive. In this paper we show that n- ' (X(a , b, c ) ) > 0 if satisfy certain bounds. In

particular, this gives finite generation of S(P(5 ,b , c)) for b 5 23 and S ( P ( 6 , b, c)).

In the last section, we also discuss the elusive case a = 5 and give an example where

n-' is -m and yet the symbolic algebra is finitely generated in all characteristics. .Also.

we give certain tight bounds for the ratio which ensure finite generation.

It is known that symbolic algebras of ideals are not always finitely ge:.?erated from

the examples in [N], [R] and [Cl]. In fact, they can be finitely generated in every positive

'Partially supported by NSF

Copyright O 1991 by Marcel Dekker. I n c

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2 5 5 8 S R I N I V A S A N

characteristic and fail t o be finitely generated in characteristic zero as shown in [C2].

However, ic still remains t o be seen i f the symbolic algebras o f monomial primes are all

finitely generated. T h e big obstacle now seems t o be the case a = 5. Finally. the smallest

triple in the lexicographic order for which finite generation is yet t o be settled is (5.24.37).

However, S (P(5 ,24 , c ) is finitely generated for any c other than 37.

Let k be a field and a < b < c be natural numbers. Let @ : k [ x . y , z] --+ k [ t ] be

the homomorphism defined b y @ ( x ) = t a , G ( y ) = t b . and @ ( z ) = t c . T h e kernal o f @ is

called the monomial prime P ( a , b, c ) . Thus f E P ( a , b, c ) i f and only i f f ( t n . t b . t C ) = 0 .

For any prime ideal P o f a ring R, P (* ) denotes the n th symbolic power o f P. Thus

P(*) consists o f all those elements f in the ring for which there exists x 7: P such that

sf E Pn . S ( P ) = $,,, P ( " ) is the symbolic algebra o f P. Returning to monomial primes - P ( a , b, c ) , whenever there is no danger o f confusion, we write P and S ( P ) for P ( a , b. c ) and

S ( P ( a , b, c ) ) respectively. S o w let P = proj(k[x, y , z ] ) be the weighted projective space

with deg (x ) = a , deg(y) = b, deg(z) = c. Let X be the blowup o f the closed point in

P given b y the monomial prime P ( a , 6.c) o f k [ x , y , z ] . W e say that X = X ( a . b.c) is the

(singular) surface associated to P ( a , b. c ) . For any quasihomogeneous f E P ( a , b, c ) , deg( f )

denotes the weighted degree with deg(x) = a , deg(y) = b and deg(z) = c.

PROPOSITION 1. jC2j Let a < b < c be natural numbers. There exist pairrvise coprime

numbers a', b' and c ! which are factors o f a, b and c respectively, such that S ( P ( a , b, c ) )

is finitely generated as a k-algebra i f and only i f S ( P ( a l , b', c ' ) ) is finitely generated as a

k-algebra.

For a proof see lemma 10 and corollary 1 o f [C2].

L e have the following criteria o f Cutkosky [C2]

THEOREM A. jC2j Suppose a,b,c are pairwise coprime. Let P = P ( a , b.c) be a monomial

prime. S ( P ) is finitely generated i f either

(1) The characteristic o f k is positive and there exists a quasi-homogeneous form g E P ( " )

for some n such that (degg)2 < n2abc. or

(2) T h e anti-canonical dimension o f X = X ( a , b, c ) , K - ' ( X ) is positive.

THEOREM B. [C2/ Suppose that a,b.c are pairwise coprime, with P = P ( a . b.c) and

X = X ( a , b, c) . I f either

( I ) ( a + 6 - t ~ ) ~ > abc or

(2) there exists g E P(" ) for some n such that deg(g) < n ( a + b + c ) ,

then K - ' ( X ) > 0 and S ( P ) is finitely generated.

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F I N I T E G E N E R A T I O N O F S Y M B O L I C A L G E B R A S O F MONOMIAL P R I M E S 2559

Further, in positive characteristic, the criterion in theorem A is both necEssary and

sufficient as long as at least one of a,b, or c is not a perfect square.

