Contributions to Algebra and Geometry Volume 47 (2006), No. 1, 29-51.

### On the Depth and Regularity of the Symmetric Algebra

J¨urgen Herzog Gaetana Restuccia Giancarlo Rinaldo
*FB6 Mathematik und Informatik, Universit¨at Duisburg-Essen*

*45117 Essen, Germany*
*e-mail: juergen.herzog@uni-essen.de*

*Dipartimento di Matematica, Universit`a di Messina*
*Contrada Papardo, salita Sperone, 31, 98166 Messina, Italy*
*e-mail: grest@dipmat.unime.it* *e-mail: rinaldo@dipmat.unime.it*

Abstract. Let (R,m) be a standard graded K-algebra whose defining ideal is componentwise linear. Using Gr¨obner basis techniques, bounds for the depth and the regularity of the symmetric algebra Sym(m) are given.

MSC 2000: 13H10, 13P10, 13D02

Introduction

LetRbe a standard gradedK-algebra with graded maximal idealm= (x_{1}, . . ., x_{n}).

The algebra R can be written as S/I where I ⊂ m^{2} is a graded ideal in the
polynomial ring S = K[x_{1}, . . . , x_{n}]. In this paper we want to study the depth
and the regularity of the symmetric algebra Sym(m) of the ideal m. Depth and
regularity have been extensively investigated [10] for the Rees algebra of m, while
for Sym(m) only partial results and estimates for the depth are known, see [9].

As a technique to study the symmetric algebra we use Gr¨obner bases: let <

be any term order on S, and let R^{∗} =S/in(I) and m^{∗} the graded maximal ideal
ofR^{∗}, where in(I) denotes the initial ideal of I. In Section 1 we compare Sym(m)
and Sym(m^{∗}). Denote byn the graded maximal ideal of S. For an elementf ∈S
and a graded ideal L⊂S we set

v_{L}(f) = max{j:f ∈n^{j}L},
0138-4821/93 $ 2.50 c 2006 Heldermann Verlag

and show that

reg Sym(m)≤reg Sym(m^{∗}) and depth Sym(m^{∗})≤depth Sym(m),
if

v_{in(I)}(in(f))≥v_{I}(f) for all f ∈I.

Provided K is a field of characteristic 0, we show in Proposition 1.8 that this last condition is satisfied for the reverse lexicographical term order in generic coordinates, if I is componentwise linear in the sense of [7].

Thus in order to obtain upper bounds for the regularity and lower bounds for
the depth of the symmetric algebra of the graded maximal ideal of a standard
graded algebra whose defining ideal is componentwise linear, it suffices to study
standard graded K-algebras with monomial relations. For such algebras we use
the theory of s-sequence which was introduced in [8]. Recall that Sym(m) can
be written as P/J where P = R[y_{1}, . . . , y_{n}] and J ⊂ P is generated by the
polynomialsg =Pn

i aiyi withPn

i aixi = 0. The sequence x1, . . . , xn is said to be
an s-sequence if for some term order < on the monomials in y_{1}, . . . , y_{n} which is
induced by y_{1} < y_{2} <· · ·< y_{n}, the initial ideal of J is generated by terms which
are linear in the yi.

For the computation of in(J) we cannot use the standard techniques of Gr¨ob-
ner basis theory because our base ringR is not a field. To overcome this problem
we show in Section 2 that in caseI is a monomial ideal, in(J) can be computed as
follows: write Sym(m) as S[y_{1}, . . . , y_{n}]/(I, J_{0}), and determine the initial ideal of
(I, J_{0}) with respect to a suitable term order which extends the given term order
in theyi, and is induced byx1 < x2 <· · ·< xn < y1 < y2 <· · ·< yn. This initial
ideal is of the form (I, L_{0}), and in(J) is the image of L_{0} modulo I.

With this method we characterize in Theorem 2.2 those monomial ideals for which x1, . . . , xn is an s-sequence in R. These ideals include the stable ideals.

We apply these results in Section 3 to compute the depth and the regularity of the symmetric algebra Sym(m) in caseI is strongly stable in the reverse order.

Letu be a monomial. We denote by m(u) the smallest integer i for which xi

divides u. The main results are Theorem 3.7:

regR≤reg Sym(m)≤regR+ 1, and reg Sym(m) = regR ⇐⇒ max{m(u)} ≤2, where the maximum is taken over all Borel generators u of I of maximal degree, and Theorem 3.9:

depth Sym(m) = 0, if depthR = 0, and

depth Sym(m) = depthR+ 1, if depthR >0.

Assuming charK = 0, the generic initial ideal Gin(I) of I with respect to the
reverse lexicographical induced by x_{1} < x_{1} < · · · < x_{n} is strongly stable in the
reverse order. Thus if the defining ideal of the standard graded algebra R is
componentwise linear the results of Section 1 and Section 3 imply:

reg Sym(m)≤regR+ 1, and depth Sym(m)≥depthR+ 1 if depthR >0.

1. Symmetric algebras and initial ideals

In this section we recall some basic facts about s-sequences, and discuss the sym- metric algebra of an initial ideal.

Let R be a Noetherian ring and M an R-module generated by f_{1}, . . . , f_{n}.
Then M has a presentation

R^{m} →R^{n}−→M −→0
with relation matrix and A= (a_{ij})^{i=1,... ,m}

j=1,... ,n.

The symmetric algebra Sym(M) has the presentation R[y1, . . . , yn]/J,

where J = (g_{1}, . . . , g_{m}) and g_{i} =Pn

j=1a_{ij}y_{j} with i= 1, . . . , m.

We consider P =R[y_{1}, . . . , y_{n}] a graded R-algebra assigning to each variable
yi the degree 1 and to the elements of R the degree 0. Then J is a graded ideal,
and Sym(M) a graded R-algebra.

Let <a monomial order induced by y_{1} <· · ·< y_{n}. For f ∈P, f =P

αa_{α}y^{α}
we put in(f) = aαy^{α} where y^{α} is the largest monomial with respect to the given
order such that a_{α} 6= 0. We call in(f) the initial term of f. Note that in contrast
to ordinary Gr¨obner basis theory the base ring of our polynomial ringP is not a
field. Nevertheless we may define the ideal

in(J) = (in(f) :f ∈J).

This ideal is generated by terms which are monomials in y_{1}, . . . , y_{n} with coeffi-
cients in R, and is finitely generated since P is Noetherian.

For i = 1, . . . , n we set M_{i} = Pi

j=1Rf_{j}, and let I_{i} = M_{i−1} :_{R} f_{i} = {a ∈
R:af_{i} ∈Mi−1}. We also set I_{0} = 0. Note that I_{i} is the annihilator of the cyclic
module M_{i}/Mi−1 ∼=R/I_{i}.

