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Initial ideals of truncatedhomogeneous idealsJan Snellman aa Department of Mathematics , Stockholm University , 106 91,Stockholm, SwedenPublished online: 27 Jun 2007.

To cite this article: Jan Snellman (1998) Initial ideals of truncated homogeneous ideals,Communications in Algebra, 26:3, 813-824, DOI: 10.1080/00927879808826166

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COMMUNICATIONS IN ALGEBRA, 26(3), 813-824 (1998)

INITIAL IDEALS O F TRUNCATED HOMOGENEOUS IDEALS

Jan Snellman Department of Mathematics, Stockholm Universiiy

106 91 Stockholm, Sweden e-mail: j ans@matemat i k . su . se

ABSTRACT. Denote by R the power series rlrig in countably many variables Jsrer a field K; then R' is the smallest sub-algebra of R that rontalns all homogeneous elements It is a fact that a homogeneous, finitely generated ideal J in R' hive an initial ideal g r ( J ) . w t h respect to an arbitrarj adrnlsa~hle order, that is l oud ly

finttely generated in the sense that dimK ( Z ~ i ~ ~ ~ ~ ~ J l d - , ) < ~ * 3 for all tc tal

degrees d. Furthermore, g r ( J ) is locally fin~tely generated even under the weaker hypothesis that J is homogeneous and locally finitrly generated.

In this paper, we inr,estigate the relatron between g r ( J ) and the sequencc of initial ideals of the "truncated" ideals p, ( J ) C K j r ~ , . , x,]. It is shown t nat gr(J) is reconstructible from said sequence .\lore precisel?.. i t 1s shown that for all g there exists an .h '(g) such that Tg gr(J) = T 9 gr(p,, ( J ) ) ' whenever n > A'I g): here 7 denotes the total-degree filtration.

The starting point for the investigations that lead to this article was th- question: "what is the relation between the initial ideal of an ideal generated by m forms in 71

variables, and the initial ideal of the truncation of the ideal to the polyno~rnial ring in n' variables?". Recall that a form is a homogeneous polynomial. By the truncation of a polynomial in n variables to one in n' variables we mean the polynomial that is obtained by removing any monon~ial divisible by a variable with index greater than n'. This is of co~irse the same as taking the image under the quotient epimorphism K [ Z I ,..., zn, ,xnt , ,..., zn]

(z,,! +l , . . . , ~n ) E K[x1.. . . , . , , I ] .

Computing a large number of examples, in different monomial ortierings, one notices that the following seems to hold: the initial ideals of the ideals above will differ in high degrces: bnt coincide in low degrees

Copyr~ght O 1998 by Marcel Dekker. Inc

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Conversely, if we fix a dcgree d, and assume that ra is very, very large, then, varying n', we note that for sufficiently large n' we have that the initial ideals of the ideals coincide up to degree d. So, the initial ideals of these restricted ideals is made up of two parts: the '.variable-independent" components of lower degree, and the "tail", which varies with 7;.

Now, let us assume that ri = 03: that is: the original ideal J is generated by "generalized forms'' with infinitely (countably) many variables.

In [5] the theory for calculating initial ideals for J znside the pertinent ring (called R') is developed. It is natural to a ~ k whether this ideal can be approximated degree- wise in the fashion outlined above: that is, if we for a fixed degree d can find an N ( d ) such that, for any 11 > ,V(d), the minimal rnonornial generators of the initial ideal gr(J) of J and the mini~nal ~nono~nial generators of gr(pn ( J ) ) , the initial ideal of the truncation of J , coincide up to degree d.

This article answers this question affirmatively. In fact, it is showed that we may take J to be any homogeneor~s locally finitely generated ideal, by which we mean that dimK J d < m for all d, and conclude that there exists an ~ ( d ) such ct: R',J~-, . . that gr(J)d = gr(pn (J)); whenever n > ~ ( d ) . An immediate consequence of this result (which is stated in Theorem 5.13) is that the initial ideal of J is completely determined by the the restricted ideals of J.

All rings and semigroups under consideration will be commutative. Let K be a field, and let R = K[jol. ~ 2 . ~ 3 , . . . ] ] be the power series ring over K on a de- numerable family of variables. Define R' to be the the smallest subring of R that contains all homogeneous (with respect to total degree) elements, and M to be the free (commutative) monoid on the variables XI, x2, x3,. . . . Regarding an element m E M as a finitely supported map N+ + N, we define Supp(m) C N+, and put maxsupp(m) = nlax Supp(rn) Then, for each n, we can define the subsemigroups Mn := { m E M / rnaxsupp(m) 5 n ) .

