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On the uniformity of zero-dimensional complete intersections
Journal of Algebra 391 (2013) 82–92
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Journal of Algebra www.elsevier.com/locate/jalgebra
On the uniformity of zero-dimensional complete intersections Anthony V. Geramita a,b , Martin Kreuzer c,∗ a b c
Department of Mathematics and Statistics, Queen’s University, K7L3N6 Kingston, Canada Dipartimento di Matematica, Universitá di Genova, Genova, Italy Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau, Germany
a r t i c l e
i n f o
Article history: Received 14 July 2009 Available online 26 June 2013 Communicated by Luchezar L. Avramov MSC: 14M10 13D40 14N05 Keywords: Complete intersection Cayley–Bacharach Theorem Zero-dimensional scheme Minimal distance Generalized Reed–Muller code
a b s t r a c t After showing that the General Cayley–Bacharach Conjecture formulated by D. Eisenbud, M. Green, and J. Harris (1996) [6] is equivalent to a conjecture about the region of uniformity of a zerodimensional complete intersection, we prove this conjecture in a number of special cases. In particular, after splitting the conjecture into several intervals, we prove it for the first, the last and part of the penultimate interval. Moreover, we generalize the uniformity results of J. Hansen (2003) [12] and L. Gold, J. Little, and H. Schenck (2005) [9] to level schemes and apply them to obtain bounds for the minimal distance of generalized Reed–Muller codes. © 2013 Elsevier Inc. All rights reserved.
1. Introduction In [6], D. Eisenbud, M. Green, and J. Harris formulated several conjectures extending the classical Cayley–Bacharach Theorem for finite sets of points in Pn . Specifically, their Conjecture CB12 (called the General Cayley–Bacharach (GCB) Conjecture in this paper) claims that subschemes of zero-dimensional complete intersections in Pn have special Hilbert functions. More specifically, the conjectures claim that the a-invariants of their coordinate rings are comparatively small. In the last years, the GCB Conjecture has undergone many transformations and appears to be at the heart of several difficult problems (see for instance [2,3,5,7,8,15,16]). At the same time, it has steadfastly resisted attempts to prove it completely.
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Corresponding author. E-mail addresses: [email protected] (A.V. Geramita), [email protected] (M. Kreuzer).
0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.05.027
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In this paper, we take some further steps towards a proof of the GCB Conjecture. We begin by showing that it is equivalent to a statement about the region of uniformity of a zero-dimensional complete intersection X in PnK . Here we say that X is (i , j )-uniform if every subscheme Y of degree deg(Y) = deg(X) − i satisfies HFY ( j ) = HFX ( j ), and the region of uniformity of X is the set of all (i , j ) for which X is (i , j )-uniform. Notice that we are asking this condition for every subscheme of the appropriate degree. It is well-known that there always exists at least one subscheme having the truncated Hilbert function HFY = min{HFX , deg(Y)}. Next, we slice the GCB Conjecture up into several more manageable intervals. We prove the conjecture in the first of these intervals by comparing the Hilbert function of a subscheme Y of the desired kind to the Hilbert function of a complete intersection subscheme having the same a-invariant. The necessary inequalities are obtained by combining Macaulay’s Growth Theorem for Hilbert functions and Gotzmann’s Persistence Theorem. Next we prove the conjecture in the last interval using liaison, and in some other cases using ad hoc methods. Altogether, these results are sufficient to establish the GCB Theorem for zero-dimensional complete intersections in P2K , in P3K , and for a number of cases in P4K . In the last section we generalize, to zero-dimensional level schemes, a uniformity result which was discussed in [12] and [9]. We use our GCB Theorems to improve the bounds for the minimal distance of generalized Reed–Muller codes given in those papers, and we illustrate the improvements given by the GCB Theorem (resp. the GCB Conjecture) with several examples. 2. The uniformity of a zero-dimensional scheme Unless specified otherwise, we shall adhere to the notation used in [13] and [14]. Let K be a field, let X ⊆ PnK be a zero-dimensional subscheme of degree s, let P = K [x0 , . . . , xn ] be standard graded, let I X ⊂ P be the homogeneous vanishing ideal of X, and let R = i 0 R i = P / I X be the homogeneous coordinate ring of X. The Hilbert function of X is the map HFX : Z −→ Z given by HFX (i ) = dim K ( R i ) and its first difference function is HFX (i ) = HFX (i )− HFX (i − 1). The number aX = max{i | HFX (i ) < s} is called the a-invariant of R (or of X). Thus aX + 1 is the last place where HFX (i ) = 0. Definition 2.1. Let 1 i s − 1 and 1 j aX . The scheme X is said to be (i , j )-uniform if every subscheme Y ⊆ X of degree s − i satisfies HFY ( j ) = HFX ( j ). For instance, the scheme X is (1, aX )-uniform if and only if it has the classical Cayley–Bacharach property. A non-degenerate scheme X is (s − n − 1, 1)-uniform if and only if it is in linearly general position. Furthermore, the scheme X is in uniform position if and only if it is (i , j )-uniform for all 1 j aX and all 1 i s − HFX ( j ). It is clear that an (i , j )-uniform scheme is also (i − 1, j )-uniform and (i , j − 1)-uniform. Given j, the maximal i such that X could be (i , j )-uniform is i = s − HFX ( j ). Thus the maximal region of uniformity of X is determined solely by the Hilbert function of X. It is illustrated by the dotted region in the following diagram. j .... .........
• aX
j
. .• . . . . . .
.
−. . . . . . . . . . . . . . . . . . •................
HFX ( j )
.................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0
s−1 s
.................
i
If X is in uniform position, its region of uniformity is precisely this maximal region. However, for any specific X, the region of uniformity will usually be much smaller. The main topic of this paper is
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to examine the region of uniformity of zero-dimensional complete intersections. This topic is closely related to the Cayley–Bacharach conjectures of D. Eisenbud, M. Green, and J. Harris, as we shall see next. 3. The General Cayley–Bacharach Conjecture In [6], the authors formulate the following Conjecture CB12. Let Y ⊆ PnK be a subscheme of a zero-dimensional complete intersection of hypersurfaces of degrees d1 · · · dn . If Y fails to impose independent conditions on hypersurfaces of degree m, then we have deg(Y) e · dt · dt +1 · · · dn where t and e are defined by the relations n n n (di − 1) m + 1 < (di − 1) and e = m + 2 − (di − 1) i =t
i =t −1
i =t
(Notice that we have corrected the definition of e given in [6], which was an obvious misprint.) This conjecture generalizes their Conjecture CB11 which deals with the case d1 = · · · = dn = 2. By distinguishing the possible values of t, we can divide Conjecture CB12 into several intervals. Since we are going to address them separately, let us spell out the partial conjectures individually. Conjecture 3.1 (The General Cayley–Bacharach (GCB) Conjecture). Let X ⊆ PnK be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn . It is conjectured that the following claims hold true.
( I 1 ) Let 1 e < dn−1 and Y ⊆ X be such that HFY (e + dn − 2) = 0. Then we have deg(Y) e · dn . ( I 2 ) Let 1 e < dn−2 and Y ⊆ X be such that HFY (e + dn−1 + dn − 3) = 0. Then we have deg(Y) e · dn−1 · dn . .. . ( I n−1 ) Let 1 e < d1 and Y ⊆ X be such that HFY (e + d2 + · · · + dn − n) = 0. Then we have deg(Y) e · d2 · · · dn . To see that this conjecture is equivalent to Conjecture CB12, it suffices to note that HFY (m) < deg(Y) is equivalent to HFY (m + 1) = 0. The GCB Conjecture can be phrased as a conjecture about the region of uniformity of a zerodimensional complete intersection as follows. Conjecture 3.2 (The Uniformity Conjecture). Let X ⊆ PnK be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn . It is conjectured that the following claims hold true.