It is clear that

(I_{1}y_{1}, . . . , I_{n}y_{n})⊆in(J),
and the two ideals coincide in degree 1.

Definition 1.1. *The generators* f_{1}, . . . , f_{n} *of* M *are called an* s-sequence (with
*respect to* <), if

(I_{1}y_{1}, . . . , I_{n}y_{n}) = in(J).

*If in addition* I1 ⊂I2 ⊂ · · · ⊂In*, then* f1, . . . , fn *is called a strong* s-sequence.

Since Sym(m) = P/J may be viewed as the general fiber of a 1-parameter flat family whose special fiber isP/in(J), invariants of Sym(m) =S/J compared with the corresponding invariants of P/in(J) can only be better. Thus, for example, if x1, . . . , xn is a strong s-sequence one has

depth Sym(m) ≥ depthR[y_{1}, . . . , y_{n}]/(I_{1}y_{1}, . . . , I_{n}y_{n})

≥ min{depthR/Ii+i: i= 0,1, . . . , n},

see [8, Proposition 2.5].

In the same spirit we may do the following comparison: let I ⊂ S be a graded
ideal, and m the graded maximal ideal of R = S/I. Let furthermore < be a
term order on S, in(I) the initial ideal of I, R^{∗} = S/in(I), and m^{∗} the graded
maximal ideal of R^{∗}. How are the invariants of Sym(m) and Sym(m^{∗}) related to
each other? At least for the depth there seems not to be no obvious relationship
as the following examples demonstrate.

Example 1.2. Let < be the lexicographical order induced by y_{3} > y_{2} > y_{1} >

x3 > x2 > x1, and let I = (x1x3−x^{2}_{2}, x1x2−x^{2}_{1}). Then in(I) = (x1x2, x1x3, x^{3}_{2}),
depth Sym(m) = 1, and depth Sym(m^{∗}) = 2.

On the other hand, letI= (x_{2}x_{3}−x^{2}_{1}, x_{2}x_{3}−x^{2}_{3}). Then in(I) = (x^{2}_{3}, x_{2}x_{3}, x^{2}_{1}x_{3}, x^{2}_{1}x^{2}_{2}),
depth Sym(m) = 1, and depth Sym(m^{∗}) = 0.

These examples show that we need some extra hypotheses. Let L ⊂ S be any graded ideal and f ∈L, f 6= 0. We set

v_{L}(f) = max{j:f ∈n^{j}L}.

For systematic reasons we set v_{L}(f) = ∞, iff = 0.

Let R(n) = L

jn^{j}t^{j} be the Rees ring of the graded maximal ideal n of S.

Then

R(n) =S[x_{1}t, . . . , x_{n}t]⊂S[t].

The function v_{L}: L→Z has the following interpretation:

Lemma 1.3. *Let* f_{1}, . . . , f_{m} *be a homogeneous system of generators of* L, and
*consider the ideal* C = (L, Lt) = (f_{1}, . . . , f_{m}, f_{1}t, . . . , f_{m}t) *in* R(n). Let f ∈L *be*
*a homogeneous element and* a∈Z*,* a ≥0. Then

f t^{a}∈C ⇐⇒ a≤v_{L}(f) + 1.

*Proof.* Suppose that a ≤ v_{L}(f) + 1. We may assume that f 6= 0 and a > 0.

Otherwise it is trivial that f t^{a} ∈ C. Let j = vL(f). Then f ∈ n^{j}L. Hence
f =Pm

i=1g_{i}f_{i}with allg_{i} homogeneous of degree≥j. Fora∈Zwith 0< a≤j+1
we writef t^{a} =P

i=1,... ,mg_{i}t^{a−1}f_{i}t. Note that ifg ∈S is homogeneous of degreek,
then gt^{a} ∈R(n) if and only ifa ≤k. Therefore g_{i}t^{a−1} ∈R(n) for all i, and hence
f t^{a}∈C.

Conversely suppose that f t^{a} ∈ C. We note that R(n) is bigraded, C is a
bigraded ideal, and f t^{a} is bihomogeneous, if we assign the following bidegrees to
the generators of R(n):

degx_{i} = (1,0) and degx_{i}t= (0,1) for all i= 1, . . . , n.

Thus we can writef t^{a} as a linear combination
f t^{a}=

n

X

i=1

g_{i}t^{a}f_{i}+

n

X

i=1

h_{i}t^{a−1}f_{i}t,

of the generators of C with bihomogeneous coefficients g_{i}t^{a}, h_{i}t^{a−1} ∈ R(n). It
follows that deggi ≥ a and deghi ≥ a− 1. Here we use the convention that
the zero-polynomial has degree ∞. Assuming that a > j + 1, we have f =
Pn

i=1(g_{i} +h_{i})f_{i} with g_{i}+h_{i} ∈n^{j+1}, a contradiction.

With the introduced notation we have
Theorem 1.4. *Suppose that*

v_{in(I)}(in(f))≥v_{I}(f) *for all* f ∈I.

*Then*

reg Sym(m)≤reg Sym(m^{∗}), *and* depth Sym(m^{∗})≤depth Sym(m).

This theorem is again a consequence of the fact that under the given hypotheses,
Sym(m) may be viewed as the general fiber of a 1-parameter flat family whose
special fiber is Sym(m^{∗}). Indeed, write

Sym(m) = R[y1, . . . , yn]/J and Sym(m^{∗}) =R^{∗}[y1, . . . , yn]/J^{∗}.
Letf_{1}, . . . , f_{m} be a set of generators ofI. Writef_{i} =Pn

j=1f_{ij}x_{j} fori= 1, . . . , m,
and set

J_{0} = ({

n

X

j=1

f_{ij}y_{j}}_{i=1,... ,m}∪ {x_{i}y_{j} −x_{j}y_{i}}1≤i<j≤n).

Similarly, we define J_{0}^{∗}. Then J =J_{0}modI, and J^{∗} =J_{0}^{∗}mod in(I). Hence
Sym(m) =S[y_{1}, . . . , y_{n}]/(I, J_{0}) and Sym(m^{∗}) =S[y_{1}, . . . , y_{n}]/(in(I), J_{0}^{∗}).

Let (M, <) be the totally ordered set of monomials ofS =K[x1, . . . , xn] where<

is the given monomial order. We define a degree-function d:S[y_{1}, . . . , y_{n}]→ M:

letf ∈S[y_{1}, . . . , y_{n}],f =P

ν,µc_{ν,µ}x^{ν}y^{µ}. Then

d(f) = max{x^{ν+µ}: c_{ν,µ} 6= 0},
and we call

in_{d}(f) = X

ν,µ xν+µ=d(f)

c_{ν,µ}x^{ν}y^{µ}.

the *initial polynomial* of f (with respect tod).