If > is an admzsszble order on M, that is, a total order that respects the multi- plicative structure (so that 1 1s the smallest element, and m > m' - trn > tm') and is such that zl > z.1 > x:3 > . . . , it is shown in [5] that for each f E R', the set Mon(f) c M of all nionornials (also called power products) of f have a maximal element with respect to >. This monomial is called theleading monomial o f f and is denoted by Lpp( f ).

Let I be an ideal of R'. The initial ideal gr(I) is the monomial ideal generated by all leading monomials of elements in I .

We denote by 1 f 1 the total degree of f . There is a natural filtration on R by T k R = { f E R / 1 f 1 < I ; ) . This restricts to a filtration on R', as well as on I , and gr(1). We shall sometimes use the following notation, as an alternative: I<d := T ~ I , I<d := Td-ll and 4 := Td l \ Td-'I.

-

Note also that R'is positively graded; R' = ut>o Rt, whereas R = n,>o 8. For any positive integer rl. the power series ring K[[x l , . . . : xn]] is both a subal-

gebra and a quotient of R. since RIA, rr K [ [ x l , . . . , x,,]], where .4, C R is the ideal generated by

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1 TRUNCATED HOMOGENEOUS IDEALS 815

Therefore, we ran define an K-algebra epirnorpliisrn p,, ( the n' th truncatim homo- morphism) by the composite R + RjA,, = K [ [ x l . . . . ,x,]]. Note that for rn E M, p, (m) = in if maxsupp(m) 5 n , and 0 otherwise. Thus p,, (M) = Mn,

Clearly, the inverse limit of the inverse system

is equal to R. If we consider only the col~erent sequences of bounded degre:, we find that these elements form a ring isomorphic: to R'. On the other hand, sins::: for each

T , we lave t a t p ( R ) = K . . ,.L,], we also have t,hat p,, ( K [ i l . . . . xn+,] ) = . . K [ x l . . . . . x,], so the inverse system (1) contains as a subsystem all polyno riial rings K [ z l . . . . ,x,,j. We put R := lp K [ z l . . . . ,I,,]; it is easy to show that

For completeness, we consider also the direct limits (under inclusion) of the po- lynomial rings K [ z l . . . . , x,,], and of the power series rings K [ [ x l , . . . , x,]] I t is not hard to prove that 1 9 K [ z l , . . . . z,] R' S R R whereas 1 2 K[[:?(, . . . , z,]] contains I s K [ x l . . . . , x,] strictly, but does not contain, nor is it contair~ed in, the

ring R'. If I c R' is an ideal, then so is p,, (I) c K [ z l , . . . , z,] for any positi1.e l a . The

latter ideal is said to be the n-th truncation of I. We also say that it is a irwlcation of I ; furthermore, we call the inverse: surjective system

a co-filtration of I; we use the same term for the inverse (not surject~ve!) s js tem that we get by extending (2) t o R' by rneans of the natural injections K [ z l , . . . . E , , ] v R'.

3. TRUNCATION AND INITIAL IDEALS

Of critical importance when approxinlnting g r ( I ) with the mono,:rlial ideals gr(p,, ( I ) ) will be the relation between these and the ideals p,, (gr(1)) . In tl is section. we show that although the operations of t r u n c a t i o ~ ~ and fornlir~g initial ideals does not co~nnlute, there is a useful relation between the two that we can explcit; this re- lation is quite similar t o the way that "specialization", or more generally, extension of scalars, interacts with the operation of forn~irig initial ideals (ser [4, 11).

3.1. Truncations and leading monomials.

Lemma 3.1. If f E R' \ (0) and n = maxsupp Lpp( f ) then we h a w that Lpp( f ) = Lpp(pk ( f ) ) whenever k _> n.

Lemma 3.2. If f E R f , m = L p p ( f ) , p E M then Lpp(fp) = i ~ b p .

Lemma 3.3. If J is an ideal i n R', then p, ( g r ( J ) ) C gr(p, (J)) for a h n . If J is a monomial ideal then equality holds.