(U 1 ) Let 1 e < dn−1 . Then X is (edn − 1, aX − e − dn + 3)-uniform. (U 2 ) Let 1 e < dn−2 . Then X is (edn−1 dn − 1, aX − e − dn−1 − dn + 4)-uniform. ... (U n−1 ) Let 1 e < d1 . Then X is (ed2 · · · dn − 1, aX − e − d2 − · · · − dn + n + 1)-uniform. Here we have aX = d1 + · · · + dn − n − 1. To see that the Uniformity Conjecture is equivalent to the GCB Conjecture, we consider the ith interval and let Y ⊂ X be a subscheme of degree edn−i +1 · · · dn − 1. The GCB Conjecture says that we should have HFY (e + dn−i +1 + · · · + dn − i − 1) = 0. Let Y = X \ Y be the residual subscheme. Then we have
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HFY (aX + 1) − (e + dn−i +1 + · · · + dn − i − 1) = HFX (aX + 1) − (e + dn−i +1 + · · · + dn − i − 1) − HFY (e + dn−i +1 + · · · + dn − i − 1) = HFX (aX − e − dn−i +1 − · · · − dn + i + 2) Since this has to hold for all subschemes Y of X of degree s − edn−i +1 · · · dn + 1, we are in fact conjecturing that X is (edn−i +1 · · · dn − 1, aX − e − dn−i +1 − · · · − dn + i + 2)-uniform. Let us illustrate these conjectures with a couple of examples. In both cases the thick line marks the maximal possible region of uniformity and the dotted region corresponds to the conjectured region of uniformity. Example 3.3. Let X be a complete intersection of type (4, 7) in P2K . Then the Uniformity Conjecture says that the region of uniformity of X should contain the dotted region in the following picture. j .... ........
9 •...
3
.........•.. .................... •. ...........................•. ... ...................................•... ....................................•.. ...................................... ................................................................. •.. .................... ... ....................................•. ................................................................................................. ... ............................................. .... .........................•... ...................................................................................................................................... ..................•. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 6
13
20
25
.................
i
28
Example 3.4. Let X be a complete intersection of type (2, 3, 5) in P3K . Then the Uniformity Conjecture says that the region of uniformity of X should contain the dotted region in the following picture. j .... .........
7 •...
3
.........•.. ............................. •. .............................................•. ... .....................................................•... ........................................... ............. ... ......................................................•.. ..................................................................... ............................................... .................................. ............................................................................................... •.. . . . . . . . . . . . . . . . . . . . . . . . . .... ...........................•. 4
9
14
21
26
.................
i
30
Finally, we strengthen the GCB Conjecture and the Uniformity Conjecture by introducing the following Artinian version. Conjecture 3.5 (The Artinian Uniformity (AU) Conjecture). Let P = K [x1 , . . . , xn ] be standard graded, let ( f 1 , . . . , f n ) ∈ P n be a homogeneous regular sequence, let I = f 1 , . . . , f n be the ideal it generates, and let di = deg( f i ) for i = 1, . . . , n. Assume that we have d1 · · · dn , and let J ⊇ I be an ideal in P . It is conjectured that the following claims hold true.
( A 1 ) Let 1 e < dn−1 , and suppose that HF P / J (e + dn − 2) = 0. Then we have dim K ( P / J ) e · dn . ( A 2 ) Let 1 e < dn−2 , and suppose that HF P / J (e + dn−1 + dn − 3) = 0. Then we have dim K ( P / J ) e · dn−1 · dn . .. .
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( An−1 ) Let 1 e < d1 , and suppose that HF P / J (e + d2 + · · · + dn − n) = 0. Then we have dim K ( P / J ) e · d2 · · · dn . Let us verify that this version implies the Uniformity Conjecture. Since we are exclusively dealing with Hilbert functions, we may enlarge the base field as we please. Thus we shall assume that it is infinite and that (possibly after a suitable homogeneous linear change of coordinates) the indeterminate x0 is a non-zerodivisor for the homogeneous coordinate ring R = P / I X . The ring A = R /(x0 ) is called the Artinian reduction of R. It is of the form A = P / I with I ∼ = ( I X + (x0 ))/(x0 ). Its Hilbert function is HF A = HFX , the first difference function of HFX . If we apply the AU Conjecture to ideals of the form J ∼ = ( I Y + (x0 ))/(x0 ) in P , where Y is a subscheme of X, we obtain the Uniformity Conjecture. Since not every ideal in A is of the form ( I Y + (x0 ))/(x0 ) for some subscheme Y ⊆ X, the AU Conjecture is stronger than the GCB Conjecture and the Uniformity Conjecture. Therefore we shall study it in the following sections. It will also come in handy to consider the following dual version of the AU Conjecture. Conjecture 3.6 (The Artinian Dual (AD) Conjecture). In the setting of the AU Conjecture, it is conjectured that the following statements hold true for i = 1, . . . , n − 1.
( D i ) Let 1 e < dn−i and suppose that ( J / I )c = 0 for c = (d1 − 1) + · · · + (dn−i −1 − 1) + (dn−i − e ). Then we have dim K ( J / I ) e · dn−i +1 · · · dn . To show that the AD Conjecture is equivalent to the AU Conjecture, we argue as follows. The a-invariant of a graded zero-dimensional K -algebra is the largest degree in which its Hilbert function is not zero. It is well-known that a P / I = d1 + · · · + dn − n. Let us denote J / I by J , and let c = e + dn−i +1 + · · · + dn − i − 1. Then condition ( A i ) reads as follows:
( A i ) Let 1 e < dn−i , and suppose that HF A / J (c ) = 0. Then we have dim K ( A / J ) e · dn−i +1 · · · dn . The annihilator ideal J = Ann A ( J ) satisfies HFJ (i ) = HF A / J (a P / I − i ) for all i ∈ Z (see [4]). Hence
we have ( A / J )c = 0 if and only if J c˜ = 0 in degree c˜ = a P / I − c = (d1 + · · · + dn − n) − e − dn−i +1 − · · · − dn + i + 1 = (d1 − 1) + · · · − (dn−i −1 − 1) + (dn−i − e ). Furthermore, we have dim K ( A / J ) = dim K ( J ). Keeping this in mind, we see that ( A i ) is equivalent to the following condition.
( D i ) Let 1 e < dn−i , and suppose that J c˜ = 0. Then we have dim K (J ) e · dn−i +1 · · · dn . Finally, we note that the fact that the ring A is a 0-dimensional complete intersection implies J = Ann A ( J ), i.e. every ideal is the annihilator of some ideal. By changing ˜J into J in condition ( D i ), the claimed equivalence follows. 4. The first and the last interval In the following we prove the AU Conjecture in some special cases. We begin by showing that it holds true in the first interval ( A 1 ). A similar result was shown in [5] by entirely different means. In the following we let P = K [x1 , . . . , xn ] be standard graded, and we let A = P / I be a zerodimensional complete intersection ring of type (d1 , . . . , dn ), where d1 · · · dn . In other words, we assume that I is generated by a homogeneous regular sequence ( f 1 , . . . , f n ) in P whose degree sequence is (d1 , . . . , dn ). Proposition 4.1 (The first interval). Let A = P / I be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn , let 1 e dn−1 , and let J ⊇ I be a homogeneous ideal in P such that HF P / J (e + dn − 2) = 0. Then we have dim K ( P / J ) edn . In other words, the AU Conjecture is true in the interval A 1 and for e = 1 in the interval A 2 .
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Proof. First we show that HF P / J (dn − 1) e. By the hypothesis, this is clear for e = 1. So, let e 2 and assume that HF P / J (dn − 1) < e. Given any number i dn − 1 dn−1 − 1 e − 1 such that HF P / J (i ) e − 1 i, we apply Macaulay’s Growth Theorem (cf. [14, 5.5.28]). The binomial representation of m = HF P / J (i ) in base i is
m [i ] =
i i
+ ··· +
i −m+1 i −m+1
and therefore HF P / J (i + 1) (m[i ] )+ + = HF P / J (i ). Since we have
HF P / J (dn − 1) < e
and
HF P / J (dn − 1) + (e − 1) 1
there exists a number i ∈ {dn − 1, dn , . . . , e + dn − 3} for which HF P / J (i ) = HF P / J (i + 1). Thus we have a case of maximal growth in Macaulay’s Growth Theorem. By Gotzmann’s Persistence Theorem (cf. J = J i +1 satisfies HF P /J ( j ) = HF P /J (i ) for all j i. Therefore we [10] and [1, Thm. 3.6]), the ideal
get dim( P / J ) = 1, in contradiction to the fact that the residue classes of the elements of the regular sequence generating I are contained in J because of i + 1 dn . Next we show that HF P / J is greater than or equal to the function H given by H (i ) .... ........