This function satisfies the following conditions: for allf, g∈S[y1, . . . , yn] one has (a) d(f+g)≤max{d(f), d(g)} and d(f+g) = max{d(f), d(g)} if d(f)6=d(g);

(b) d(f g) = d(f)d(g).

Let L ⊂ S[y_{1}, . . . , y_{n}] be an ideal. Let in_{d}(L) denote the ideal in S[y_{1}, . . . , y_{n}]
generated by all initial polynomials in_{d}(f) with f ∈L.

Recall the following concept: given a linear function ω : Z^{m} → Z we define
the *weight* of a term u = λx^{a} in K[x_{1}, . . . , x_{m}] to be ω(a). Different terms
may have the same weight. Nevertheless we may define for any polynomial f ∈
K[x_{1}, . . . , x_{m}] the *initial polynomial*in_{ω}(f) of f with respect to ω to be the sum
of all terms inf which have maximal weight, and we denote byω(f) this maximal
weight. Finally, if L⊂K[x_{1}, . . . , x_{m}] is an ideal one sets

inω(L) = ({inω(f) : f ∈I}).

We shall need the following result:

Lemma 1.5. *For any ideal* L ⊂ S[y_{1}, . . . , y_{n}] = K[x_{1}, . . . , x_{n}, y_{1}, . . . , y_{n}] *there*
*exists a weight function* ω: Z^{2n} →Z *such that* in_{d}(L) = in_{ω}(L).

*Proof.* Let f_{1}, . . . , f_{m} ∈ L. Just as in ordinary Gr¨obner basis theory one has
the following criterion: we consider the relations of in_{d}(f_{1}), . . . ,in_{d}(f_{m}), i.e. the
m-tupels r = (r1, . . . , rm) with ri ∈ S[y1, . . . , ym] such that Pm

i=1riind(fi) = 0.

LetR be a generating set of relations of in_{d}(f_{1}), . . . ,in_{d}(f_{m}). Then the following
conditions are equivalent:

(a) the initial polynomials in_{d}(f_{1}), . . . ,in_{d}(f_{m}) generate in_{d}(L);

(b) for eachr ∈ R, the polynomialf =Pm

i=1r_{i}f_{i} can be rewritten as
g_{1}f_{1}+g_{2}f_{2}+· · ·+g_{m}f_{m},

such that d(f)≥d(g_{i}f_{i}) fori= 1, . . . , m.

The same criterion holds if we replace everywhere d byω.

Suppose now that in_{d}(f_{1}), . . . ,in_{d}(f_{m}) generate in_{d}(L), and that we can find
a weight function ω such that

(i) in_{ω}(f_{i}) = in_{d}(f_{i}) for i= 1, . . . , m;

(ii) ω(f)≥ω(g_{i}f_{i}) for i= 1, . . . , m for all (the finitely many) equations in (b).

Then the above criterion and (ii) imply that the polynomials in_{ω}(f_{1}), . . . ,in_{ω}(f_{m})
generate in_{ω}(L). Therefore (i) yields in_{d}(L) = in_{ω}(L).

Now we show how we can chooseω such that (i) and (ii) are satisfied. Given
a polynomial h ∈ S[y_{1}, . . . , y_{n}], let h_{i}, i = 1,2, . . . be the terms in h such that
d(h)> d(h_{i}). Then we define the (finite) setP_{h} ={(d(h), d(h_{i})) :i= 1,2, . . .} of
pairs of monomials inS.

Now we consider the finite set of pairs of monomials Sm

i=1P_{f}_{i} in S, and add
to this union the sets of pairsP_{f}∪Sm

i=1P_{g}_{i}_{f}_{i} as well as all the pairs (d(f), d(g_{i}f_{i}))
(i= 1, . . . , m) which correspond to the finitely many relations in R. Altogether
this is a finite set of pairs of monomials (u, v) inS withu > v for each pair. Then
[5, Proposition 15.16] asserts that there exists a weight function ω_{0}: Z^{n}→Zsuch
that for each of the pairs (u, v) above, we have ω0(u)> ω0(v).

The weight function ω: Z^{2n} →Z we are looking for is defined as follows: for
(ν, µ)∈Z^{2n} with ν, µ ∈Z^{n} we set

ω(ν, µ) =ω0(ν+µ).

We note that for a monomialu=x^{ν}y^{µ} we haveω(u) =ω_{0}(d(u)). Thus our choice
of ω guarantees that the conditions (i) and (ii) are satisfied.

In the proof of the next lemma we need the following notation: for a monomial
u in S we set m(u) = inf{i : x_{i} divides u}, and u^{0} = u/x_{m(u)}. In particular,
u=u^{0}x_{m} wherem =m(u).

The following crucial lemma together with the previous lemma will imply Theo- rem 1.4.

Lemma 1.6. (in(I), J_{0}^{∗})⊂in_{d}(I, J_{0}), and equality holds if
v_{in(I)}(in(f))≥v_{I}(f) *for all* f ∈I.

*Proof.* Letf_{1}, . . . , f_{m}be a Gr¨obner basis ofI, and letu_{i} = in(f_{i}) fori= 1, . . . , k.

Then

(in(I), J_{0}^{∗}) = (u_{1}, . . . , u_{k}, u^{0}_{1}y_{m}_{1}, . . . , u^{0}_{k}y_{m}_{k},{x_{i}y_{j} −x_{j}y_{i}}1≤i<j≤n)
where m_{i} =m(u_{i}) for i= 1, . . . , k.

On the other hand, write f_{i} =Pn

j=1f_{ij}x_{j} for i= 1, . . . , k. Then
(I, J_{0}) = (f_{1}, . . . , f_{k},

n

X

j=1

f_{1j}y_{j}, . . . ,

n

X

j=1

f_{kj}y_{j},{x_{i}y_{j}−x_{j}y_{i}}1≤i<j≤n).

It is clear that ui ∈ ind(I, J0) since ind(fi) = in<(fi) = ui. We also have xiyj −
x_{j}y_{i} ∈in_{d}(I, J_{0}) for all iand j since in_{d}(x_{i}y_{j} −x_{j}y_{i}) =x_{i}y_{j}−x_{j}y_{i}.

For each i the presentation f_{i} = Pn

j=1f_{ij}x_{j} may be chosen such that each
monomial appearing in fi appears in exactly one of the summands fijxj. Then
if the leading term u_{i} of f_{i} appears in the summand f_{ij}x_{j}, then (u_{i}/x_{j})y_{j} =
in_{d}(Pn

`=1f_{i`}y_{`}). Thus (u_{i}/x_{j})y_{j} ∈ in_{d}(I, J_{0}). However since x_{m}_{i}y_{j} −x_{j}y_{m}_{i} ∈
ind(I, J0), we also have u^{0}_{i}ymi ∈ind(I, J0). This shows that

(in(I), J_{0}^{∗})⊂in_{d}(I, J_{0}).