I Proof. Slnce we are comparing mono~nlal ideals, we need only rherk the inclusion for monom~als. Let m be a typical element of p,, ( g r ( J ) ) n M, that Ir, m E M n ,

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m = Lpp(f) and f E J. We must prove that I n E gr(p, ( . I ) ) , that is, that Lpp(f) = Lpp(pr, (g)) for son~e g E R'. By Le~nms 3.1, y = J suffices.

When J is a monomial ideal, so is p, (J), licrlce p,, (g r ( J ) ) = pn ( J ) = gr(pn (J)) . 0

The following corollary is imrnediatc.:

Corol lary 3.4. If J is an ideal in R', then p, (gr(.J))' C g ( p n (,J))e for all n. If J is a morlomial ideal then equality holds.

Remark 3.5. Similar results appear in (41 and [I. Proposition 3.41.

E x a m p l e 3.6. The inclusion p,, (gr(J)) C gr(p, ( J ) ) may be strict. Let

where the set { at,, a, 1 1 < i , j < 3 ) is algebraically independent1 with respect to the prime subfield of K. In fact. ( f , g) = (A1 . hz) where each tru~ication of the hi's are generic forms in the sense of [2, 31. Then gr(f1, f 2 ) = (x:, 21x2, x l x ~ , x i ) and pz (g r ( f l , f>) ) = ( ~ : , X I X ~ , X ; ) S gr(pz ( f 1 ~ f 2 ) ) = ( 2 ' : , 5 1 ~ 2 , 2 : ) .

Corol lary 3.7. If J is an ideal of R'. then pn-1 (gr(p,, ( J ) ) ) C gr(p,,-l ( J ) ) , thus we get a sequence

Proof. Applying Lemma 3.3, we have that

We have seen (for instance, from Example 3.6) that the truncated initial ideal gr(pn ( I ) ) may contain mononlials that ;..re not ~ I I gr(1). Conversely, if rn E g r ( I ) n M has maximal support greater than n. then obviously 7n @ gr(pn ( I ) ) .

What about the monomials that lie in gr(p, (I)) for all sufficiently large TL? Do theyt by necessity, belong to gr(I)'! - Definition 4.1. If I is an ideal of R'. let gr(I) = lJ,>o (7,>,gr(p, ( I ) )e , that is. - gr(I) n M consists of those monomials that lie in gr(p,yjI)) for all sufficiently large N. - Lemma 4.2. If I zs an ideal of R ' , then gr(I) is u monomial ideal in R'

Propos i t ion 4.3. If I is an ideal of R', then g r ( I ) > gr ( I ) . If I is a monomial ideal, then equality holds.

Proof. It is enough to verify that g r ( I ) 0 M C g 7 ) n M. Let m E gr(I) rJ M, that is, m = Lpp(f) where f E J. For large enough 71 (more precisely, for n > maxsupp(m)), Lemma 3.1 shows that Lpp(f) = Lpp(p, ( f ) , hence m E gr(J).

'For the definition of algebraic (in)depen(le~~ce, see the discussion in [7] on "algcbraische abhangigkeit" and "irreduzible Mengeii'

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TRUNCATED HOMOGENEOUS IDEALS 8 17

, - If I is a mononlial ideal. arid 7 1 1 E gr ( J ) O M , then for large n (for n > 1L for some

N ) we have that In = tp with t E M : p E M n gr(p,, (J)). Writing p = L p p ( ~ , ( f ) ) for some f E J we have that T ~ L = t Lpp(pl, ( f ) ) .

We can assume t,hat not only is n > N. but also TI > n~axsupp(m). Thc~n, we get from Lemma 3.1 that Lppjp, ( f ) ) = Lpp(f) . Hence m = t Lpp(f) which shows that m E gr(J) . 0

5. APPROXIMATING T H E INITIAL. IDEAL O F A LOCALLY FINITELY GEN3RATED

lDEAL

This section contains the main result of this article. The reader is reniinded that 7 denotes the total-degree filtration on R and its K-subalgebras.

5.1. Existence of a locally finite Grobner basis.

L e m m a 5.1. For a (proper) hornoyeneou.s ideal J C R', the follo.wing trre equrva- lent:

( i )

(iz)

(iii)

A

Qg : dimK J ,

z;z; f4 J ~ - J < m,

There exists n countable, homogeneous generatzrlg set S of J such !hat for all positive integers d , the set Sd = { s E S I 1st = d ) is finite, There exists a countable generating set S of J such that for all posi!r ve i n t eg~rs d , the set Sd = { s E S 1 Is/ = d ) is finite.

homogeneous ideal J fulfillirig the ~onditions of Lemma 5.1 is called locally finitely generated. Countable subsets of R that contains only finitely nmriy ele!nents of a given total degree are called locally finite. Note that, in particu ar, finitely generated hon~ogeneous ideals are locally finitely generated.