..... ............................................................................................................• ..• ..... ..... ..... ..... ..... ..... ..... ..... . ..... . . . ..... ... . . . . ..... .... . ..... . . ..... ... . . . . ..... ... . . ..... . . ... ..... . . . . ..... .... . . ..... . .. . . . • •..... ... ... ..
e −
1
e −1
dn −1
e +dn −2
.................
i
This follows again from Macaulay’s Growth Theorem: we have already seen that HF P / J (i ) has to be strictly decreasing in the range dn − 1 i e + dn − 2. In the range 1 i e − 1, if ever HF P / J (i ) i, then Macaulay’s Growth Theorem implies HF P / J ( j ) i for all j i. But then HF P / J (dn − 1) i e − 1 yields a contradiction. It remains to consider the range e − 1 i dn − 1. Suppose that HF P / J (i ) < e for one of these i. Again Macaulay’s Growth Theorem shows HF P / J ( j ) < e for all j i, contradicting HF P / J (dn − 1) e. Altogether, we have shown dim K ( P / J ) = i 0 HF P / J (i ) i 0 H (i ) = edn . 2 Corollary 4.2. The AU Conjecture is true for n = 2. In particular, the GCB Conjecture and the Uniformity Conjecture hold true for complete intersections in P2K . Proof. For n = 2, there is only one interval A 1 to consider.
2
In those cases in which we are able to prove the GCB Conjecture we shall call it henceforth the GCB Theorem. Next we prove the AU Conjecture in the last interval ( A n−1 ). A different proof for this result is contained in [5]. For a homogeneous ideal I ⊆ P , we define the depth sequence (d1 , . . . , d ) of I by inductively choosing the minimal number di for which there exists a homogeneous regular sequence of type (d1 , . . . , di ) in I . Its length is the depth of I with respect to (x1 , . . . , xn ). Clearly, a homogeneous ideal J containing I has a depth sequence which is componentwise less than or equal to the depth sequence of I . Now we are ready to state and prove the desired result.
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Proposition 4.3 (The last interval). Let A = P / I be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn , let 1 e d1 − 1, and let J ⊇ I be a homogeneous ideal in P such that HF P / J (e + d2 + · · · + dn − n) = 0. Then we have dim K ( P / J ) ed2 · · · dn . In particular, the AU Conjecture holds true in the last interval. Proof. Let J = { f ∈ P | f · J ⊆ I } be the linked ideal. Then we have
HF P / J (d1 − e ) = HF P / I a P / I − (d1 − e ) − HF P / J a P / I − (d1 − e )
= HF P / I (e + d2 + · · · + dn − n) − HF P / J (e + d2 + · · · + dn − n) < HF P / I (e + d2 + · · · + dn − n) Thus we see that J d −e = 0, and hence the depth sequence of J is componentwise less than or 1 equal to (d1 − e , d2 , . . . , dn ). In particular, the ideal J contains a zero-dimensional complete intersection ideal of type (d1 − e , d2 , . . . , dn ). This implies dim K ( P / J ) (d1 − e )d2 · · · dn , and therefore dim K ( P / J ) ed2 · · · dn . 2 Corollary 4.4. The AU Conjecture is true for n = 3. In particular, the GCB Conjecture and the Uniformity Conjecture hold true for complete intersections in P3K . Proof. For n = 3, there are two intervals ( A 1 ) and ( A 2 ) to consider. The first interval is dealt with by Proposition 4.1, and the second interval by the preceding proposition. 2 Corollary 4.5. The AU Conjecture is true for n = 4 and d1 = d2 = 2. In particular, the GCB Conjecture and the Uniformity Conjecture hold true for complete intersections of type (2, 2, d3 , d4 ) in P4K , where 2 d3 d4 . Proof. For n = 4, there are three intervals ( A 1 ), ( A 2 ), and ( A 3 ) to consider. The first interval is dealt with by Proposition 4.1, the second interval consists only of the case e = 1 which is treated in the same proposition, and in the third interval the conjectures hold by the preceding proposition. 2 5. The penultimate interval In this section we want to study the AU Conjecture in the interval ( A n−2 ). We begin with a couple of basic simplifications. Proposition 5.1 (Reduction to the Gorenstein case). Suppose that we want to prove one case of the AU Conjecture, i.e. we want to prove it for a specific ( A i ) and a fixed value of e ∈ {1, . . . , dn−i − 1}. Then we may assume that P / J is a Gorenstein ring. Proof. By the hypothesis, the largest degree j for which we have ( P / J ) j = 0 satisfies the inequality j e + dn−i +1 + · · · + dn − i − 1 =: e˜ . By replacing J by J + P e˜ +1 , we may assume that j = e + dn−i +1 + · · · + dn − i − 1. Now suppose that dim K ( P / J ) j 2. Let f be a homogeneous polynomial of degree j whose residue class is a non-zero element of the socle of P / J , and let J = J + f . Then J continues to satisfy the hypothesis of the AU Conjecture and dim K ( P / J ) j = dim K ( P / J ) j − 1. By repeating this process a number of times, we find an ideal J ⊇ J for which the largest degree j with ( P / J ) j = 0 is j = e + dn−i +1 + · · · + dn − i − 1 and for which dim K ( P / J ) j = 1. If P / J is not a Gorenstein ring, there exists a degree k < j such that P / J contains non-zero socle elements of degree k. Again we let f ∈ P k be a polynomial whose residue class is such a socle element, and we set J = J + f . Then P / J satisfies the hypotheses of the AU Conjecture and dim K ( P / J ) = dim K ( P / J ) − 1. By repeating this process, we arrive at an ideal J such that P / J has
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a 1-dimensional socle, and this socle is its homogeneous component of (largest) degree e + dn−i +1 + · · · + dn − i − 1. Since dim K ( P / J ) is less than or equal to dim K ( P / J ), it suffices to prove the AU Conjecture for J . 2 In view of this proposition, we may assume that the Hilbert function of P / J is symmetric and that J = I : g with a homogeneous polynomial g ∈ P γ of degree γ = d1 + · · · + dn−i − e − n + i + 1. By using liaison as in the proof of Proposition 4.3, we see that the conjecture is equivalent to the following “dual” version. Lemma 5.2. Assume that we are in the situation of the interval ( A n−2 ) in the AU Conjecture, i.e. that 1 e < d2 , and let g ∈ P d1 +d2 −e−1 \ I d1 +d2 −e−1 . Then the following statements are equivalent. (a) The ideal J = I : g satisfies dim K ( P / J ) ed3 · · · dn . J = I + g satisfies dim K ( P / J ) (d1 d2 − e )d3 · · · dn . (b) The ideal J are related Proof. This follows immediately from the fact that the Hilbert functions of P / J and P / by HF P /J (i ) = HF P / I (a P / I − i ) − HF P / J (a P / I − i ) for all i 0 since the two ideals are linked. 2 Now we are ready to prove the main result of this section. Proposition 5.3 (The second part of the penultimate interval). Let A = P / I be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn , let d1 e d2 − 1, and let J ⊇ I be a homogeneous ideal in P such that HF P / J (e + d3 + · · · + dn − n + 1) = 0. Then we have dim K ( P / J ) ed3 · · · dn . In particular, the AU Conjecture is true in the penultimate interval ( A n−2 ) for d1 e < d2 . Proof. By Proposition 5.1, we may assume that P / J is a Gorenstein ring with a P / J = e + d3 + · · · + dn − n + 1 and that J = I : g for some homogeneous polynomial g of degree γ = d1 + d2 − e − 1. Suppose the claim of the proposition is not true. Let ( I , J ) be a counterexample for which d1 is minimal. Let us show first that gcd( f 1 , g ) = 1. For a contradiction, assume that f 1 = h1 h2 and g = h2 g˜ for some polynomials h1 , h2 , g˜ with deg(h2 ) 1. Then h1 g = f 1 g˜ ∈ I implies h1 ∈ J . Hence the ideal I = h1 , f 2 , . . . , f n has initial degree α ( I ) < α ( I ). It is a complete intersection, since ph1 ∈ f 2 , . . . , f n
with p ∈ P implies ph1 h2 = p f 1 ∈ f 2 , . . . , f n , and thus p ∈ f 2 , . . . , f n . But I ⊆ J , and we still have HF P / J (e + d3 + · · · + dn − n + 1) = 0 and deg(h1 ) < e d2 − 1. Since deg(h1 ) is smaller than d1 , the pair ( I , J ) is not a counterexample to the claim of the proposition, i.e. we have dim K ( P / J ) e d3 · · · dn . This contradicts the assumption that ( I , J ) is a counterexample, and thus the claim gcd( f 1 , g ) = 1 is proved. Next we observe that d1 γ d2 − 1. The ideal J = I : J = I + g contains both the regular sequence ( f 1 , f 2 , . . . , f n ) and the regular sequence ( f 1 , g ). Thus its depth sequence is componentwise less than or equal to (d1 , γ , d3 , . . . , dn ), and hence we have dim K ( P / J ) d1 γ d3 · · · dn . Now e d1 implies d21 − d1 ed1 − e, and therefore d1 γ = d1 (d1 + d2 − e − 1) d1 d2 − e. Altogether, we find
dim K ( P / J ) d1 γ d3 · · · dn (d1 d2 − e )d3 · · · dn and the lemma implies that ( I , J ) is not a counterexample to the claim of the proposition. This contradicts our assumption and finishes the proof. 2 6. An application to coding theory In [12], J. Hansen proved a uniformity result for reduced zero-dimensional complete intersection in P2K and used it to derive some bounds for the minimal distances of certain linear codes. Later, in [9], his result was generalized to reduced zero-dimensional complete intersections in PnK . Below