We suppose now that v_{in(I)}(in(f))≥ v_{I}(f) for all f ∈I. Note first that the ideal
B = ({x_{i}y_{j} −x_{j}y_{i}}_{1≤i<j≤n}) is contained in the ideal (in(I), J_{0}^{∗}) as well as in the
ideal in_{d}(I, J_{0}). Thus in order to show that in_{d}(I, J_{0}) ⊂ (in(I), J_{0}^{∗}) it suffices to
show that in_{d}(I, J_{0})/B ⊂(in(I), J_{0}^{∗})/B.

Letϕ: S[y_{1}, . . . , y_{n}]→R(n) be the epimorphism given by
ϕ(y_{i}) = x_{i}t for i= 1, . . . , n.

It is known and easy to see thatB= Kerϕ. Hence we haveR(n)∼=S[y_{1}, . . . , y_{n}]/B.

Therefore, iff_{1}, . . . , f_{m} is a reduced Gr¨obner basis of I, then

(in(I), J_{0}^{∗})/B =ϕ(in(I), J_{0}^{∗}) = (in(f_{1}), . . . ,in(f_{m}),in(f_{1})t, . . . ,in(f_{m})t),
and

(I, J_{0})/B =ϕ(I, J_{0}) = (f_{1}, . . . , f_{m}, f_{1}t, . . . f_{m}t).

Now let f ∈ (I, J_{0}). We want to prove that ϕ(in_{d}(f)) ∈ ϕ(in(I), J_{0}^{∗}). Since
(I, J_{0}) is bigraded, we may assume thatf is bihomogeneous of bidegree (b, a). Set

|α|=Pn

i α_{i} forα= (α_{1}, . . . , α_{n}). Thenf =P

ν,µc_{νµ}x^{ν}y^{µ}with|ν|=band|µ|=a
for all ν and µ in the sum, and ϕ(f) = gt^{a} where g =P

ν,µc_{νµ}x^{ν+µ} belongs to I
and is of degree a+b. It follows that eitherϕ(in_{d}(f)) = 0 or ϕ(in_{d}(f)) = in(g)t^{a}.
In the first case there is nothing to prove. In the second case, note first that by
Lemma 1.3,a≤v_{I}(g)+1, sincegt^{a}=ϕ(f)∈ϕ(I, J_{0}) = (f_{1}, . . . , f_{m}, f_{1}t, . . . , f_{m}t).

Since by assumption v_{in(I)}(in(g)) ≥ v_{I}(g), we obtain that a ≤ v_{in(I)}(in(g)) + 1.

Again applying Lemma 1.3 we conclude that

ϕ(in_{d}(f)) = in(g)t^{a}∈(in(f_{1}), . . . ,in(f_{m}),in(f_{1})t, . . . ,in(f_{m})t) = ϕ(in(I), J_{0}^{∗}),

as desired.

*Proof.* [Proof of Theorem 1.4] By Lemma 1.5 there exists a weight functionωsuch
that in_{d}(I, J_{0}) = in_{ω}(I, J_{0}). Applying [5, Theorem 15.17] we obtain the following
inequalities of graded Betti-numbers

βij(I, J0)≤βij(ind(I, J0)) for all i, j.

The assumptions of Theorem 1.4 and Lemma 1.6 imply that (in(I), J_{0}^{∗}) = in_{d}(I, J_{0}),
and henceβij(I, J0)≤βij(in(I), J0∗) for alli, j. This yields the desired inequalities
for the depth and regularity of the symmetric algebras under consideration.

The last result of this section describes a case in which the hypotheses of Theorem
1.4 are satisfied. For a given term order < and f ∈S, we denote by inm_{<}(f) (or
simply inm(f)) the initial monomial off.

Proposition 1.7. *Let* I ⊂ S *be a graded ideal, and* < *a term order. Suppose*
*there exists a minimal system of homogeneous generators* f_{1}, . . . , f_{m} *of* I *with the*
*property that for each integer*t*the set of polynomials*{fi: degfi ≤t}*is a Gr¨obner*
*basis of the ideal they generate. Then* v_{in(I)}(in(f)) ≥ v_{I}(f) *for all homogeneous*
*polynomials* f ∈I.

*Proof.* Letf_{1}, . . . , f_{m} be a minimal system of homogeneous generators ofI satis-
fying the conditions as described in the proposition. Let f ∈I be a homogeneous
polynomial, and let vI(f) = j. Then f ∈ m^{j}I; hence there exist homogeneous
polynomials g_{i} ∈m^{j} such that f =Pm

i=1g_{i}f_{i} and degg_{i}f_{i} = degf for all i.

Let u = min{inm(g_{i}f_{i}) : i = 1, . . . , m}. Then u ≤ inm(f). Assume u <

inm(f). Let S = {i: inm(gifi) = u}. Then P

i∈Sin(gi) in(fi) = P

i∈Sin(gifi) =

0. Let t = max{degf_{i}: i ∈ S}, and suppose without loss of generality that
degf_{i} ≤ t for i ≤ r and degf_{i} > t for i > r. Since by assumption f_{1}, . . . , f_{r} is
a Gr¨obner basis and S ⊂ {f1, . . . , fr}, there exist homogeneous polynomials hi

such that X

i∈S

g_{i}f_{i} =

r

X

i=1

h_{i}f_{i} with u <inm(h_{i}f_{i}) and degh_{i}f_{i} = degf for i= 1, . . . , r.

Replacing P

i∈Sg_{i}f_{i} in the sum Pm

i=1g_{i}f_{i} by Pr

i=1h_{i}f_{i}, we can rewrite f as
f =

m

X

i=1

g_{i}^{0}fi with u <inm(g^{0}_{i}fi) for all i.

Note furthermore that degh_{i} ≥degf−t= degg_{i}_{0} ≥j, where i_{0} ∈S is the index
with degf_{i}_{0} =t. Thus we see that all h_{i} ∈m^{j} for i= 1, . . . , r, and henceg_{i}^{0} ∈m^{j}
for i= 1, . . . , m.

After finitely many steps of rewritingf we may assume that inm(f) = inm(g_{i}f_{i})
for some i. Then we conclude that v_{in(I)}(in(f))≥ j, since in(g_{i}f_{i}) = in(g_{i}) in(f_{i})

and in(gi)∈m^{j}.

Recall that a graded idealI ⊂S is called*componentwise linear, if each component*
I_{j} of I generates an ideal with linear resolution.

Let I be a componentwise linear ideal. Fix an integer t, and let I≤t be the
ideal generated by all componentsIj withj ≤t. ThenI≤tis again componentwise
linear. In fact, (I≤t)_{j} =I_{j} for j ≤t, while for j > t one has (I≤t)_{j} =Sj−tI_{t}. Thus
all components of I≤t generate ideals with linear resolution.

