HILBERT REGULARITY OF Z-GRADED MODULES OVER POLYNOMIAL RINGS

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1 JOURNAL OF COMMUTATIVE ALGEBRA Volume 9, Number, Summer 7 HILBERT REGULARITY OF Z-GRADED MODULES OVER POLYNOMIAL RINGS WINFRIED BRUNS, JULIO JOSÉ MOYANO-FERNÁNDEZ AND JAN ULICZKA ABSTRACT Let M be a finitely generated Z-graded module over the standard graded polynomial ring R = K[X,, X d ] with K a field, and let H M (t) = Q M (t)/ ( t) d be the Hilbert series of M We introduce the Hilbert regularity of M as the lowest possible value of the Castelnuovo-Mumford regularity for an R-module with Hilbert series H M Our main result is an arithmetical description of this invariant which connects the Hilbert regularity of M to the smallest k such that the power series Q M ( t)/( t) k has no negative coefficients Finally, we give an algorithm for the computation of the Hilbert regularity and the Hilbert depth of an R-module Introduction This note may be considered as part of a program that aims at estimating numerical invariants of a graded module M over a polynomial ring K[X,, X d ] (K is a field) in terms of the Hilbert series H M (t) For the notions of commutative algebra we refer the reader to Bruns and Herzog [] Well-known examples of such estimates are the bounds of Bigatti [] and Hulett [6] on the Betti numbers or the bounds of Elias, Robbiano and Valla [4] on the number of generators for ideals primary to m = (X,, X d ) A more recent result is the upper bound on depth M (or, equivalently, a lower bound on projdim M) given by the third author [], namely, the Hilbert depth Hdepth M It is defined as the maximum AMS Mathematics subject classification Primary 3D4, Secondary 3D45 Keywords and phrases Hilbert regularity, boundary presentation of a rational function, nonnegative power series, Hilbert depth The second author was partially supported by the Spanish government, Ministerio de Economía, Industria y Competitividad (MINECO), grant Nos MTM C3-3 and MTM C3--P, as well as by the University Jaume I, grant No P-B5- and the German Research Council, grant No GRK-96 Received by the editors on September, 4, and in revised form on January, 5 DOI:6/JCA Copyright c 7 Rocky Mountain Mathematics Consortium 57

2 58 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA value of depth N for a module N with H M (t) = H N (t) We must emphasize that we will always consider the standard grading on R under which all indeterminates have degree As soon as this hypothesis is dropped, matters become extremely difficult as witnessed by the paper [9] of the second and third author The objective of this paper is to bound the Castelnuovo-Mumford regularity reg M in terms of H M (t) Of course, the bound is the lowest possible value of reg N for a module N with H M (t) = H N (t), which we term Hilbert regularity Hreg M Both Hilbert depth and Hilbert regularity can be computed in terms of Hilbert decompositions introduced by Bruns, Krattenthaler and Uliczka [3] for arbitrary gradings; for a method computing Hilbert depth for Z n -graded modules, see Ichim and the second author [7] The approach by Hilbert decompositions is related to Stanley depth and Stanley regularity, see Herzog [5] for a survey Stanley regularity for quotients by monomial ideals was considered by Jahan [8] Also, Herzog introduced Hilbert regularity via decompositions Write H M (t) = Q(t)/( t) d with d = dim M and Q Z[t] (we may certainly assume that M is generated in degrees ) Then Hdepth M = d m, where m is the smallest value of all natural numbers j such that Q(t)/( t) j is a positive power series, ie, a power series with nonnegative coefficients [] Note that the Hilbert series Q(t)/( t) d has nonnegative coefficients Hilbert regularity cannot always be described in such a simple way, but it is closely related to the smallest k for which Q( t)/( t) k is positive, see Theorems 47 and 4 Our main tool for the analysis of Hilbert series is H(t) = k f i t i ( t) n + ctk ( t) n + g j t k ( t) d j, which we call (n, k)-boundary presentations since the pairs of exponents (u, v) occurring in the numerator and denominator of the terms t i / ( t) n, t k /( t) n, and t k /( t) d j occupy the lower and the right boundary of a rectangle in the u-v-plane whose right lower corner is (k, n)