In section 3, we will be using the following theorem of Huneke [ H I ]

THEOREM C. [ H l j Suppose P = P ( a . b . c ) is generated by the 2x2 minors of the 2x3

matrix

( :: ;) Then S ( P ) is generated in degree 2 if r '< s.

2. FINITE GENERATION OF S ( P )

Let R = k[x, y, z ] , and P = P ( a , b, c) be a monomial prime. By Proposition 1, in

order to show finite generation of S ( P ) , it suffices to consider the case wherr: a.b and c

are pairwise coprime. Also. without loss of generality, we may assume a < 0 < c. .ill

integers w , q . 1, r , ro in this section are nonnegative. In the proof of the followi,:ig theorem,

the tntegers q , 1 and r are obtained via the division algorithm.

THEOREM 1. Let a.b.c be pairwise coprime. Let q,l and r be nonnegative integers such

that c = qb + la + r. where r < a and la + r < b. Let w be the smallest posi:ive integer

such that w b - r - 0 mod a. Then

(I) If w + q > a - 2, then S(P) is finitely generated in all characteristics.

(2) If u: + q < fi then S(P) is finitely generated in all positive characteristics.

(3) If u! = 1, then S(PJ is finitely generated in characteristic zero also.

PROOF: We will first prove (1). Suppose w + q > a - 2. Then

= l a + r - w b + a b

= l a t ha - ( w b - r )

Further, w < a implies that a h - u,b > 0. Therefore. w b - r

b + c = ( w + q - ( a - l ) ) b + ( 1 + b)a - (- a )a

so that either there exists g = yr - x i E P for some t , or g = z - yrxW E P. 111 either case,

deg(g ) 5 c + b < ( a + b + c). Thus by Theorem B, S(P) is finitely generated.

Sow we will prove (2). Suppose ( w + q ) 5 a. Since ( w + q jb c mod a. there

exists a form y W + q - z x t E P . Xow

(w + qI2b2 = ( w + q ) 2 b ( b ) 5 aqbb < abc

so by Theorem A. S(P) is finitely generated in positive characteristic.

Finially. if w = 1. (u: + q ) b = b + qb < b+ c. By theorem B, S ( P ) is finitely generated

in all characteristics.

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2560 S R I N I V A S A N

COROLLARY 1. U c > (a-3)b or a is even and c > (a-4)b, then S(P) is finitely generated

in all characteristics.

PROOF: Without loss of generality. we may assume that a > 4 since S(P) is finitely gener-

ated for a < 4 by Theorem 3.14 [ H 2 ] .

Case 1: c > (a - 3)b, so that q 2 a - 3. By Theorem 1. S(P) is finitely gmerated if

w + q > a - 2. Hence S (P) is finitely generated if w > 1.

Now let w = 1, SO that w + q = a - 2. Then

Hence (w + q)2b2 < abc. Now ( W + q)b - c = ~b - r - la 2 0 mod a. So, there exists

a form yw+q - zxi in P. for some integer t 2 0, whose degree is (w + q)b. So by theorem

A, S(P) is finitely generated.

Case 2: Let a be even and c > (a - 4)b. We have q 2 a - 4. By Theorem 1, S (P) is

finitely generated if w + q > a - 2. Since c E (w + q)b mod a. and a is even, jw + q) must

be odd. Thus, we just need to show that S(P) is finitely generated if w + q = a - 3 with

w = 1.

( W + q)2b = (a - 3)'h = (a - 4)(a - 3)b + (a - 3)b

< c(a - 3) + (a - 3)b = ac - 3c + (a - 3)b

< ac - 3c + 3c . since a > 4

< ac

Thus (w + q)2b2 < abc and by Theorem A, S(P) is finitely generated.

THEOREM 2. Let a = 6 with b and c arbitrary. Let P = P(6, b, c). Then S(P) is finitely

generated in all characteristics.