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we shall see that the region of uniformity predicted by the GCB Conjecture is usually much bigger than what was shown in these papers. In fact, our next result points out that the weaker uniformities shown there hold in a much more general setting, namely for level schemes. Here we say that a zero-dimensional scheme X ⊆ PnK is level if the Artinian reduction R of its homogeneous coordinate ring R satisfies socle( R ) = R aX +1 . Proposition 6.1. Let X ⊆ PnK be a level scheme and s = deg(X). Then the following statements hold true. (a) The scheme X is (1, aX )-uniform, i.e. it has the Cayley–Bacharach property. (b) If X is (i , j )-uniform for some 1 i s and some 1 j aX , then X is (i + 1, j − 1)-uniform. (c) For i = 1, . . . , aX , the scheme X is (i , aX + 1 − i )-uniform. Proof. First we show (a). Let f Y ∈ R be a minimal separator of a subscheme Y ⊂ X of degree deg(Y) = deg(X) − 1. Then we have ¯f Y ∈ socle( R ), and therefore deg( f Y ) = aX + 1. Hence X has the Cayley–Bacharach property. To prove (b), we let Y ⊆ X be a subscheme of degree s − i. Let J be the ideal of Y in R. Then HFY ( j ) = HFX ( j ) implies J j = 0. Now consider a subscheme Y ⊂ Y of degree s − i − 1, and let J be the ideal of Y in R. We have dim K ( J ) = i and dim K ( J ) = i + 1, and we distinguish two cases. The first case is ( J ) j −1 = 0. Then we also have ( J ) j = 0 because otherwise ( J ) j −1 ⊆ socle( R ) contradicts j − 1 < aX + 1 and the property of R being a level algebra. Now dim K ( J ) dim K ( J ) j −1 + dim K ( J ) j + dim K ( J ) 2 + dim K ( J ) yields a contradiction. Hence we are left with the case ( J ) j −1 = 0. In this case we get HFY ( j − 1) = HFX ( j − 1). Since every subscheme of X of degree s − i − 1 arises in this way, it follows that X is (i + 1, j − 1)-uniform. Finally, we note that claim (c) is an immediate consequence of (a) and (b). 2 Using the main result of [11], we can immediately translate this proposition into estimates for the minimal distance of certain linear codes. Given a finite field K , a reduced zero-dimensional subscheme X = { p 1 , . . . , p s } ⊆ PnK , and a number j ∈ {1, . . . , aX }, the image of the map
Φ j : R j −→ K s defined by Φ j ( f ) = ( f ( p 1 ), . . . , f ( p s )) is called the jth generalized Reed–Muller code C j (X) associated to X. Corollary 6.2. Let K be a finite field, and let X ⊆ PnK be a reduced level scheme. Then the jth generalized Reed–Muller code C j (X) associated to X is an [s, k, d]-code where k = HFX ( j ) and d aX + 2 − j. Proof. By [11, Thm. 8], we have d = 1 + max{i | X is (i , j )-uniform}. Thus the claim follows from part (c) of the proposition. 2 We end this paper with three examples which show that, in the case of a reduced zerodimensional complete intersection, the uniformity (and therefore the minimal distance bounds) coming from the GCB Theorem usually exceeds the uniformity shown in the preceding proposition for the relevant numbers j. In each picture the region of uniformity provided by Proposition 6.1 is dotted and the additional region of uniformity coming from the GCB Theorem is marked by asterisks. Example 6.3. Let X be a zero-dimensional complete intersection of type (5, 5) in P2K .
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j . ..........
8 •...
3
..........•.. ..... . ..........................• ...... .............. ... ......................................• ....................................•.. .................... ............. ... . ............................................................... .............................................• ... ....................∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗... ...∗..∗∗∗∗∗∗∗∗∗∗∗.............................................. ....................................• .. ..................∗.....∗ .. .....................∗....∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ........................................................................... ................................∗ •.. ..∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗...∗ ..∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ..... ... .. ..∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗.. . .................•. 3
9
19
14
.................
i
25
Notice that Corollary 4.2 implies that X satisfies the Uniformity Conjecture, i.e. it has at least the marked region of uniformity. Example 6.4. Let X be a zero-dimensional complete intersection of type (3, 3, 4) in P3K . j .... ........
7 •...
3
...........•.. ....... . .. ..................................• ...... .......... ....................................................... •.. ............. . .......................................................................•.. .................................................. . ............. ...∗∗∗∗∗∗∗∗∗∗∗ ......................................................................... . . ...................................... ...................∗........∗∗∗∗∗∗∗∗ •.. ....................∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗.... ......................................................•.. ...........................∗...∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗..... ............................. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ .. . . . . . . . . . . ∗...∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ . • 6
18
11
23
26
32
.................
i
36
Notice that Corollary 4.4 implies that X satisfies the Uniformity Conjecture, i.e. it has at least the marked region of uniformity. Example 6.5. Let X be a zero-dimensional complete intersection of type (2, 2, 3, 3) in P4K . j ..... ........
6 •... 3
...........•.. ............................................. •.. ...... ... ...... .......................................................................................................• ... .............∗∗∗∗∗∗∗...∗ .........................................................................................•.. ...∗∗∗.∗........................... ..............∗.....∗ ......................................................................... ............. ...∗∗∗∗∗∗∗∗∗∗∗∗∗∗....∗ .....∗ .....∗ .....∗ .....∗ .....∗ .....∗ .....∗ ......∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗......∗ ............................∗∗∗∗∗∗∗∗∗∗∗∗ •.. . . . . . . . . ....∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗....∗ ....................................•. 5
13
17
23
31
.................
i
36
Notice that Corollary 4.5 implies that X satisfies the Uniformity Conjecture, i.e. it has at least the marked region of uniformity. Acknowledgments The second author thanks the Department of Mathematics and Statistics at Queen’s University (Kingston, Canada) for its kind hospitality during part of the preparation of this paper. This research was supported by NSERC. Both authors are extremely grateful to the anonymous referee for such a very careful reading of our paper. References [1] A. Bigatti, A.V. Geramita, J.C. Migliore, Geometric consequences of extremal behaviour in a theorem of Macaulay, Trans. Amer. Math. Soc. 346 (1994) 203–235. [2] G. Caviglia, D. Maclagan, Some cases of the Eisenbud–Green–Harris conjecture, Math. Res. Lett. 15 (2008) 427–433. [3] S. Cooper, Growth conditions for a family of ideals containing regular sequences, J. Pure Appl. Algebra 212 (2007) 122–131. [4] E. Davis, A.V. Geramita, F. Orecchia, Gorenstein algebras and the Cayley–Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985) 593–597. [5] A.A. du Plessis, C.T.C. Wall, Singular hypersurfaces, versality, and Gorenstein algebras, J. Algebraic Geom. 9 (2000) 309–322.
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[6] D. Eisenbud, M. Green, J. Harris, Cayley–Bacharach theorems and conjectures, Bull. Amer. Math. Soc. 33 (1996) 295–324. [7] C.A. Francisco, Almost complete intersections and the Lex-Plus-Powers Conjecture, J. Algebra 276 (2004) 737–760. [8] C.A. Francisco, B.P. Richert, Lex-plus-powers ideals, in: E. Peeva (Ed.), Syzygies and Hilbert Functions, in: Lect. Notes Pure Appl. Math., vol. 254, CRC Press, 2007, pp. 113–144. [9] L. Gold, J. Little, H. Schenck, Cayley–Bacharach and evaluation codes on complete intersections, J. Pure Appl. Algebra 196 (2005) 91–99. [10] G. Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Rings, Math. Z. 158 (1978) 61–70. [11] J. Hansen, Points in uniform position and maximum distance separable codes, in: Zero-Dimensional Schemes, Ravello, 1992, de Gruyter, Berlin, 1994, pp. 205–211. [12] J. Hansen, Linkage and codes on complete intersections, Appl. Algebra Engrg. Comm. Comput. 14 (2003) 175–185. [13] M. Kreuzer, L. Robbiano, Computational Commutative Algebra 1, Springer, Heidelberg, 2000. [14] M. Kreuzer, L. Robbiano, Computational Commutative Algebra 2, Springer, Heidelberg, 2005. [15] J. Mermin, S. Murai, The lex-plus-powers conjecture holds for pure powers, Adv. Math. 226 (4) (2011) 3511–3539, preprint, available at http://arxiv.org/abs/0801.0391v1, 2008. [16] B.P. Richert, A study of the lex plus powers conjecture, J. Pure Appl. Algebra 186 (2004) 169–183.