The following proposition is of vital importance for what is to follorv Although the result agrees with the irltuitio11. and the naive idea of an iductive proof (assume that we have a finite. partial Grobner basis up to degree d; add normal fsrms of the unprocessed generators of degrre d + 1. as well as normal forms of S-polynornials of degree d+ 1 of elcnlents in the partial Grohncr basis; we have added a fiuite nunlber of elements, so the partial Grobner basis up to degree d + 1 is finite) (:,in be made to work, there are some tricky details, in particular with the proper definition of normal forms. The interested reader may consult 15).

Proposi t ion 5.2. If J zs locally finztely generated then so is g r ( J )

In what follows. J will (unless otherwise stated) be a homogeneous. locally fini- tely generated ideal of R'. We will prove that the initial ideals gr(p, (J)), easily computable by standard Grijbner basis techniques, approxinlate g r ( J ) . This result is surrl~narized in Theorem 5.13.

5.2. A generat ing se t of gr(.J) n M. By Proposition 5.2 and Lemnil 5.1 we can find a locally finite Grobner basis F = { f l , f> f3 , . . . } of J, where f, is b~mogeneous, and there exists positive integers ~ ( 1 ) < a(2) < a(3) < a(4) < ... such that for each total degree d. J f , l < d i 5 a (d) .

We may assume that F is minimal and reduced. Then, the set of leading mono- mials of F is a minimal generating set for gr(J) :

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Definition 5.3. Put m, = Lpp(f,) for all 2 and let B = {m,). Then we have that (B) = gr (J ) n M and (B),! = g r ( J ) . where we use the notational apparatus of [5]: (B) denotes the sernigroup ideal generated by B in M. and denotes the (nionornial) ideal generated by B in R'.

As a notational convenience, we denote, for any d, by Bd and by BCd the sets - Bd = { m E B I ) V L / = d ) = {ma(d- l j .+ l r . . .

B5,i = T ~ B = { 7 r ~ E B ( 1n~( 5 d } = {ml: m2. 7n3 , . . . , ma(d))

Lemma 5.4. Td gr (J ) = Td (Bsd),1. Furthermore, the K vector space

5.3. The necessary number of active variables. It will be of great importance to keep track of how many "active" variables are needed up to a givrn degree. The following definition makes this notion more precise.

When approximating Td g r ( J ) with Td gr(pn ( J ) ) , we certainly need at least as many active variables, that is, a t least as large n, as when approximating with TdPn (gr(J)) . The latter quantity, that is, the least N(d) such that Tdp, (gr(J)) = Td g ( ~ ) whenever n 2 N (d), is of course determined by B.

Definition 5.5. The "the necessary number of active variables up to degree d . N ( d ) , is defined as N(d) = rnax ( { maxsupp(m) / m E BSd } ) 5.4. Restricting B. It is clear that

Po ( B ) C PI ( B ) C P2 (B) C P3 ( B ) C . . . (3)

For an infinite B. (3) will not stabilize. However:

We now use (3) anti (4) to construct ascending chains of idcals in R': from (3) we get the (non-stabilizing) sequence

PO (gr(J?)' C pi (gr(J))e C ~2 (gr(J)Ie C p,3 (gr(.J))' C . . . (5)

and from (4) the stabilizing sequence

Lemma 5.7. The stable ualue Tdp~r(dj ( ~ T ( J ) ) ~ is equal to 7Ligr(J).

5.5. Relating the truncated initial ideals and the initial ideal.

5.5.1. We know from Corollary 3.4 that for all n, p, (g r ( J ) )e c gr(pn ( J ) ) e , in particular, Tdp, (g r ( J ) )e C Td gr(pn ( J ) ) e . For n 2 N ( d ) we get that Td p ( J ) =

Tdpn ( u ( J ) ? ~ c Tdgr(p,, ( J ) I e .