3 HILBERT REGULARITY OF Z-GRADED MODULES 59 Using the description of Hilbert regularity in terms of Hilbert decompositions, one easily sees that Hreg M is the smallest k for which a (, k)-boundary representation with nonnegative coefficients f i, c, g j exists (Without the requirement of nonnegativity the smallest such k is deg H M (t)) The bridge to power series expansions of Q( t)/( t) k are given by the fact that the coefficients g j appear in such expansions The paper is structured as follows We introduce Hilbert regularity in Section and discuss boundary representations in Section 3 Hilbert regularity is then determined in Section 4, whereas Section 5 contains an algorithm that computes Hilbert depth and Hilbert regularity simultaneously Hilbert regularity Let K be a field, and let M be a finitely generated graded module over a positively graded K-algebra R The Castelnuovo-Mumford regularity of M is given by reg M = max { i + j : H i m(m) j }, where m is the maximal ideal of R generated by the elements of positive degree In the sequel, we only consider R = K[X,, X d ] In this case, a theorem of Eisenbud and Goto [, 43] yields reg M = max { j i : Tor R i (K, M) j }, where K is naturally identified with R/m Definition The (plain) Hilbert regularity of a finitely generated graded R-module is { } Hreg M := min r N there is an fg gr R-module N with H N = H M and reg N = r Let F i be a graded free module over K[X,, X i ], i =,, d, considered as an R-module via the retraction R K[X,, X i ] that sends X i+,, X d to The module F F d is called a Hilbert decomposition of M if the Hilbert functions of M and F F d coincide This leads us to the following:

4 6 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA Definition The decomposition Hilbert regularity of M is dechreg M = min {reg N : H N (t) = H M (t)}, where now N ranges over direct sums F F d, ie, over the Hilbert decompositions of M It is in particular clear that dechreg M Hreg M As we shall see below both numbers coincide in our setting of standard graded polynomial rings However, both definitions make sense in much more generality if one replaces the K[X,, X i ] by graded retracts of K[X,, X d ], see [3] In the more general setting the equality is a completely open problem, for regularity as well as for depth In fact, proving equality for depth in the multigraded setting would come close to proving the Stanley conjecture for depth, see [5] Remark 3 (a) The notion of Hilbert decomposition is the same as that in [3], except that the F i are further decomposed into cyclic modules there (b) Hilbert depth and Hilbert regularity are companions in the following sense: the Hilbert depth determines the smallest width of a Betti table admitting the given Hilbert series, Hilbert regularity determines the smallest such possible height The Betti table is given in terms of the graded Betti numbers β i,j = dim K Tor R i (K, M) j by β, β, β p,p β,r β,r+ β p,r+p, where p = projdim M and r = reg M The decomposition Hilbert regularity can be described in terms of positive representations P = (Q d,, Q ) of the Hilbert series: H M (t) = Q d(t) ( t) d + + Q (t) ( t) + Q (t), where each Q i is a polynomial with nonnegative coefficients Such polynomials will be called nonnegative It is well known that there is

5 HILBERT REGULARITY OF Z-GRADED MODULES 6 always a Hilbert decomposition of M This simple fact will be proved (again) in Proposition 5 Let F F d be a Hilbert decomposition of M Then, we have H Fi = Q i (t)/( t) i with a nonnegative polynomial Q i, and we immediately get a positive representation of the Hilbert series Conversely, given a positive representation of the Hilbert series, we find a direct sum F F d by choosing F i as the free module over K[X,, X i ] that has a ij basis elements of degree i where Q i = j a ij t j Moreover, reg F i = deg Q i, and therefore, we have Proposition 4 dechreg M = min P max deg Q i, P = (Q d,, Q ), i where P ranges over the positive representations of H M (t) For Hilbert depth, we can similarly give a plain or a decomposition definition: the Hilbert depth of M is defined as: { } Hdepth M := max r N there is an fg gr R-module N with H N = H M and depth N = r The Hilbert depth of M turns out to coincide with the arithmetical invariant p (M) := max {r N ( t) r H M (t) is nonnegative}, called the positivity of M, see [, Theorem 3] The inequality Hdepth M p(m) follows from general results on Hilbert series and regular sequences The converse can be deduced from the main result of [, Theorem ], which states the existence of a representation H M (t) = dim M Q j (t) ( t) j with nonnegative Q j Z[t, t ]