We now assume that charK = 0. Choose generic coordinates x1, . . . , xn, and
let < be the degree reverse lexicographical term order induced by x_{n} > xn−1 >

· · · > x_{1}. Let f_{1}, . . . , f_{m} be a minimal homogeneous set of generators of I such
that inm(f1) ≤ inm(f2) ≤ · · · ≤ inm(fm). It follows from [7, Theorem 1.1] that
such a minimal system of generators of I is Gr¨obner basis of I. Therefore, since
for each integer t, the idealI≤tis componentwise linear it follows that f_{1}, . . . , f_{m}_{1}
is a Gr¨obner basis of I≤t, where mt = max{i: degfi ≤t}. Hence we may apply
Proposition 1.7 and Theorem 1.4, and obtain

Corollary 1.8. *Suppose* charK = 0. Let I ⊂ S *be componentwise linear ideal.*

*Choose generic coordinates*x_{1}, . . . , x_{n}*, and let* <*be the degree reverse lexicograph-*
*ical term order induced by* x_{1} < x_{2} <· · ·< x_{n}*. Then*

reg Sym(m)≤reg Sym(m^{∗}), *and* depth Sym(m^{∗})≤depth Sym(m).

2. Algebras with monomial relations whose maximal ideal is generated by an s-sequence

LetK be a field,S =K[x_{1}, . . . , x_{n}] be a polynomial ring, andI ⊂S a monomial
ideal. We denote by G(I) = {u_{1}, . . . , u_{r}} the unique minimal monomial set of
generators of I, and by m the graded maximal ideal ofR =S/I.

For a monomial uinS we set m(u) = inf{i:x_{i}|u} andu^{0} =u/x_{m(u)}. Then in
particular, u_{i} =u^{0}x_{m}_{i} where m_{i} =m(u_{i}).

LetJ0 ⊂S[y1, . . . , yn] be the ideal which is generated by

G_{0} ={u^{0}_{i}y_{m}_{i} :i= 1, . . . , r} ∪ {x_{i}y_{j}−x_{j}y_{i} : 1≤i < j ≤n},

and let J ⊂ P = R[y_{1}, . . . , y_{n}] be the ideal which is generated by the residue
classes of the elements inG_{0} modulo I.

Then we have

Sym_{R}(m)∼=P/J ∼=S[y_{1}, . . . , y_{n}]/(I, J_{0}).

We fix a term order < onP induced by y_{1} < y_{2} <· · ·< y_{n} for which we want to
compute in<(J). To this end we extend the given order on the monomials in the
variables y_{i} to a term order ≺ order satisfying x_{1} < x_{2} <· · · < x_{n} < y_{1} < y_{2} <

· · ·< y_{n}. SinceI is a monomial ideal it follows that
in≺(I, J_{0}) = (I, L^{0}),

for some monomial ideal L^{0} ⊂S[y1, . . . , yn]. Let L⊂P be the image of L^{0}. Then
Lemma 2.1. in_{<}(J) = L.

*Proof.* For a graded module M we denote byH_{M}(t) =P

idim_{K}M_{i}t^{i} the Hilbert
series of M. We claim that

(1) L⊂in(J);

(2) H_{P/J}(t) =H_{P/}_{in(J)}(t).

By (1) and (2) it follows that L = in(J) if and only if H_{P/L}(t) = H_{P/J}(t). In
order to prove this equality of Hilbert series we use Macaulay’s theorem (see [3,
Corollary 4.2.4]) and the isomorphism P/J ∼=S[y1, . . . , yn]/(I, J0), and get

H_{P/J}(t) = H_{S[y}_{1}_{,... ,y}_{n}_{]/(I,J}_{0}_{)}(t) =H_{S[y}_{1}_{,... ,y}_{n}_{]/}_{in(I,J}_{0}_{)}(t)

= H_{S[y}_{1}_{,... ,y}_{n}_{]/(I,L}^{0}_{)}(t) = H_{P/L}(t),
as desired.

Proof of (1): We view S[y_{1}, . . . , y_{n}] as a Z^{n}-graded K-algebra by setting for
i = 1, . . . , n, degxi = degyi = (0, . . . ,0,1,0, . . . ,0) where the entry 1 is at the
i-th position. Notice that (I, J_{0}) is multi-homogeneous.

Letg ∈(I, J_{0}) be a multi-homogeneous element. We will show that if in(g)6=

0, then in(g) = in(g) where ¯f denotes the residue class modulo I of an element
f ∈S[y_{1}, . . . , y_{n}]. From this observation assertion (1) will follow.

Let g = P

av_{a}y^{a} where the sum is taken over all a ∈ N^{n} and where the
coefficients va are monomials in the variables x1, . . . , xn, with all but finitely
many v_{a} are zero.

Let in(g) = v_{a}_{0}y^{a}^{0}, and assume that v_{a}_{0} 6∈ I. Then in(g) 6= 0, and g =
P

avay^{a}. Suppose in(g)6= in(g). Then there exists a1 such that va1y^{a}^{1} > va0y^{a}^{0}.

Then this means that y^{a}^{1} > y^{a}^{0}. Since v_{a}_{0}y^{a}^{0} and v_{a}_{1}y^{a}^{1} have the same multi-
degree and since y_{j} > x_{i} for all i and j, it follows that v_{a}_{1}y^{a}^{1} > v_{a}_{0}y^{a}^{0}, a contra-
diction.

Proof of (2): We show that for each multi-degree a the multi-graded components
J_{a} and in(J)_{a} have the same K-dimension.

If g ∈R[y1, . . . , yn] with in(g) =uy^{c} with u∈R, then we let inm(g) =y^{c} be
the initial term of this g.

Now letg_{1}, . . . , g_{s}aK-basis ofJ_{a}where theg_{j} are multi-graded with degg_{i} =
a. We may assume that inm(g1) ≥ inm(g2) ≥ · · · ≥ inm(gs). We claim that we
can modify this K-basis such that inm(g_{1}) > inm(g_{2}) > · · · > inm(g_{s}). In fact,
suppose that forg_{1}, . . . , g_{m}, (m≤s) we have inm(g_{1}) = inm(g_{2}) =· · ·= inm(g_{m}).

Then since the gi multi-homogeneous all of same degree a this implies that there
exist λ_{i} ∈ K, λ_{i} 6= 0, such that in(g_{i}) = λ_{i}in(g_{1}) = in(λ_{i}g_{1}) for i = 1, . . . , m.

We replace g_{1}, . . . , g_{s} by g_{1}^{0}, . . . , g^{0}_{s} where g_{1}^{0} =g_{1}, g^{0}_{i} =g_{i}−λ_{i}g_{1}, for i= 2, . . . , m
and g^{0}_{i} = gi for i = m+ 1, . . . , s. Then g_{1}^{0}, . . . , g_{s}^{0} is again a basis of Ja and
inm(g^{0}_{1})>inm(g^{0}_{i}) for all i.