Contents lists available at SciVerse ScienceDirect
Journal of Algebra www.elsevier.com/locate/jalgebra
On the uniformity of zero-dimensional complete intersections Anthony V. Geramita a,b , Martin Kreuzer c,∗ a b c
Department of Mathematics and Statistics, Queen’s University, K7L3N6 Kingston, Canada Dipartimento di Matematica, Universitá di Genova, Genova, Italy Fakultät für Informatik und Mathematik, Universität Passau, 94030 Passau, Germany
a r t i c l e
i n f o
Article history: Received 14 July 2009 Available online 26 June 2013 Communicated by Luchezar L. Avramov MSC: 14M10 13D40 14N05 Keywords: Complete intersection Cayley–Bacharach Theorem Zero-dimensional scheme Minimal distance Generalized Reed–Muller code
a b s t r a c t After showing that the General Cayley–Bacharach Conjecture formulated by D. Eisenbud, M. Green, and J. Harris (1996) [6] is equivalent to a conjecture about the region of uniformity of a zerodimensional complete intersection, we prove this conjecture in a number of special cases. In particular, after splitting the conjecture into several intervals, we prove it for the first, the last and part of the penultimate interval. Moreover, we generalize the uniformity results of J. Hansen (2003) [12] and L. Gold, J. Little, and H. Schenck (2005) [9] to level schemes and apply them to obtain bounds for the minimal distance of generalized Reed–Muller codes. © 2013 Elsevier Inc. All rights reserved.
1. Introduction In [6], D. Eisenbud, M. Green, and J. Harris formulated several conjectures extending the classical Cayley–Bacharach Theorem for finite sets of points in Pn . Specifically, their Conjecture CB12 (called the General Cayley–Bacharach (GCB) Conjecture in this paper) claims that subschemes of zero-dimensional complete intersections in Pn have special Hilbert functions. More specifically, the conjectures claim that the a-invariants of their coordinate rings are comparatively small. In the last years, the GCB Conjecture has undergone many transformations and appears to be at the heart of several difficult problems (see for instance [2,3,5,7,8,15,16]). At the same time, it has steadfastly resisted attempts to prove it completely.
*
Corresponding author. E-mail addresses: [email protected] (A.V. Geramita), [email protected] (M. Kreuzer).
0021-8693/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jalgebra.2013.05.027
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In this paper, we take some further steps towards a proof of the GCB Conjecture. We begin by showing that it is equivalent to a statement about the region of uniformity of a zero-dimensional complete intersection X in PnK . Here we say that X is (i , j )-uniform if every subscheme Y of degree deg(Y) = deg(X) − i satisfies HFY ( j ) = HFX ( j ), and the region of uniformity of X is the set of all (i , j ) for which X is (i , j )-uniform. Notice that we are asking this condition for every subscheme of the appropriate degree. It is well-known that there always exists at least one subscheme having the truncated Hilbert function HFY = min{HFX , deg(Y)}. Next, we slice the GCB Conjecture up into several more manageable intervals. We prove the conjecture in the first of these intervals by comparing the Hilbert function of a subscheme Y of the desired kind to the Hilbert function of a complete intersection subscheme having the same a-invariant. The necessary inequalities are obtained by combining Macaulay’s Growth Theorem for Hilbert functions and Gotzmann’s Persistence Theorem. Next we prove the conjecture in the last interval using liaison, and in some other cases using ad hoc methods. Altogether, these results are sufficient to establish the GCB Theorem for zero-dimensional complete intersections in P2K , in P3K , and for a number of cases in P4K . In the last section we generalize, to zero-dimensional level schemes, a uniformity result which was discussed in [12] and [9]. We use our GCB Theorems to improve the bounds for the minimal distance of generalized Reed–Muller codes given in those papers, and we illustrate the improvements given by the GCB Theorem (resp. the GCB Conjecture) with several examples. 2. The uniformity of a zero-dimensional scheme Unless specified otherwise, we shall adhere to the notation used in [13] and [14]. Let K be a field, let X ⊆ PnK be a zero-dimensional subscheme of degree s, let P = K [x0 , . . . , xn ] be standard graded, let I X ⊂ P be the homogeneous vanishing ideal of X, and let R = i 0 R i = P / I X be the homogeneous coordinate ring of X. The Hilbert function of X is the map HFX : Z −→ Z given by HFX (i ) = dim K ( R i ) and its first difference function is HFX (i ) = HFX (i )− HFX (i − 1). The number aX = max{i | HFX (i ) < s} is called the a-invariant of R (or of X). Thus aX + 1 is the last place where HFX (i ) = 0. Definition 2.1. Let 1 i s − 1 and 1 j aX . The scheme X is said to be (i , j )-uniform if every subscheme Y ⊆ X of degree s − i satisfies HFY ( j ) = HFX ( j ). For instance, the scheme X is (1, aX )-uniform if and only if it has the classical Cayley–Bacharach property. A non-degenerate scheme X is (s − n − 1, 1)-uniform if and only if it is in linearly general position. Furthermore, the scheme X is in uniform position if and only if it is (i , j )-uniform for all 1 j aX and all 1 i s − HFX ( j ). It is clear that an (i , j )-uniform scheme is also (i − 1, j )-uniform and (i , j − 1)-uniform. Given j, the maximal i such that X could be (i , j )-uniform is i = s − HFX ( j ). Thus the maximal region of uniformity of X is determined solely by the Hilbert function of X. It is illustrated by the dotted region in the following diagram. j .... .........
• aX
j
. .• . . . . . .
.
−. . . . . . . . . . . . . . . . . . •................
HFX ( j )
.................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .• . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
0
s−1 s
.................
i
If X is in uniform position, its region of uniformity is precisely this maximal region. However, for any specific X, the region of uniformity will usually be much smaller. The main topic of this paper is
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to examine the region of uniformity of zero-dimensional complete intersections. This topic is closely related to the Cayley–Bacharach conjectures of D. Eisenbud, M. Green, and J. Harris, as we shall see next. 3. The General Cayley–Bacharach Conjecture In [6], the authors formulate the following Conjecture CB12. Let Y ⊆ PnK be a subscheme of a zero-dimensional complete intersection of hypersurfaces of degrees d1 · · · dn . If Y fails to impose independent conditions on hypersurfaces of degree m, then we have deg(Y) e · dt · dt +1 · · · dn where t and e are defined by the relations n n n (di − 1) m + 1 < (di − 1) and e = m + 2 − (di − 1) i =t
i =t −1
i =t
(Notice that we have corrected the definition of e given in [6], which was an obvious misprint.) This conjecture generalizes their Conjecture CB11 which deals with the case d1 = · · · = dn = 2. By distinguishing the possible values of t, we can divide Conjecture CB12 into several intervals. Since we are going to address them separately, let us spell out the partial conjectures individually. Conjecture 3.1 (The General Cayley–Bacharach (GCB) Conjecture). Let X ⊆ PnK be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn . It is conjectured that the following claims hold true.
( I 1 ) Let 1 e < dn−1 and Y ⊆ X be such that HFY (e + dn − 2) = 0. Then we have deg(Y) e · dn . ( I 2 ) Let 1 e < dn−2 and Y ⊆ X be such that HFY (e + dn−1 + dn − 3) = 0. Then we have deg(Y) e · dn−1 · dn . .. . ( I n−1 ) Let 1 e < d1 and Y ⊆ X be such that HFY (e + d2 + · · · + dn − n) = 0. Then we have deg(Y) e · d2 · · · dn . To see that this conjecture is equivalent to Conjecture CB12, it suffices to note that HFY (m) < deg(Y) is equivalent to HFY (m + 1) = 0. The GCB Conjecture can be phrased as a conjecture about the region of uniformity of a zerodimensional complete intersection as follows. Conjecture 3.2 (The Uniformity Conjecture). Let X ⊆ PnK be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn . It is conjectured that the following claims hold true.