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TRUNCATED HOMOGENEOUS IDEALS 819

Remurk 5.8. It is not necessarily the case that gr(p,, ( J ) ) " C ( J ) ) ' , nor that

Td d p , (J)Ie c Tdgr(p,,+l ( J ) I e . 5.5.2. In most caes , the inclusion Td gr(J) = TdP,y(,) (gr(J))' C Td g r ( ~ ~ j ? J ( ~ ) ( J ) ) e will be strict. Thus, we may perforni the K-vector space decomposition 7dgr(P,w(d) ( J ) ) e = r d g r ( ~ ) @ Q N ( d ) where, in general, QN(d) is "on-zer). In fact, we can make this decomposition for any 11 2 N ( d ) : obtaining a sequence of K-vector spaces Q,,. Our next aim is to prove that, there exists an integer j i r (d ) , ?h~: sufficient rlumber of active variables up to degree d , such that n 2 ~ ( d ) ==+ d j , = 0. For such n's we will then have that Td gr(p,, ( , J ) ) e = Tdp, ( p ( J ) ) ' = Td g(.,'). We can complete diagram (7) and get

5.6. Reducing S-polynomials. It is provrd iu [5] that every S-polynomial (or, rat,her, S-power series) of elernerits of the choosen locally finite Grot~,ier basis F reduce to zero with respect to F. That is to say. each such S-poly~io:nial can be expressed as an admiss ible combznat ion of elements in F. We now fix a choice of such admissible combinations.

For any 1 5 i < j , choose ~ , , ~ , l , ai,,,~, . . . az,J,a(l f l 1 ) E R' such that

(the right-hand side is an atlnlissible conibirlat~on). Furthermore. we car also ensure that no ~ , , ~ , k have higher total degree than S ( f,, f,).

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Put A = { a,,,,k 1 1 <_ i < j , Ic < cx(lfJl) } , arid define, for any total degree d, A<d = { a,,l,k E -4 1 JS( f,, f,)l 5 d ) . That is, A<d consists of those a,,,,k that are involved in reducing those S-polynomials of elernerits in F that have total degree 5 d. Since F is locally finite there are only finitely inany such S-polynomials. We conclude that A<d is finite; this will be of utniost inlportance.

-

Definition 5.9. Let s[d) = rnax ( N ( d ) , rrlax { rmxsupp(Lpp(a)) / a E ASd }) and call this nuniber "the sufficient nurnber of active variables up to degree d . We remark that this nurnt)er unfortunately dcpends not only on J but also on the choice of A.

Remark 5.10. By construction, we have t,hat ;V(d) < ~ ( d ) . L e m m a 5.11. If P.Q E R', and zf

Proof. Assume, to simplify things, that P and (2 are nionic, with leading power products p and q respectively, and that t,he least cmxnon multiple of p and q is m. Then S(P , Q) = :P - :Q, and p, (S (P . Q ) j = F p n ( P ) - :pn (Q). On the other hand, p, (P) = p and p,, ( Q ) = q , hence S(p,, ( P ) . j~,, (C))) = :P - YQ. 0

5.7. Trunca t ing admissible combinations. Fix a total degree d. If 1 < z < j 5 d d ) then iJ,i, lJ,l.iS(f,.f,)l < d and

hence for such 71, every admissible colnbination such as (9), reducing to zero an S-polynomial of elements of F with total degree _< d, restricts to an admissible combination in K[zl . . . . . x,].

5.8. T h e m a i n theorem.

L e m m a 5.12. Let I be a homogeneous ideal zn K[xl , . . . : z,] generated by a finite, homogeneous set G. Let t be a positive zn tqrr , and suppose that all S-polynomials of elements in Gt except those that have total degree higher than t , reduce to zero with respect to G . Then, for each g 5 t , (g r ( I ) )g = ( i r ~ ( F ) ) ~ , ~

Proof. The result is well-known: there is a sirriple proof of it in [ 5 ] . El

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TRUNCATED HOMOGENEOUS IDEALS 82 1

This result can inirnediately bc gclieralized t,o the case of a locally fillitely ge- nerated ideal, simply by applying the corollary to the subideal generated by those (finitely many) elements of the locally finite generating set that have total degree < t . -

From the discussion above. w know that p,, ( F 5 d ) is such a "partid Grobner basis" for p,, (J) C K[xi , . . . , I,], (when n > N(d)) . and hence we conclutLe that

'Td gr(p,, ( J ) ) n .U = T ~ M " B < , ~ (10)