After renumbering we may assume that

inm(g^{0}_{1})>inm(g^{0}_{2})≥inm(g_{3}^{0})≥ · · · ≥inm(g_{s}^{0}).

Applying the same argument to g_{2}^{0}, . . . , g_{s}^{0} and using induction on s, the claim
follows.

In particular, the initial terms in(g_{1}), . . . ,in(g_{s}) are linearly independent over
K. Therefore we conclude that dim_{K}in(J)_{a} ≥dim_{K}J_{a}. The opposite inequality

is proved similarly.

Now we are ready to prove the main result of this section.

Theorem 2.2. *Let* K *be a field,* S =K[x_{1}, . . . , x_{n}] *the polynomial ring,* I ⊂S *a*
*monomial ideal with*G(I) ={u_{1}, . . . , u_{r}}, andR =S/I. Then for any term order

< *induced by* x_{1} < x_{2} < · · · < x_{n} < y_{1} < y_{2} <· · · < y_{n}*, the following conditions*
*are equivalent:*

(a) G = {u_{1}, . . . , u_{r}} ∪ {u^{0}_{1}y_{1}, . . . , u^{0}_{r}y_{r}} ∪ {x_{i}y_{j} −x_{j}y_{i}: 1 ≤ i < j ≤ n} *is a*
*Gr¨obner basis of* (I, J_{0}).

(b) *For all* u∈G(I) *and all* j > m(u^{0}) *either*
(i) u^{0}x_{j}/x_{m(u}^{0}_{)}∈I, or

(ii) *there exists* v ∈G(I) *such that either*
m(v) =m(u), or

m(v) =m(u^{0}) *and* v^{0} *divides* u^{0}x_{j}/x_{m(u}^{0}_{)}*.*

*If the equivalent conditions hold, then the elements* x_{1}, . . . , x_{n} *form an*s-sequence
*in* R.

*Proof.* Note that G is a Gr¨obner basis of (I, J_{0}) if and only if all S-pairs of G
reduce to zero. The only S-pairs of G which do not trivially reduce to zero are

the S-pairs S(u^{0}y_{m(u)}, x_{i}y_{j}−x_{j}y_{i}) withu∈G(I), i < j and x_{i} divides u^{0}. In that
case we have

S(u^{0}y_{m(u)}, x_{i}y_{j}−x_{j}y_{i}) = u^{0}xj

x_{i} y_{m(u)}y_{i}.
If i > m(u^{0}), then

u^{0}x_{j}

x_{m(u}^{0}_{)}ym(u)ym(u^{0}) = u^{0}x_{j}

x_{i} ym(u)yi− u^{0}x_{j}

x_{i}x_{m(u}^{0}_{)}ym(u)(xm(u^{0})yi−xiym(u^{0})).

Therefore it suffices to see in which cases (u^{0}xj/xm(u^{0}))ym(u)ym(u^{0}) reduces to zero.

Since all integers k for which x_{k} divides u^{0}x_{j}/x_{m(u}^{0}_{)} are ≥ m(u^{0}), the relations
x_{i}y_{j}−x_{j}y_{i} with i < j can not be used for further reductions.

Therefore it follows that (u^{0}xj/xm(u^{0}))ym(u)ym(u^{0}) reduces to zero if and only
eitheru^{0}x_{j}/x_{m(u}^{0}_{)} ∈I orv^{0}y_{m(v)}divides (u^{0}x_{j}/x_{m(u}^{0}_{)})y_{m(u)}y_{m(u}^{0}_{)}for somev ∈G(I).

This is exactly condition (b).

Corollary 2.3. *Suppose condition* (b) *of Theorem* 2.2 *is satisfied. Then*
(a) in(I, J0) = (u1, . . . , ur, u^{0}_{1}ym1, . . . , u^{0}_{r}ymr,{xiyj}1≤i<j≤n);

(b) in(J) *is generated by the residue classes modulo* I *of the set of monomials*
{u^{0}_{1}y_{m}_{1}, . . . , u^{0}_{r}y_{m}_{r}} ∪ {x_{i}y_{j}}_{1≤i<j≤n}.

*In particular, the annihilator ideals of* x_{1}, . . . , x_{n} *are*
I_{j} = [(x_{1}, . . . , xj−1) +L_{j}] modI
*with* L_{j} = ({u^{0}: u∈G(I) *and* m(u) = j}) *for* j = 1, . . . , n.

As a first application of Theorem 2.2 we have

Proposition 2.4. *Let* I *be a monomial ideal generated in degree* 2. Then follow-
*ing conditions are equivalent:*

(a) x_{1}, . . . , x_{n} *is an* s-sequence in R;

(b) *for all monomials* x_{i}x_{j} ∈ I *with* i ≤ j *and for all* k > i *either* x_{i}x_{k} ∈I *or*
x_{j}x_{k} ∈I.

*If the equivalent conditions hold, then* Sym(m) *is a Koszul algebra.*

*Proof.* It is obvious that for a monomial ideal which is generated in degree 2 the
condition (b) in this proposition is equivalent to condition (b) in Theorem 2.2.

Therefore we have the equivalence of (a) and (b).

If the equivalent conditions hold, then the Gr¨obner basis of the defining ideal J of Sym(m) is generated by quadratic forms. It is well known that this implies

that Sym(m) is Koszul.

3. Algebras defined by stable monomial ideals

As in the previous section we letI ⊂S =K[x_{1}, . . . , x_{n}] be a monomial ideal, and
denote by R the standard graded K-algebra S/I, and by mthe graded maximal
ideal of R. Without loss of generality we may assume that I ⊂m^{2}.

Let I be a monomial ideal. We say that I is *stable (resp. strongly stable) in*
*the reverse order*if for all monomialsu∈I, one has thatu^{0}x_{j} ∈I for all j > m(u)
(resp. (u/x_{k})x_{j} ∈I for all j > k and all k such that x_{k} divides u).

Note that if we renumber the variables so that x_{i} becomes xn−i+1 for i =
1, . . . , n, then an ideal which is stable in the reverse order becomes an ideal which
is stable in the usual sense.

Proposition 3.1. x_{1}, . . . , x_{n} *is a strong* s-sequence in R *if* I *is a stable ideal in*
*the reverse order.*

*Proof.* Let u ∈ G(I). Since I is stable in the reverse order, we have that
u^{0}x_{j} ∈ I for all j > m(u). Therefore there exists v ∈ G(I), and a monomial
w such that u^{0}x_{j} = vw. Since x_{m(u}^{0}_{)} divides u^{0}, it follows that x_{m(u}^{0}_{)} divides
vw. In case, x_{m(u}^{0}_{)} divides w, one has u^{0}x_{j}/x_{m(u}^{0}_{)} ∈ I, and condition (b)(i) of
Theorem 2.2 is satisfied. On the other hand, ifx_{m(u}^{0}_{)} dividesv, then v/x_{m(u}^{0}_{)} =v^{0}
and u^{0}x_{j}/x_{m(u}^{0}_{)} =v^{0}w, and condition (b)(ii) of Theorem 2.2 is satisfied, therefore
x_{1}, . . . , x_{n} is an s-sequence.