(U 1 ) Let 1 e < dn−1 . Then X is (edn − 1, aX − e − dn + 3)-uniform. (U 2 ) Let 1 e < dn−2 . Then X is (edn−1 dn − 1, aX − e − dn−1 − dn + 4)-uniform. ... (U n−1 ) Let 1 e < d1 . Then X is (ed2 · · · dn − 1, aX − e − d2 − · · · − dn + n + 1)-uniform. Here we have aX = d1 + · · · + dn − n − 1. To see that the Uniformity Conjecture is equivalent to the GCB Conjecture, we consider the ith interval and let Y ⊂ X be a subscheme of degree edn−i +1 · · · dn − 1. The GCB Conjecture says that we should have HFY (e + dn−i +1 + · · · + dn − i − 1) = 0. Let Y = X \ Y be the residual subscheme. Then we have
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HFY (aX + 1) − (e + dn−i +1 + · · · + dn − i − 1) = HFX (aX + 1) − (e + dn−i +1 + · · · + dn − i − 1) − HFY (e + dn−i +1 + · · · + dn − i − 1) = HFX (aX − e − dn−i +1 − · · · − dn + i + 2) Since this has to hold for all subschemes Y of X of degree s − edn−i +1 · · · dn + 1, we are in fact conjecturing that X is (edn−i +1 · · · dn − 1, aX − e − dn−i +1 − · · · − dn + i + 2)-uniform. Let us illustrate these conjectures with a couple of examples. In both cases the thick line marks the maximal possible region of uniformity and the dotted region corresponds to the conjectured region of uniformity. Example 3.3. Let X be a complete intersection of type (4, 7) in P2K . Then the Uniformity Conjecture says that the region of uniformity of X should contain the dotted region in the following picture. j .... ........
9 •...
3
.........•.. .................... •. ...........................•. ... ...................................•... ....................................•.. ...................................... ................................................................. •.. .................... ... ....................................•. ................................................................................................. ... ............................................. .... .........................•... ...................................................................................................................................... ..................•. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 6
13
20
25
.................
i
28
Example 3.4. Let X be a complete intersection of type (2, 3, 5) in P3K . Then the Uniformity Conjecture says that the region of uniformity of X should contain the dotted region in the following picture. j .... .........
7 •...
3
.........•.. ............................. •. .............................................•. ... .....................................................•... ........................................... ............. ... ......................................................•.. ..................................................................... ............................................... .................................. ............................................................................................... •.. . . . . . . . . . . . . . . . . . . . . . . . . .... ...........................•. 4
9
14
21
26
.................
i
30
Finally, we strengthen the GCB Conjecture and the Uniformity Conjecture by introducing the following Artinian version. Conjecture 3.5 (The Artinian Uniformity (AU) Conjecture). Let P = K [x1 , . . . , xn ] be standard graded, let ( f 1 , . . . , f n ) ∈ P n be a homogeneous regular sequence, let I = f 1 , . . . , f n be the ideal it generates, and let di = deg( f i ) for i = 1, . . . , n. Assume that we have d1 · · · dn , and let J ⊇ I be an ideal in P . It is conjectured that the following claims hold true.
( A 1 ) Let 1 e < dn−1 , and suppose that HF P / J (e + dn − 2) = 0. Then we have dim K ( P / J ) e · dn . ( A 2 ) Let 1 e < dn−2 , and suppose that HF P / J (e + dn−1 + dn − 3) = 0. Then we have dim K ( P / J ) e · dn−1 · dn . .. .
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( An−1 ) Let 1 e < d1 , and suppose that HF P / J (e + d2 + · · · + dn − n) = 0. Then we have dim K ( P / J ) e · d2 · · · dn . Let us verify that this version implies the Uniformity Conjecture. Since we are exclusively dealing with Hilbert functions, we may enlarge the base field as we please. Thus we shall assume that it is infinite and that (possibly after a suitable homogeneous linear change of coordinates) the indeterminate x0 is a non-zerodivisor for the homogeneous coordinate ring R = P / I X . The ring A = R /(x0 ) is called the Artinian reduction of R. It is of the form A = P / I with I ∼ = ( I X + (x0 ))/(x0 ). Its Hilbert function is HF A = HFX , the first difference function of HFX . If we apply the AU Conjecture to ideals of the form J ∼ = ( I Y + (x0 ))/(x0 ) in P , where Y is a subscheme of X, we obtain the Uniformity Conjecture. Since not every ideal in A is of the form ( I Y + (x0 ))/(x0 ) for some subscheme Y ⊆ X, the AU Conjecture is stronger than the GCB Conjecture and the Uniformity Conjecture. Therefore we shall study it in the following sections. It will also come in handy to consider the following dual version of the AU Conjecture. Conjecture 3.6 (The Artinian Dual (AD) Conjecture). In the setting of the AU Conjecture, it is conjectured that the following statements hold true for i = 1, . . . , n − 1.
( D i ) Let 1 e < dn−i and suppose that ( J / I )c = 0 for c = (d1 − 1) + · · · + (dn−i −1 − 1) + (dn−i − e ). Then we have dim K ( J / I ) e · dn−i +1 · · · dn . To show that the AD Conjecture is equivalent to the AU Conjecture, we argue as follows. The a-invariant of a graded zero-dimensional K -algebra is the largest degree in which its Hilbert function is not zero. It is well-known that a P / I = d1 + · · · + dn − n. Let us denote J / I by J , and let c = e + dn−i +1 + · · · + dn − i − 1. Then condition ( A i ) reads as follows:
( A i ) Let 1 e < dn−i , and suppose that HF A / J (c ) = 0. Then we have dim K ( A / J ) e · dn−i +1 · · · dn . The annihilator ideal J = Ann A ( J ) satisfies HFJ (i ) = HF A / J (a P / I − i ) for all i ∈ Z (see [4]). Hence
we have ( A / J )c = 0 if and only if J c˜ = 0 in degree c˜ = a P / I − c = (d1 + · · · + dn − n) − e − dn−i +1 − · · · − dn + i + 1 = (d1 − 1) + · · · − (dn−i −1 − 1) + (dn−i − e ). Furthermore, we have dim K ( A / J ) = dim K ( J ). Keeping this in mind, we see that ( A i ) is equivalent to the following condition.
( D i ) Let 1 e < dn−i , and suppose that J c˜ = 0. Then we have dim K (J ) e · dn−i +1 · · · dn . Finally, we note that the fact that the ring A is a 0-dimensional complete intersection implies J = Ann A ( J ), i.e. every ideal is the annihilator of some ideal. By changing ˜J into J in condition ( D i ), the claimed equivalence follows. 4. The first and the last interval In the following we prove the AU Conjecture in some special cases. We begin by showing that it holds true in the first interval ( A 1 ). A similar result was shown in [5] by entirely different means. In the following we let P = K [x1 , . . . , xn ] be standard graded, and we let A = P / I be a zerodimensional complete intersection ring of type (d1 , . . . , dn ), where d1 · · · dn . In other words, we assume that I is generated by a homogeneous regular sequence ( f 1 , . . . , f n ) in P whose degree sequence is (d1 , . . . , dn ). Proposition 4.1 (The first interval). Let A = P / I be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn , let 1 e dn−1 , and let J ⊇ I be a homogeneous ideal in P such that HF P / J (e + dn − 2) = 0. Then we have dim K ( P / J ) edn . In other words, the AU Conjecture is true in the interval A 1 and for e = 1 in the interval A 2 .
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Proof. First we show that HF P / J (dn − 1) e. By the hypothesis, this is clear for e = 1. So, let e 2 and assume that HF P / J (dn − 1) < e. Given any number i dn − 1 dn−1 − 1 e − 1 such that HF P / J (i ) e − 1 i, we apply Macaulay’s Growth Theorem (cf. [14, 5.5.28]). The binomial representation of m = HF P / J (i ) in base i is
m [i ] =
i i
+ ··· +
i −m+1 i −m+1
and therefore HF P / J (i + 1) (m[i ] )+ + = HF P / J (i ). Since we have
HF P / J (dn − 1) < e
and
HF P / J (dn − 1) + (e − 1) 1
there exists a number i ∈ {dn − 1, dn , . . . , e + dn − 3} for which HF P / J (i ) = HF P / J (i + 1). Thus we have a case of maximal growth in Macaulay’s Growth Theorem. By Gotzmann’s Persistence Theorem (cf. J = J i +1 satisfies HF P /J ( j ) = HF P /J (i ) for all j i. Therefore we [10] and [1, Thm. 3.6]), the ideal
get dim( P / J ) = 1, in contradiction to the fact that the residue classes of the elements of the regular sequence generating I are contained in J because of i + 1 dn . Next we show that HF P / J is greater than or equal to the function H given by H (i ) .... ........