6 6 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA The decomposition version, or positivity, is close to Stanley decompositions and Stanley depth The same holds true for Hilbert regularity, as we shall now show; our proof will also confirm the equivalence of the two notions of Hilbert depth Proposition 5 There exists a Hilbert decomposition of regularity equal to reg M and depth equal to depth M Proof If M is a free R-module, there is nothing to prove: M is already in Hilbert decomposition form Now, suppose that M is not free Let m be the maximal degree of a generator of M Then m reg M, and we can choose elements v,, v n M of degree m such that n = rank M and v,, v n are linearly independent (This is a well-known general position argument; we may have to pass to an infinite field K, but that is no problem) We set F n = Rv + + Rv n For the sake of Hilbert series computations we can replace M by F n M/F n Note that depth M/F n = depth M since depth M < depth F n by assumption on M and standard depth arguments We obtain dim M/ F n < n since rank M/F n = as an R-module For the regularity, we observe that M/F n is generated in degrees m and dim M/F n < n Since F n is free, Tor R j (K, M/F n ) = Tor R j (K, M) for j, and therefore, is the only critical homological degree for the regularity of M/F n There is a homogeneous exact sequence Tor R (K, F n ) = Tor R (K, M) Tor R (K, M/F n ) Tor R (K, F n ) This is Tor R (K, F n ) i = and Tor R (K, M) i = Tor R (K, M/F n ) i, except for i m Thus, the only critical arithmetical degree is m However, we subtract from the highest shift in homological degree in order to compute regularity, and it does not affect the inequality reg M/F n reg M if Tor R (K, M/F n ) i for some i m On the other hand, reg M max(reg F n, reg M/F n ), and altogether we conclude that reg M/F n = reg M

7 HILBERT REGULARITY OF Z-GRADED MODULES 63 Let S = R/ Ann M, and choose a degree Noether normalization R in S We first view M/F n as a module over R Then, reg R M/F n = reg S M/F n = reg R M/F n, since regularity does not change under finite graded extensions Now, we can identify R with one of the algebras K[X,, X i ] for some i < n Hence, we can proceed by induction considering M/F n Eventually, the procedure ends when the dimension of the Noether normalization has reached the depth of M, since the quotient of M then attained is free over the Noether normalization, and we are in the case of a free module Remark 6 The proof shows that regularity may be considered as a measure for filtrations = U U U q = M in which U i+ /U i is always a free module over some polynomial subquotient of R: there exists such a filtration in which each free module is generated in degree reg M, but there is no such a filtration in which all base elements have smaller degree A similar statement holds for depth Corollary 7 Let M be a finitely generated graded R-module Then, Hreg M = dechreg M In fact, if N is a module whose regularity attains the minimum, we can replace it by a Hilbert decomposition as in Proposition 5 A specific example follows Let M be the first syzygy module of the maximal ideal in the polynomial ring K[X,, X 5 ] It has been shown [3, Theorem 35] that it has multigraded Hilbert depth 4 It follows that the standard graded Hilbert depth is also 4, but this is much easier to see; the Hilbert series is () t t 3 + 5t 4 t 5 ( t) 5 = t ( t) 4 + t 4 ( t) 4 + 4t4 ( t) 5 Thus, we can get away with the worst denominator ( t) 4 for the Hilbert depth

8 64 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA Let us look at the Hilbert regularity: the decomposition () t t 3 + 5t 4 t 5 ( t) 5 = 4t ( t) 5 + 3t ( t) 4 + t ( t) 3 + t ( t) shows that Hreg M = It cannot be smaller since M has no generators in degree < On the other hand, the decomposition () is the only one with regularity, since the powers of the series /( t) are linearly independent, and it comes from a filtration as in the proof of Proposition 5 (In this example, Hreg M could be determined more easily since Hreg M and reg M = ) This shows that, in general, optimization of depth and regularity cannot simultaneously occur More generally, if M is a module with all generators in degree r and of regularity r, then Hreg M = reg M However, in general, Hilbert regularity is smaller than regularity Let N be the sum of the modules in the Hilbert decomposition () Then Hreg N < reg N holds, as () shows A simple lower bound is as follows Proposition 8 Let M be a finitely generated graded R-module Then Hreg M deg H M (t) In fact, for j > Hreg M, the Hilbert polynomial and the Hilbert function of M coincide, and the smallest number k such that the Hilbert polynomial and the Hilbert function coincide in all degrees j > k is k = deg H M (t), the degree of H M as a rational function; see [, 4] 3 Boundary presentation In this section, we introduce the fundamental tool for examination of the Hilbert regularity Definition 3 Let H(t) = Q(t)/( t) d For integers n d and k, an (n, k)-boundary presentation of H is a decomposition of H in the form (3) H(t) = k f i t i ( t) n + ctk ( t) n + g j t k ( t) d j with f i, c, g j Z