We then get that

rd ( , I ) ) ' n .u = T ~ . u B , ~ (11)

and that

rd i d p , ( J ) ) ' = 'Tdgr(J) (12)

This is the desired result! I t implies inimediately that the K-vector space Q,. defined previously, is zero. IVc sunlnlarize our results in the following tht,orem:

Theorem 5.13 (Degree-wise approximation of initial ideals). If J is a l x a l l y fini- tely generated ideal in R ' , thrn for all total degrees d we have that

~ ( d . 72) := r d p n ( g r j ~ ) ) ' c ~ * g r ( ~ , ( J ) ) ~ =: ~ ( d , n) (13)

Furthermore, there ens ts zntegers N(d), whzch we call "the necessary number of actzve varrables up to degree d " . and zntegers ~ ( d ) , vthzch we call "th,. suficzent number of actwe vartubks up to degree d " , such that.

(2) If 71 < N(d) then

L(d ,n) 5 r d g r ( J ) (14)

R ( d n ) T~ g ( ~ ) (15)

(2%) If N(d) < n < .V(d) f h t n

L (d. 1 2 ) = 'Td gr( J)

R(d. n) > ' T d g r ( ~ )

5.9. Some consequences of the approximation theorem.

Corollary 5.14. The followzrq are equivalent:

(i) gr(J) is finitely generated. (iz) gr(p, ( J ) j e stabzlzze when n tends t o infinity.

Furtermore, 27 the equzvalent condztions hold, then J is finitely gelaerated

Proof. If g r ( J ) is finitely generated, it is generated by a finite set of monomials. Therefore, there exists an integer N such that all these ~r~onornials art3 contained in

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822 SNELLMAN

M"'. Hence. the semigroup g r ( J ) n M is generated in M1'. and gr(p, ( J ) ) e ) = gr(J) whenever 7~ 2 N.

Conversely, if there exists an integer AT S I I C ~ that gr(p, ( J ) j C = gr(pk ( J ) ) e wlle- never 11. k 2 N , then by Theorml 5.13 this coIllnlorl value is g r ( J ) . We concltlde that g r ( J j f i M is generated in M" and consequently that g r ( J ) is generated in K[zi. . . . . :c,,]. It follows that g r ( J ) is finitely generated.

The last assertion is clear: a Grobner basis is also a gcncratirig set. 0 - Coro l l a ry 5.15. g r ( J ) = g r ( J ) - Proof. By Proposition 4.3, the inclusion > holds. Now let m E g r ( J ) il M. so that 711 E gr(p,, ( J ) ) n M for all sufficiently large n. Denote by 9 the degree of m. We now from Theorem 5.13 that g r ( J ) g = grjp,, (J)) for all sufficiently large n. But 771 E gr(p, ( J ) ) for all such n, hence 7n E g r ( J ) . 0 - Ques t ion 5.16. For arbitrary ideals I c R', is zt true that gr ( I ) = gr(1) ?

Coro l l a ry 5.17. g r ( J ) is completely de t~rmined by the ideals gr(p, ( J ) ) ' , and hertce by the ideals gr(p, ( J ) ) . Therefore, g r ( J ) zs determined b y the zdeals p,, ( J ) .

Ques t ion 5.18. Is J itself determined by the zdeals p,, ( J ) ?

Note that this question has a negative answer for non-locally finitely generated ideals: if I = (xl, x2. xs. . . . ) whereas I' = I + ( x i + x:, + ~3 + . . . ) then I # I' but pn (I) = on (1') = ( x l , . . . . 2,) for all n.

The author has recently proved [6 ] that the answer to Question 5.18 is "yes": locally finitely generated i d ~ a l s are determined by their truncations. The idea of the proof is to topologize R by the separated filtration giver1 1)y the kernels of the truncation hon~omorphisms. and then show that in this topology, locally finitely generated ideals are closed.

5.9.1. The mysterious ~ ( d ) . Theorem 5.13 is unsatisfactory in one aspect: it does not really tell us how to compute T d g r ( J ) from the initial idcals of the restricted ideals p, ( J ) . since it does not provide any hints as how to f i d the number ~ ( d ) . Wr can. of rourse, use the methods of [5] to find it, but that illvolves calculating rd g r ( J ) directly.