..... ............................................................................................................• ..• ..... ..... ..... ..... ..... ..... ..... ..... . ..... . . . ..... ... . . . . ..... .... . ..... . . ..... ... . . . . ..... ... . . ..... . . ... ..... . . . . ..... .... . . ..... . .. . . . • •..... ... ... ..
e −
1
e −1
dn −1
e +dn −2
.................
i
This follows again from Macaulay’s Growth Theorem: we have already seen that HF P / J (i ) has to be strictly decreasing in the range dn − 1 i e + dn − 2. In the range 1 i e − 1, if ever HF P / J (i ) i, then Macaulay’s Growth Theorem implies HF P / J ( j ) i for all j i. But then HF P / J (dn − 1) i e − 1 yields a contradiction. It remains to consider the range e − 1 i dn − 1. Suppose that HF P / J (i ) < e for one of these i. Again Macaulay’s Growth Theorem shows HF P / J ( j ) < e for all j i, contradicting HF P / J (dn − 1) e. Altogether, we have shown dim K ( P / J ) = i 0 HF P / J (i ) i 0 H (i ) = edn . 2 Corollary 4.2. The AU Conjecture is true for n = 2. In particular, the GCB Conjecture and the Uniformity Conjecture hold true for complete intersections in P2K . Proof. For n = 2, there is only one interval A 1 to consider.
2
In those cases in which we are able to prove the GCB Conjecture we shall call it henceforth the GCB Theorem. Next we prove the AU Conjecture in the last interval ( A n−1 ). A different proof for this result is contained in [5]. For a homogeneous ideal I ⊆ P , we define the depth sequence (d1 , . . . , d ) of I by inductively choosing the minimal number di for which there exists a homogeneous regular sequence of type (d1 , . . . , di ) in I . Its length is the depth of I with respect to (x1 , . . . , xn ). Clearly, a homogeneous ideal J containing I has a depth sequence which is componentwise less than or equal to the depth sequence of I . Now we are ready to state and prove the desired result.
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Proposition 4.3 (The last interval). Let A = P / I be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn , let 1 e d1 − 1, and let J ⊇ I be a homogeneous ideal in P such that HF P / J (e + d2 + · · · + dn − n) = 0. Then we have dim K ( P / J ) ed2 · · · dn . In particular, the AU Conjecture holds true in the last interval. Proof. Let J = { f ∈ P | f · J ⊆ I } be the linked ideal. Then we have
HF P / J (d1 − e ) = HF P / I a P / I − (d1 − e ) − HF P / J a P / I − (d1 − e )
= HF P / I (e + d2 + · · · + dn − n) − HF P / J (e + d2 + · · · + dn − n) < HF P / I (e + d2 + · · · + dn − n) Thus we see that J d −e = 0, and hence the depth sequence of J is componentwise less than or 1 equal to (d1 − e , d2 , . . . , dn ). In particular, the ideal J contains a zero-dimensional complete intersection ideal of type (d1 − e , d2 , . . . , dn ). This implies dim K ( P / J ) (d1 − e )d2 · · · dn , and therefore dim K ( P / J ) ed2 · · · dn . 2 Corollary 4.4. The AU Conjecture is true for n = 3. In particular, the GCB Conjecture and the Uniformity Conjecture hold true for complete intersections in P3K . Proof. For n = 3, there are two intervals ( A 1 ) and ( A 2 ) to consider. The first interval is dealt with by Proposition 4.1, and the second interval by the preceding proposition. 2 Corollary 4.5. The AU Conjecture is true for n = 4 and d1 = d2 = 2. In particular, the GCB Conjecture and the Uniformity Conjecture hold true for complete intersections of type (2, 2, d3 , d4 ) in P4K , where 2 d3 d4 . Proof. For n = 4, there are three intervals ( A 1 ), ( A 2 ), and ( A 3 ) to consider. The first interval is dealt with by Proposition 4.1, the second interval consists only of the case e = 1 which is treated in the same proposition, and in the third interval the conjectures hold by the preceding proposition. 2 5. The penultimate interval In this section we want to study the AU Conjecture in the interval ( A n−2 ). We begin with a couple of basic simplifications. Proposition 5.1 (Reduction to the Gorenstein case). Suppose that we want to prove one case of the AU Conjecture, i.e. we want to prove it for a specific ( A i ) and a fixed value of e ∈ {1, . . . , dn−i − 1}. Then we may assume that P / J is a Gorenstein ring. Proof. By the hypothesis, the largest degree j for which we have ( P / J ) j = 0 satisfies the inequality j e + dn−i +1 + · · · + dn − i − 1 =: e˜ . By replacing J by J + P e˜ +1 , we may assume that j = e + dn−i +1 + · · · + dn − i − 1. Now suppose that dim K ( P / J ) j 2. Let f be a homogeneous polynomial of degree j whose residue class is a non-zero element of the socle of P / J , and let J = J + f . Then J continues to satisfy the hypothesis of the AU Conjecture and dim K ( P / J ) j = dim K ( P / J ) j − 1. By repeating this process a number of times, we find an ideal J ⊇ J for which the largest degree j with ( P / J ) j = 0 is j = e + dn−i +1 + · · · + dn − i − 1 and for which dim K ( P / J ) j = 1. If P / J is not a Gorenstein ring, there exists a degree k < j such that P / J contains non-zero socle elements of degree k. Again we let f ∈ P k be a polynomial whose residue class is such a socle element, and we set J = J + f . Then P / J satisfies the hypotheses of the AU Conjecture and dim K ( P / J ) = dim K ( P / J ) − 1. By repeating this process, we arrive at an ideal J such that P / J has
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a 1-dimensional socle, and this socle is its homogeneous component of (largest) degree e + dn−i +1 + · · · + dn − i − 1. Since dim K ( P / J ) is less than or equal to dim K ( P / J ), it suffices to prove the AU Conjecture for J . 2 In view of this proposition, we may assume that the Hilbert function of P / J is symmetric and that J = I : g with a homogeneous polynomial g ∈ P γ of degree γ = d1 + · · · + dn−i − e − n + i + 1. By using liaison as in the proof of Proposition 4.3, we see that the conjecture is equivalent to the following “dual” version. Lemma 5.2. Assume that we are in the situation of the interval ( A n−2 ) in the AU Conjecture, i.e. that 1 e < d2 , and let g ∈ P d1 +d2 −e−1 \ I d1 +d2 −e−1 . Then the following statements are equivalent. (a) The ideal J = I : g satisfies dim K ( P / J ) ed3 · · · dn . J = I + g satisfies dim K ( P / J ) (d1 d2 − e )d3 · · · dn . (b) The ideal J are related Proof. This follows immediately from the fact that the Hilbert functions of P / J and P / by HF P /J (i ) = HF P / I (a P / I − i ) − HF P / J (a P / I − i ) for all i 0 since the two ideals are linked. 2 Now we are ready to prove the main result of this section. Proposition 5.3 (The second part of the penultimate interval). Let A = P / I be a zero-dimensional complete intersection of type (d1 , . . . , dn ), where d1 · · · dn , let d1 e d2 − 1, and let J ⊇ I be a homogeneous ideal in P such that HF P / J (e + d3 + · · · + dn − n + 1) = 0. Then we have dim K ( P / J ) ed3 · · · dn . In particular, the AU Conjecture is true in the penultimate interval ( A n−2 ) for d1 e < d2 . Proof. By Proposition 5.1, we may assume that P / J is a Gorenstein ring with a P / J = e + d3 + · · · + dn − n + 1 and that J = I : g for some homogeneous polynomial g of degree γ = d1 + d2 − e − 1. Suppose the claim of the proposition is not true. Let ( I , J ) be a counterexample for which d1 is minimal. Let us show first that gcd( f 1 , g ) = 1. For a contradiction, assume that f 1 = h1 h2 and g = h2 g˜ for some polynomials h1 , h2 , g˜ with deg(h2 ) 1. Then h1 g = f 1 g˜ ∈ I implies h1 ∈ J . Hence the ideal I = h1 , f 2 , . . . , f n has initial degree α ( I ) < α ( I ). It is a complete intersection, since ph1 ∈ f 2 , . . . , f n
with p ∈ P implies ph1 h2 = p f 1 ∈ f 2 , . . . , f n , and thus p ∈ f 2 , . . . , f n . But I ⊆ J , and we still have HF P / J (e + d3 + · · · + dn − n + 1) = 0 and deg(h1 ) < e d2 − 1. Since deg(h1 ) is smaller than d1 , the pair ( I , J ) is not a counterexample to the claim of the proposition, i.e. we have dim K ( P / J ) e d3 · · · dn . This contradicts the assumption that ( I , J ) is a counterexample, and thus the claim gcd( f 1 , g ) = 1 is proved. Next we observe that d1 γ d2 − 1. The ideal J = I : J = I + g contains both the regular sequence ( f 1 , f 2 , . . . , f n ) and the regular sequence ( f 1 , g ). Thus its depth sequence is componentwise less than or equal to (d1 , γ , d3 , . . . , dn ), and hence we have dim K ( P / J ) d1 γ d3 · · · dn . Now e d1 implies d21 − d1 ed1 − e, and therefore d1 γ = d1 (d1 + d2 − e − 1) d1 d2 − e. Altogether, we find