To conclude the proof we have to showI_{1} ⊂I_{2} ⊂ · · · ⊂I_{n}. By Corollary 2.3,
I_{j} = [(x_{1}, . . . , xj−1) +L_{j}] modI.

Letv ∈(x1, . . . , xj−1)+Lj. Ifxidividesv fori= 1, . . . , j, thenv ∈(x1, . . . , xj)+

L_{j+1}. Therefore we assume that v ∈ L_{j}, and that i > j whenever x_{i} divides v.

Then v = u^{0}w with u ∈ G(I), m(u) = j and m(u^{0}) > j. Since I is stable,
u^{0}xj+1 ∈ I, and since m(u^{0}) > j, m(u^{0}xj+1) = j + 1. Thus u^{0}xj+1 = gh where
g ∈G(I),his a monomial andm(gh) =j+1. Ifm(g) = j+1, theng^{0} ∈L_{j+1}, and
hencev =g^{0}hwbelongs to L_{j+1}. Otherwise, m(h) =j+1, and thenv =gh^{0}w∈I.

Therefore,vmodI = 0 ∈Ij+1.

Examples 3.2. (a) Let I = (x_{1}, . . . , x_{n})^{d}. Then I is stable in the reverse order,
and so by Proposition 3.1 the sequence x1, . . . , xn is an s-sequence in R. Using
Corollary 2.3 we see that the annihilator ideals of x_{1}, . . . , x_{n} are

I_{j} = (x_{1}, . . . , xj−1,(x_{j+1}, . . . , x_{n})^{d−1}) modI.

In particular,x_{1}, . . . , x_{n}is a strongs-sequence. Using the formulas [8, Proposition
2.4] we get

dim Sym(m) =n and e(Sym(m)) = d−1,

wheree(Sym(m)) denotes the multiplicity. Note that by the Huneke-Rossi formula [11] one has dim Sym(m) = n in general.

(b) The ideal I = (x_{1}x_{2}, x_{1}x_{3}, x_{2}x_{4}, x_{3}x_{4}) satisfies condition (b) of Theorem 2.2,
but is not stable in the reverse order.

For the proof of the next result we need a few lemmata on stable ideals and basic facts on graded resolutions.

Lemma 3.3. *Suppose that* I *is strongly stable in the reverse order. Then for*
j = 1, . . . , n, the ideals (I, L_{j}) *with* L_{j} = (u^{0})u∈G(I),m(u)=j *as defined in Corollary*
2.3 *are strongly stable in the reverse order.*

*Proof.* Letv ∈(I, L_{j}) be a monomial, and suppose thatx_{i}dividesv. For allk > i
we want to show that (v/xi)xk ∈(I, Lj). Since I is strongly stable in the reverse
order, we may assume that v ∈L_{j}. Then there exists a monomial u∈G(I) with
m(u) = j such thatv =u^{0}w for some monomial w.

If xi|w, then (v/xi)xk = u^{0}(w/xi)xk ∈ Lj. If xi|u^{0}, then xi|u and hence
(u/x_{i})x_{k} ∈I, since I is strongly stable in the reverse order.

Moreover, m((u/x_{i})x_{k}) = j. Hence (u/x_{i})x_{k} = gh where g ∈ G(I) and h is a
monomial, such that either m(g) =j orm(h) =j. If m(h) =j, then (v/xi)xk =
gh^{0}w∈I, and if m(g) =j, then (v/x_{i})x_{k}=g^{0}hw∈L_{j}.
Lemma 3.4. *Let* I *and* J *be graded ideals in* S *such that* Tor^{S}_{1}(S/I, S/J) = 0.

*Then* reg(I+J)≥regI, and equality holds if and only if J *is generated by linear*
*forms.*

*Proof.* Let F be the graded minimal free resolution of S/I, and G the graded
minimal free resolution of S/J. Then Tor^{S}_{1}(S/I, S/J) = 0 implies that F⊗G is
the graded minimal free resolution of S/(I+J).

LetP =P

i,jβ_{i,i+j}(S/I)x^{i}y^{i+j} be the graded Poincar´e series of S/I and Q=
P

i,jβi,i+j(S/J)x^{i}y^{i+j} the graded Poincar´e series of S/J. ThenP Q is the graded
Poincar´e series of S/(I +J). One has regS/I = deg_{y}P, where deg_{y}P denotes
the y-degree of P. Similarly, regS/J = deg_{y}Q and regS/(I+J) = deg_{y}P Q =
deg_{y}P + deg_{y}Q. It follows that regS/I = regS/J if and only deg_{y}Q = 0, and
this is the case if and only if J is generated by linear forms.

For a graded ideal I ⊂ S and an integer j we denote by I≥j the ideal generated by all homogeneous elements f ∈I with degf ≥j.

Lemma 3.5. *Let* I ⊂S *be a graded ideal. Then the natural map*
Tor_{i}(I≥j, K)_{i+j} →Tor_{i}(I, K)_{i+j}

*is surjective for all* i *and* j.

*Proof.* The short exact sequence

0−→I≥j −→I −→I/I≥j −→0 induces the long exact sequence

Tor_{i}(I≥j, K)_{i+j} −→Tor_{i}(I, K)_{i+j} −→Tor_{i}(I/I≥j, K)_{i+j}.

Note that (I/I≥j)_{k} = 0 for k ≥j. Let K(x;I/I≥j) be the Koszul complex of the
sequence x = x_{1}, . . . , x_{n} with values in I/I≥j. Then K_{i}(x;I/I≥j)_{i+j} = 0. Now
since Tor_{i}(I/I_{≥j}, K)_{i+j} ∼= H_{i}(x;I/I_{≥j})_{i+j}, we conclude that H_{i}(x;I/I_{≥j})_{i+j} = 0,

as desired.

Lemma 3.6. *Let* I ⊂ J ⊂S *be graded ideals, and* f ∈S *a linear form. Suppose*
*that* f *is a non-zerodivisor on* S/I *and on* S/J *and that for some* j,

Tor_{i}(I, K)_{i+j} →Tor_{i}(J, K)_{i+j}
*is surjective for all* i. Then

Tor_{i}((I, f), K)_{i+j} →Tor_{i}((J, f), K)_{i+j}
*is surjective for all* i.