PROOF: We will use the notation intrdouced in Theorem 1. By Proposition 1, we may

assume that 6, b and c are pairwise coprime. In view of Theorem 1, it suffices to consider

the case when w + q < a - 2 = 4. Suppose q = 1. Since w + q is odd, we have w = 2. Thus

c = b + la + ro and 2b - ro = 0 mod 6. Since (b, 6) = 1, b 1 or 5 mod 6. If b r 1 mod 6,

then ro = 2. Therefore (c, 6) = 3, which is impossible. If b z 2 mod 6. then = 4, and

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F I N I T E G E N E R A T I O N OF SYMBOLIC ALGEBRAS OF MONOMIAL P R I M E S 2561

(c. 6 . ) = 6 which is a coqtradiction. Thus q can never be 1. Thus q 2 2 = 6 - 4 and

c > qb 2 (6 - 4)b. Hence by corollary 1, S(P) is finitely generated in all characlwistics.

3 . a = 5 AND T H E GESERAL CASE

In this section, let P = P(5, b.c) be the kernal of the map @ : k[x, y , r --+ k [ t ]

given by @ ( r ) = t 5 , @(y) = t b , @(z) = t C . We will discuss the finite generation of S ( P ) =

ento P(") , Without loss of generality, we may assume that 5. b and c aw pairwise

coprime, and 5 < b < c.

THEOREM 3. en,, P(") is finitely generated if b 5 23, or if: $ (+;.i - &) = - 31- 163

( .598.. . , ,675.. . ).

PROOF: We will use the notation of Theorem 1. ( 5 + b + ~ ) ~ 2 5bc if b 5 20 or if c < 36-30,

Thus we just need to consider the case c < 3b - 30. By Corollary 1, S(P) is finitely generatd

if c > 26. Hence we may suppose that c < 2b. Thus c = b + 51 + ro, and wb- ro i: 0 mod 5.

By Theorem 1, we only need to consider the case & < w + 1 < 3. Hence w = 2. So. we 3 b - c

h a v e y 3 - 2 x 7 E P a n d y 2 z - x E P . Hence P = 12(M), where

unless P has any other term of lower degree, in which case we get finite generation of S(P) 3 b - c 2 c - b

by criterion (2) of Theorem B. Let cu = y3 - z x 7 , P = t2 - y x s , y = .r 5 - yZz.

%P,7 E P . Since deg(cu) = 3b, we get finite generation of S(P) if b < %. Thus S(13) is finitely

generated if

Now by Theorem C, S(P) is generated in degree 2 if 2c - b 5 3b - c or 3c 5 4b. Hence

3c > 4b.

Since deg(P) = 2c and deg(y ) = 2b + c , it can be seen that

S(P) is then finitely generated if

Thus S(P) is finitely generated if

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2562 S R I N I V A S A N

From ( I ) , we may assume 9b > 5c. By Theorem 1, we can assume 2b > c. Hence 7b > 4c.

It can be seen that 1 -(j3z- + ay2) E ~ ( ~ 1 . z

Thus S(P) is finitely generated if

Thus S(P) is finitely generated if

From ( I ) , (2), and (3), and that facts 9b # 5c (otherwise 5 / b) and b and c are integers.

we get that S(P) is finitely generated, except possibly when

It remains to show S(P(5. b, c)) is finitely generated for b 5 23. Finite generation for

b < 20 follows immediately from Theorem B since (5 + b + c ) ~ > 5bc.

Now suppose b = 21. Then 9 is in the interval (&, 7 -KO) only for 32 < c 5 35.

Since 5,b and c are pairwise coprime, we need only consider c = 32 or c = 34. If c = 32.

then y2 - x2z is in P and if c = 34, then yz - x" is in P. In either case. there is an element

of degree less than a + b + c in P. Hence S(P) is finitely generated for a = 5 and b = 21 by

theorem B.

Let b = 22. Now 9 is in the "bad interval" only for 32 < c 5 36. Since 5.32 and c are

pairwise coprime, the only possible value for c is 33. But then yr - x13 E P, and hence

there is an element of degree b + c < a + b + c in the prime ideal. Hence S(P) is finitely

generated by theorem B.

Finally, let b = 23. The only values of c we must consider are 36,37 and 38. If c = 36,

then yZ - x2z E P. If c = 37, then x12 - yr E P, and if c = 38, then x 3 y - z E P. In all

these cases, there is an element of degree less than a + b + c in P, and hence S(P) is finitely

generated.

REMARKS: (1) The procedure used in this proof which is carried out up to P(3) can clearly

be continued for higher symbolic powers P ( n ) . The interval cannot be improved by going

up to n = 5. Better results could possibly be obtained by repeating this process for higher

values of n .