dim K ( P / J ) d1 γ d3 · · · dn (d1 d2 − e )d3 · · · dn and the lemma implies that ( I , J ) is not a counterexample to the claim of the proposition. This contradicts our assumption and finishes the proof. 2 6. An application to coding theory In [12], J. Hansen proved a uniformity result for reduced zero-dimensional complete intersection in P2K and used it to derive some bounds for the minimal distances of certain linear codes. Later, in [9], his result was generalized to reduced zero-dimensional complete intersections in PnK . Below
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we shall see that the region of uniformity predicted by the GCB Conjecture is usually much bigger than what was shown in these papers. In fact, our next result points out that the weaker uniformities shown there hold in a much more general setting, namely for level schemes. Here we say that a zero-dimensional scheme X ⊆ PnK is level if the Artinian reduction R of its homogeneous coordinate ring R satisfies socle( R ) = R aX +1 . Proposition 6.1. Let X ⊆ PnK be a level scheme and s = deg(X). Then the following statements hold true. (a) The scheme X is (1, aX )-uniform, i.e. it has the Cayley–Bacharach property. (b) If X is (i , j )-uniform for some 1 i s and some 1 j aX , then X is (i + 1, j − 1)-uniform. (c) For i = 1, . . . , aX , the scheme X is (i , aX + 1 − i )-uniform. Proof. First we show (a). Let f Y ∈ R be a minimal separator of a subscheme Y ⊂ X of degree deg(Y) = deg(X) − 1. Then we have ¯f Y ∈ socle( R ), and therefore deg( f Y ) = aX + 1. Hence X has the Cayley–Bacharach property. To prove (b), we let Y ⊆ X be a subscheme of degree s − i. Let J be the ideal of Y in R. Then HFY ( j ) = HFX ( j ) implies J j = 0. Now consider a subscheme Y ⊂ Y of degree s − i − 1, and let J be the ideal of Y in R. We have dim K ( J ) = i and dim K ( J ) = i + 1, and we distinguish two cases. The first case is ( J ) j −1 = 0. Then we also have ( J ) j = 0 because otherwise ( J ) j −1 ⊆ socle( R ) contradicts j − 1 < aX + 1 and the property of R being a level algebra. Now dim K ( J ) dim K ( J ) j −1 + dim K ( J ) j + dim K ( J ) 2 + dim K ( J ) yields a contradiction. Hence we are left with the case ( J ) j −1 = 0. In this case we get HFY ( j − 1) = HFX ( j − 1). Since every subscheme of X of degree s − i − 1 arises in this way, it follows that X is (i + 1, j − 1)-uniform. Finally, we note that claim (c) is an immediate consequence of (a) and (b). 2 Using the main result of [11], we can immediately translate this proposition into estimates for the minimal distance of certain linear codes. Given a finite field K , a reduced zero-dimensional subscheme X = { p 1 , . . . , p s } ⊆ PnK , and a number j ∈ {1, . . . , aX }, the image of the map
Φ j : R j −→ K s defined by Φ j ( f ) = ( f ( p 1 ), . . . , f ( p s )) is called the jth generalized Reed–Muller code C j (X) associated to X. Corollary 6.2. Let K be a finite field, and let X ⊆ PnK be a reduced level scheme. Then the jth generalized Reed–Muller code C j (X) associated to X is an [s, k, d]-code where k = HFX ( j ) and d aX + 2 − j. Proof. By [11, Thm. 8], we have d = 1 + max{i | X is (i , j )-uniform}. Thus the claim follows from part (c) of the proposition. 2 We end this paper with three examples which show that, in the case of a reduced zerodimensional complete intersection, the uniformity (and therefore the minimal distance bounds) coming from the GCB Theorem usually exceeds the uniformity shown in the preceding proposition for the relevant numbers j. In each picture the region of uniformity provided by Proposition 6.1 is dotted and the additional region of uniformity coming from the GCB Theorem is marked by asterisks. Example 6.3. Let X be a zero-dimensional complete intersection of type (5, 5) in P2K .
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j . ..........
8 •...
3
..........•.. ..... . ..........................• ...... .............. ... ......................................• ....................................•.. .................... ............. ... . ............................................................... .............................................• ... ....................∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗... ...∗..∗∗∗∗∗∗∗∗∗∗∗.............................................. ....................................• .. ..................∗.....∗ .. .....................∗....∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ........................................................................... ................................∗ •.. ..∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗...∗ ..∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ..... ... .. ..∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗.. . .................•. 3
9
19
14
.................
i
25
Notice that Corollary 4.2 implies that X satisfies the Uniformity Conjecture, i.e. it has at least the marked region of uniformity. Example 6.4. Let X be a zero-dimensional complete intersection of type (3, 3, 4) in P3K . j .... ........
7 •...
3
...........•.. ....... . .. ..................................• ...... .......... ....................................................... •.. ............. . .......................................................................•.. .................................................. . ............. ...∗∗∗∗∗∗∗∗∗∗∗ ......................................................................... . . ...................................... ...................∗........∗∗∗∗∗∗∗∗ •.. ....................∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗.... ......................................................•.. ...........................∗...∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗..... ............................. ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ .. . . . . . . . . . . ∗...∗∗∗∗∗∗∗∗∗∗∗∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ . • 6
18
11
23
26
32
.................
i
36
Notice that Corollary 4.4 implies that X satisfies the Uniformity Conjecture, i.e. it has at least the marked region of uniformity. Example 6.5. Let X be a zero-dimensional complete intersection of type (2, 2, 3, 3) in P4K . j ..... ........
6 •... 3
...........•.. ............................................. •.. ...... ... ...... .......................................................................................................• ... .............∗∗∗∗∗∗∗...∗ .........................................................................................•.. ...∗∗∗.∗........................... ..............∗.....∗ ......................................................................... ............. ...∗∗∗∗∗∗∗∗∗∗∗∗∗∗....∗ .....∗ .....∗ .....∗ .....∗ .....∗ .....∗ .....∗ ......∗.....∗.....∗.....∗.....∗.....∗.....∗.....∗......∗ ............................∗∗∗∗∗∗∗∗∗∗∗∗ •.. . . . . . . . . ....∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗....∗ ....................................•. 5
13
17
23
31
.................
i
36
Notice that Corollary 4.5 implies that X satisfies the Uniformity Conjecture, i.e. it has at least the marked region of uniformity. Acknowledgments The second author thanks the Department of Mathematics and Statistics at Queen’s University (Kingston, Canada) for its kind hospitality during part of the preparation of this paper. This research was supported by NSERC. Both authors are extremely grateful to the anonymous referee for such a very careful reading of our paper. References [1] A. Bigatti, A.V. Geramita, J.C. Migliore, Geometric consequences of extremal behaviour in a theorem of Macaulay, Trans. Amer. Math. Soc. 346 (1994) 203–235. [2] G. Caviglia, D. Maclagan, Some cases of the Eisenbud–Green–Harris conjecture, Math. Res. Lett. 15 (2008) 427–433. [3] S. Cooper, Growth conditions for a family of ideals containing regular sequences, J. Pure Appl. Algebra 212 (2007) 122–131. [4] E. Davis, A.V. Geramita, F. Orecchia, Gorenstein algebras and the Cayley–Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985) 593–597. [5] A.A. du Plessis, C.T.C. Wall, Singular hypersurfaces, versality, and Gorenstein algebras, J. Algebraic Geom. 9 (2000) 309–322.
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