*Proof.* Let F be the graded minimal free resolution of S/I, and G the graded
minimal free resolution of S/J. Let α: F → G be the complex homomorphism
induced by I ⊂J. We denote by αi,j the jth graded component of αi. Then the
map Tor_{i}(I, K)_{i+j} → Tor_{i}(J, K)_{i+j} can be identified with ¯α_{i,i+j}: (F_{i}/mF_{i})_{i+j} →
(G_{i}/mG_{i})_{i+j}, where ¯α_{i,i+j} denotes thei+jth graded component of ¯α_{i} =α_{i}⊗S/m.

The resolution of S/(I, f) is given by F⊗Hwhere H is the complex
0 −−−→ S(−1) −−−→^{f} S −−−→ 0.

Similarly, the resolution ofS/(J, f) is given byG⊗H. Hence the inclusion (I, f)⊂ (J, f) can be lifted by the complex homomorphism α⊗id. Thus for all i and j we have

(α⊗id)_{i,i+j} =α_{i,i+j}⊕αi−1,i+j−1: (F_{i})_{i+j}⊕(Fi−1)i+j−1 −→(G_{i})_{i+j}⊕(Gi−1)i+j−1,
which induces the homomorphisms

(α⊗id)_{i,i+j}: (F_{i}/mF_{i})_{i+j}⊕(Fi−1/mFi−1)i+j−1→(G_{i}/mG_{i})_{i+j}⊕Gi−1/mGi−1)i+j−1.
Since (α⊗id)_{i,i+j} = ¯α_{i,i+j} ⊕α¯_{i−1,i+j−1} and since ¯α_{i,i+j} is surjective for all i, it
follows that (α⊗id)_{i,i+j} is surjective, as desired.

Theorem 3.7. *Let*R =S/I *where*I *is a strongly stable ideal in the reverse order,*
*let* u_{1}, . . . , u_{r} *be the Borel generators of* I *and* d = max{deg(u_{i}) : i = 1, . . . , r}.

*Then*

(a) regR≤reg Sym_{R}(m)≤regR+ 1;

(b) regR= reg Sym_{R}(m)⇐⇒max{m(u_{i}) : deg(u_{i}) =d} ≤2.

*Proof.* (a) By the Eliahou-Kervaire resolution of I (see [6]) the regularity of I
equals d since I is stable in the reverse order. Hence it amounts to show that
d≤reg(I, J0)≤d+ 1.

Since the highest degree of a generators of (I, J_{0}) is d it follows that d ≤
reg(I, J_{0}). In order to prove the upper inequality, it suffices to show that reg in(I,
J0)≤d+ 1 since reg(I, J0)≤reg in(I, J0).

Forj = 1, . . . , n we consider the ideal

Kj = (I, I_{1}^{0}y1, I_{2}^{0}y2, . . . , I_{j}^{0}yj),

whereI_{j}^{0} = (x_{1}, . . . , xj−1) + (I, L_{j}), and we setK_{0} =I. Recall from Corollary 2.3
that the ideals I_{j} =I_{j}^{0}modI are the annihilator ideals of x_{1}, . . . , x_{n}.

We will show by induction on j that regKj ≤ d+ 1. This implies the upper
bound, since by Corollary 2.3 we have in(I, J_{0}) =K_{n}.

Since K_{0} =I is strongly stable in the reverse order, we have regK_{0} =d.

Now letj >0 and assume that regKj−1 ≤d+ 1. We have
K_{j} = (Kj−1, I_{j}^{0}y_{j}) = (Kj−1, I_{j}^{0})∩(Kj−1, y_{j}),

and I_{1}^{0} ⊂ I_{2}^{0} ⊂ · · · ⊂ I_{j}^{0}, since I_{1} ⊂ I_{2} ⊂ · · · ⊂ I_{j} by Proposition 3.1. It follows
that (Kj−1, I_{j}^{0}) =I_{j}^{0}. Hence we obtain the exact sequence

0−→K_{j} −→I_{j}^{0} ⊕(Kj−1, y_{j})−→(I_{j}^{0}, y_{j})−→0.

This together with Lemma 3.4 implies that

regK_{j} ≤ max{regI_{j}^{0},reg(Kj−1, y_{j}),reg(I_{j}^{0}, y_{j}) + 1}

= max{regKj−1,regI_{j}^{0} + 1}.

By induction hypothesis regK_{j−1} ≤ d + 1. Hence it remains to show that
regI_{j}^{0} ≤ d.

For a monomial ideal H we denote by H^{≥j} the ideal generated by all mono-
mials u ∈ H with m(u) ≥ j. Then we have I_{j}^{0} = (x_{1}, . . . , x_{j−1}) + (I, L_{j})^{≥j}.
Therefore, by Lemma 3.4, regI_{j}^{0} = reg(I, L_{j})^{≥j}. In Lemma 3.3 it is shown that
(I, L_{j}) is strongly stable in the reverse order. It is clear that then also (I, L_{j})^{≥j}
is strongly stable in the reverse order, and that the highest degree of a Borel
generator of (I, L_{j})^{≥j} is≤d. This implies that reg(I, L_{j})^{≥j} ≤d, as desired.

(b) Letm= max{m(u_{i}) : deg(u_{i}) =d}, and assumem≤2. Sinced≤reg(I, J_{0})≤
reg in(I, J_{0}), and since in(I, J_{0}) = K_{n} it suffices to prove that reg(K_{n}) = d. In
fact, we show by induction onj that regK_{j} ≤dforj = 0, . . . , n. We first consider
the case m = 1. The induction begin is trivial because K_{0} =I. The assumption
m = 1 implies that (I, L_{j})^{≥j} is generated in degree ≤ d−1 for all j, and this
implies reg(I, L_{j})^{≥j} ≤d−1 for allj. Arguing as in the proof of (a) it follows that
regK_{j} ≤d for all j.

Now assume that m = 2. Again we show by induction on j that regK_{j} ≤ d.

For j = 0 the assertion is trivial. We must also consider the case j = 1. Since
K_{1} = (I, I_{1}^{0}y_{1}) = (I, L_{1})∩(I, y_{1}) we obtain the exact sequence

0−→K_{1} −→(I, L_{1})⊕(I, y_{1})−→(I, L_{1}, y_{1})−→0.

For all j this yields the long exact sequence

−→ Tor_{i+1}((I, L_{1}), K)_{j}⊕Tor_{i+1}((I, y_{1}), K)_{j} −→Tor_{i+1}((I, L_{1}, y_{1}), K)_{j}

−→ Tori(K1, K)j −→Tori((I, L1), K)j⊕Tori((I, y1), K)j.

We need to show that Tor_{i}(K_{1}, K)_{j} = 0 for j > d+i. Since (I, L_{1}) and I are
strongly stable ideals in the reverse order, generated in degree≤d, it follows that