(2) The first 11 triples (a,b,c) in the lexicographic order for which finite generation of

S(P(a, b, c ) ) is yet to be settled are (5,24,37). (5,26,43), (5,2T. 41), (5,29.47), (5,31.48).

(5,32,51), (5,33,49), (5,37,56), (5,37,61), (5,38,59) and (5,39.62). However, in all these

cases, S (P(a , b,c)) is finitely generated for every other value of c. These are all the triples

( 5 , b, c) for k 5 40 for which fimte generation is not known.

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FINITE GENERATIOX OF SYMBOLIC ALGEBRAS OF MONOMIAL PRIMES 2563

As a result of Theorem 3, we give an example of a monomial prime P such that S ( P )

is finitely generated, but n - ' ( S ( P ) ) = -m. Let P(a , b ,c) be a monomial prime. Define

H,,, = { f E P(,) I f is quasi-homogeneous with d e g ( f ) = m ) .

Recall that for quasihomogeneous f. deg( f ) is the degree off with the weighting oleg(x) = a.

deg(y) = b and deg(z) = c. Then

~ - l ( X ( a , b . c ) ) = -m if and only if Hn(a+b+c),n = 0 for a11 n > 0. We will need the

following lemma.

L E M M A 1. Suppose that there exists an irreducible, qua..sihomogeneous form f E P(") such

that d e g ( f ) < n a . Then for any nonzero quasihomogeneous form h E P(" ' ) ,

That is, i f f E Hk," with k < & is irreducible, then H r , , = 0 for all 1 and m such that

l < $ .

PROOF: This is a restatement of proposition 2 (2) of [C2]

EXAMPLE 1: Let P = P(5,77,101). Since $ > 6.85.. . , by theorem 3, S(F') is finitely

generated. Now P is generated by xjl - y2z, y3 - x z 6 t and z2 - yxz5.

Hence ( y 3 - x 2 6 t ) 2 - x(z5l - y2z)(zZ - yxZ5) = y6 - 3y3x262 + yx77 + ; :3yZx E P Z .

Thus f = y5 - 3xZ6yZt + x7' + yxt3 E P ( 2 ) . It can be seen that f is irrc:ciucible, and

d e g ( f ) = 5(77) = 385. Thus f E H385.2, and 388 < 2 Now b:i the lemma,

Hn(a+b+c),n = 0 for all n > 0. Thus n-'(,Y) = -m. This shows that the co::iverse of 2 of

Theorem A [Theorem 2 of C2] is not true in general.

REFERENCES

[Cl] S.D. Cutkosky, Weil divlsors and symbolic algebras. Duke Math. J. 57 ( 1388) 175-183.

[C2] S.D. Cutkosky. Symbolic algebras of monomials primes. Jour~ial fiir die reine und

angewandte Mathematik, to appear.

[El S. Eliahou, Courbes monomiales et algebre de Rees symboliques. These, LrniversitC de

Geneve 1983.

[H-U] J . Herzog and B. Clrich, Self-linked curve singularities, preprint.

[Hb] S. Huckaba, Analytic spread modulo an element and symbolic Rees tt'gebras, Jour.

AIg. 128 (1990) 306-320.

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2564 SRINIVASAN

[HI] C. Huneke, On the finite generation of symbolic blowups, math Z. 179 (1982), 465-572.

[H2] C. Huneke. Hilbert functions and symbolic powers, Michigan Math J . 34 (1987) 293-

318.

[N] M. Nagata, On the 14th problem of Hilbert, Amer. Jour. Math 81 (1959) 766-772.

[R] P. Roberts, A prime ideal in a polynomial ring whose symbolic blow-up is not noethe-

rian. Proc. Amer. Math. Soc. 94 (1985) 589-592.

[Sc] P. Schenzel, Examples of Noetherian symbolic blow-up rings, Revue Rournaine de

Mathematiques pures et applique& 33 (1988) 375-383.

[V] W. Vasconcelos, On the structure of certain ideal transforms, Llath Z. 198 (1988)

435-448.

Rece ived : Sep t embe r 1990

R e v i s e d : J a n u a r y 1991

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