9 HILBERT REGULARITY OF Z-GRADED MODULES 65 If c =, the boundary presentation is called corner-free Note that Q(t)/( t) d may be viewed as a (d, deg Q)-boundary presentation of H If deg Q d, there is also a (d deg Q, )-boundary presentation: let Q( t) = i q it i, then H(t) = Q(t) deg Q ( t) d = q i ( t) i deg Q q i ( t) d = ( t) d i In the sequel, the polynomial Q( t) will be needed several times; therefore, we introduce the notation for an arbitrary Q Z[t] Example 3 Let Q(t) := Q( t) H(t) = t + 3t t 3 ( t) 3 A (, 3)-boundary presentation of H is given by H(t) = t + t t + t3 ( t) + t 3 ( t) 3 The term boundary presentation is motivated by visualization of a decomposition of a Hilbert series A decomposition Q(t) d ( t) d = j a ij t j ( t) i can be depicted as a square grid with the box at position (i, j) labeled by a ij In the case of an (n, k)-boundary presentation the nonzero labels in this grid form the bottom and the right edges of a rectangle with d n + rows and k + columns The coefficient in the corner

10 66 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA Figure Two boundary presentations of ( t + 3t t 3 )/( t) 3 (d n, k) plays a dual role since it belongs to both edges; therefore, it is denoted by an extra letter Next, we deduce a description for the coefficients in a boundary presentation Lemma 33 Let H(t) = Q(t)/( t) d be a series with (n, k)-boundary presentation (3) Moreover, let Then Q(t) ( t) d n = a i t i and f i = a i for i =,, k, Q(t) ( t) k = b i t i k c = a k b i = b d n a i, g j = b j for j =,, d n Proof Multiplication of (3) by ( t) n yields k Q(t) ( t) d n = f i t i + ct k + g j t k ( t) d n j Hence, the f i agree with the first k coefficients of the power series a it i, while a k = c + g j

11 HILBERT REGULARITY OF Z-GRADED MODULES 67 j α j α + j β j β + i i+ i i+ Figure Next, we look at (3) with t substituted by t: k Q( t) f i ( t) i c( t)k t d = t n + t n + This time, multiply by t d /( t) k and obtain k ( t) k = Q(t) Q( t) = ( t) k g j ( t) k t d j f i t d n ( t) k i + ctd n + g j t j ; hence, g j = b j for j =,, d n and c = b d n k f i Since the coefficients in the power series expansion of a rational function are unique, Lemma 33 has an immediate consequence: Corollary 34 The coefficients in an (n, k)-boundary presentation of H(t) = Q(t)/( t) d are uniquely determined (3) In the rest of this section we will make extensive use of the relation t i ( t) j = ti+ ( t) j + t i, j > ( t) j Repeated application of this relation allows us to transform an (n, k)- boundary presentation of a rational function H into an (n, k), respectively, (n, k + )-boundary presentation We give a formula for the coefficients of the new boundary presentation in terms of the old coefficients next

12 68 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA Lemma 35 Let H(t) = k f i t i ( t) n + ctk ( t) n + g j t k ( t) d j be an (n, k)-boundary presentation Then, there exists a corner-free (n, k+)-boundary presentation; its coefficients f (k+), g (k+) are given by { f (k+) f i for i =,, k i = c + r= g r for i = k g (k+) j = j g r for j =,, d n r= If n >, then there is also a corner-free (n, k)-boundary presentation with coefficients f (n ), g (n ) given by f (n ) i = g (n ) j = i f r for i =,, k r= { g j for j =,, d n c + k r= f r for j = d n In particular, an expansion of a corner-free boundary presentation leads to a boundary presentation with the entries next to the corner being equal Corollary 36 Let H(t) = k f i t i ( t) n + g j t k ( t) d j be a corner-free (n, k)-boundary presentation If k >, then there exists an (n, k )-boundary presentation; its coefficients f (k ), c (k ), g (k ) are given by f (k ) i = f i for i =,, k, c (k ) = f k g,