Illstead, one would like to perfor111 calculations of gr(p, ( J ) ) with in~re~asing 71, and from inspecting the results determine when the '.stablt, value a t degree d" 11as been rcachcd. Ideally, we should be able to compute Tdgr(p,, ( J ) ) e for succesively larger values of n, and then, when this sequence s e e m to have reached its stable value, because it has not changed for k consecutive values for 71. conclude that we have indeed reached the necessary nun~bcr of active variables.

Quest ion 5.19. Does there, for each homogeneous, locally finitely generated ideal I C R' , mist a k , independent of d , such that

~ ~ g r ( ~ , ( I ) ) ~ = rd &,+I ( I ) ) ~ = . . . = rd g r(pn+k (I))e

zmplzes that 'Td gr(p, ( I ) ) e = Td g r ( I ) ?

If this fails, one would be interested in the answer to the following question:

Ques t ion 5.20. Given J and d, zs there a faster way of corrcputiny ,+(d) than by calculating a partial Grobner basis for J up to degree d ?

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TRUNCATED HOMOGENEOUS IDEALS 823

TABLE 1. Initial ideals of truncat,ions of' the gcnerlc ideal g e n e r a d by a quadratic form arid a form of dcgrce 4

Partial results, slich as for generic itlt~als, or for a restricted set ot' admissible - orders, would be mteresting, should the ~ C ' I I ~ I ~ ~ ~ r o b l e ~ r l he hard to iolve

APPENDIX A. THE L E X I C O G R A P H I C A L INITIAL I D E A L OF A GEKEIC I D E A L

G E N E R A T E D BY A Q U A D R A T I C FORM A N D A FORM O F DEGREE 4

In this section, we will calculate the initial ideal (wit11 respect to the lexicographic order) of the generic ideal I generated by a. generic quadratic form and a generic form of degree 4. By "generic ideal". we mean. as in [3. 21. that not cnly are the generators generic, but they are independent in the sense that the u111o1l of their sets of coefficients is algebraically indeperlde~lt.

Table 1 shows the initial ideals of the restricted ideals p2 (I) to (17 (I). These restricted ideals are of course generic ideals in their correspondi~lg poly r~ornial rings. In the interest of brevity, we show only gr(p,, (I)) which means that or~ly the first three initial ideals are showed in their entirety.

From this table, we see that it is very plausible that

~ ' g r ( I ) = (z:, z ls ; , zlz;z:, L~ .K~ .L~~ .X: . X ~ Z ~ Z ~ Z ~ T ~ , x I L : z ~ x ~ : c ~ z G ) .

By considering also the restricted ideals with a s many as 11 variable:;: one can be rather certain that the minimal mononiial generators of gr(I) of degree, 8 are

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SNELLMAN

ACKNO\VLEDGEMENTS The author wislies to t,hank Jhrgen Backelin and Kristina C r o ~ i a for their carefill reading of the manuscript and for their nwncrous s u g g ~ s t i o ~ i s for improverrlent of i t .

[I] Dave Bayer. .-\ndrC- Galligo. and Xrke S t ~ l l ~ ~ l a r ~ Grobnrr Bases and extens~on of scalars. In Da\,id Eisenbud and Imcnzo Kobhiano. editorb. Cornputatzor~ui Algebruzc Geometry and Gommutatzve Aigebm, volume 2-1 of Syrnposzu hfuthr~riiatzca. 1991

[2] Ralf Frbberg An inequality for Hllbert wries of graded algebras Mathernattca Scandcnavtca. 56:117-144, 3985.

[3] Ralf Frijberg and Joachim Hollnlan Nrlt~ert srr ies for rrleals generated by generic fornls. Journal of Syrnbolzc Curnputatzun. 17:149-157. 1991

[4] Patricra Gianni Properties of Grohr~er Uaies urldrr specralrzations. volulne 383 of L N C S , pages 293-297. 1987

151 Jan Sneilman Grohir r bases and normal forrii.~ In a suhring of the power series ring on r o u ~ ~ t a b l y many variables over a field Jounmi of Sijmboizc Corrrpututzon, 1997. To appear

[F] Jan Sncll~rian. Some properties of a sullrrng of the power series rlng on a countably rllfiu~te number of varrablrs over a field 1997

[7] B L Van der \Varrd~n, Aloderne Algebra Dlr: Grutldlehren der hlathematischen \\rissenscllafter~. Vrerlag von .Julius Springer. 1930

Received: November 1996

Revised: March 1997

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