13 HILBERT REGULARITY OF Z-GRADED MODULES 69 g (k ) j = { g for j = g j g j for j =,, d n If n < d, then there is also an (n +, k)-boundary presentation with coefficients f (n+), c (n+), g (n+) given by { f (n+) f for i = i = f i f i for i =,, k, c (n+) = g f k, g (n+) j = g j for j =,, d n Corollary 37 If a rational function H admits an (n, k)-boundary presentation, then there is also an (n, k )-boundary presentation for every pair (n, k ) with n n, k k; for (n, k ) (n, k), this presentation is corner-free Moreover, the coefficients of this (n, k )- boundary presentation are nonnegative, provided that the same holds for the (n, k)-boundary presentation In particular, there exists an (n, k)-boundary presentation of the series Q(t)/( t) d for every k deg Q and n =,, d Note that, in these cases, the formula of Lemma 35 provides an alternative proof for the equality of the coefficients f i and the first coefficients of Q(t)/( t) d n Analogously, if d deg Q, then the (d deg Q, )-boundary presentation can be expanded to an (n, k)- boundary presentation for n =,, d deg Q and k, also confirming the description of the g j Lemma 38 An (n, k)-boundary presentation of a rational function H(t) = Q(t)/( t) d exists if and only if k n deg H = deg Q d It is corner-free if and only if k n > deg H; otherwise the entry in the corner is given by ( ) d n a q, where a q denotes the leading coefficient of Q Proof An (n, k)-boundary presentation with n, k > can be transformed into an (n, k )-boundary presentation, as the relation

14 7 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA (3) implies: k = f i t i ( t) n + ctk k + ( t) n + ( i ) r= f r t i ( t) n ctk ( c g + k r= f r g j t k ( t) d j ( t) n + ) t k ( t) n g j t k ( t) d j Hence, the existence of a non-corner-free (n, k)-boundary presentation of H for (n, k) with k n = deg Q n and the assertion on the entry in its corner follow by induction on d n beginning with the (d, deg Q)- boundary presentation of H The remainder is clear since an (n, k)- boundary presentation which is not corner-free cannot be obtained by expanding some (n, k )-boundary presentation with n n, k k Since any (n, k)-boundary presentation with k > deg Q can be obtained as an expansion of the (d, deg Q)-boundary presentation of Q(t)/( t) d, we obtain a second description of the coefficients g j : Proposition 39 Let H(t) = Q(t) k ( t) d = f i t i ( t) n + g (k) j t k ( t) d j with k > d Then, the coefficient g (k) j for j =,, d n agrees with the (k )th coefficient of the power series expansion of Q(t)/( t) j+ In particular, for Q(t)/( t) k = n a(k) n t n and Q(t)/( t) k = n b(k) n t n, we have b (k) j = a (j+) k for k deg Q and j =,, d Proof Let j d We consider the (d j, k)-boundary presentation of H with k > deg Q Since this can be viewed as an

15 HILBERT REGULARITY OF Z-GRADED MODULES 7 expansion of the corner-free (d, deg(q) + )-boundary presentation, Q(t) ( t) d + tdeg(q)+ ( t) d, we have f (d j) k = g (k) j ; thus, by Lemma 33, the coefficient g (k) j with the (k )th coefficient of Q(t) ( t) = Q(t) d (d j) ( t) j+ agrees Expanding the (d j, k)-boundary presentation downwards does not affect g (k) j ; therefore, this equality is also valid for any (n, k)-boundary presentation with n d j The second part follows immediately from Lemma 33 4 Arithmetical characterization of the Hilbert regularity In this section, we continue our investigation of the Hilbert regularity so we restrict our attention to nonnegative series H(t) = Q(t)/( t) d As mentioned above, such a series admits a Hilbert decomposition In particular, H is the Hilbert series of some finitely generated graded R-module M; we set Hdepth H := Hdepth M and Hreg H := Hreg M It is easy to see that H also admits a boundary presentation with nonnegative coefficients In the sequel, such a boundary presentation will be called nonnegative for short Lemma 4 Let H(t) = d i=n Q i (t) ( t) i be a Hilbert decomposition, and let k = max i deg Q i Then, there exists a nonnegative (n, k)-boundary presentation of H Proof Obviously, a Hilbert decomposition can be rewritten as (4) d i=n Q i (t) d ( t) i = k j=n a ij t i ( t) j with a ij N

16 7 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA It is sufficient to show that this decomposition can be turned into one of the form p k j=n b ij t i d ( t) j + j=p+ b kj t k ( t) j with b ij > for any p with n p d Repeated application of relation (3) yields k b ij t i k ( t) j = ( i r= b rj)t i ( t) j + ( k r= b rj)t k ( t) j Since the coefficients on the right-hand side are still nonnegative, the claim follows by reverse induction on p d, starting with the vacuous case p = d Corollary 4 (a) Let H(t) = Q(t)/( t) d be a nonnegative series Then, H admits a nonnegative (, Hreg H)-boundary presentation as well as a nonnegative (Hdepth H, k)-boundary presentation with suitable k (b) If H admits a non-corner-free (, k)-boundary presentation, then it holds that Hreg H k Proof Statement (a) is clear from the definition of Hreg H, respectively, Hdepth H Part (b) follows from (a) and the fact that a non corner-free boundary presentation cannot be obtained by expansion of another boundary presentation Remark 43 By Lemma 38, a (, k)-boundary presentation of H exists if and only if k deg H; together with Corollary 4 (a), this yields another proof of Proposition 8 Corollary 4 implies that, for computations of Hilbert regularity (and also of Hilbert depth), we may exclusively consider boundary presentations This observation leads to an estimate for Hreg M in the flavor of the equality p(m) = Hdepth M In order to formulate this inequality we need the following notion:

17 HILBERT REGULARITY OF Z-GRADED MODULES 73 Definition 44 For any Q Z[t] and k N, let Q(t)/( t) k = n a(k) n t n For any d N, we set { } δ d (Q) := min k N a (k),, a(k) d nonnegative and { Q(t) } δ(q) := min k N ( t) k nonnegative Note that δ d (Q) is finite if and only if the lowest nonvanishing coefficient of Q is nonnegative, as one easily sees by induction on d By [, Theorem 47], δ(q) is finite if and only if Q, viewed as a real-valued function of one variable, takes positive values in the open interval (, ) For a finitely generated graded R-module M with Hilbert series H M (t) = the equality Hdepth M = p(m) implies Q M (t) ( t) dim M, δ(q M ) = dim M Hdepth M; thus, according to [, Proposition 55] and the Auslander-Buchsbaum theorem, δ(q M ) could be called Hprojdim M, the Hilbert projective dimension Note that Hprojdim M depends only upon Q M but not upon dim M The above-mentioned estimate for the Hilbert regularity reads as follows: Proposition 45 Let H(t) = Q(t)/( t) d be a nonnegative series Then Hreg H δ d ( Q) Proof Since Q() = Q() >, δ d ( Q) is finite Let Hreg H = k Then there exists a (, k)-boundary presentation k H(t) = d f i t i + ct k + g j t k ( t) d j

18 74 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA with nonnegative coefficients By Lemma 33, the first d coefficients of Q(t)/( t) k agree with the coefficients g j, and therefore, they are nonnegative Hence, δ d ( Q) k = Hreg H Proposition 46 Under the hypothesis of Proposition 45, in addition, we have Hreg H δ( Q) Proof An (n, k)-boundary presentation of the series Q(t)/( t) d induces an (n+m, k)-boundary presentation of Q(t)/( t) d+m, m N, with the same coefficients The (, Hreg H)-boundary presentation of Q(t)/( t) d has nonnegative coefficients; hence, the same holds for the (m, Hreg H)-boundary presentation of Q(t)/( t) d+m, and, by Corollary 37, the (, Hreg H)-boundary presentation of Q(t)/( t) d+m is also nonnegative This implies δ d+m ( Q) Hreg H for all m N, and thus, δ( Q) Hreg H, as desired Theorem 47 Under the hypothesis of Proposition 45 and the additional assumption of either (i) δ d ( Q) deg Q or (ii) deg Q d, we have Hreg H = δ d ( Q) = δ( Q) Proof In both cases, expansion of the (d, deg Q), respectively, the (d deg Q, )-boundary presentation, yields a (, δ d ( Q))-boundary presentation of H, which is nonnegative by the nonnegativity of H and the definition of δ d ( Q) Hence, δ d ( Q) Hreg H δ( Q) δ d ( Q) The next example shows that, contrary to Hdepth M p(m) in case of the Hilbert depth, the inequality Hreg H δ d ( Q) may be strict Example 48 For H(t) = t + t t 3 + t 4 ( t),

19 HILBERT REGULARITY OF Z-GRADED MODULES 75 we obtain Q(t) = Q(t) Therefore, Q(t) t = Q(t) t = + t + t + t 3 + t n n 4 implies δ ( Q) = = Hprojdim H The (, )-boundary presentation of H is given by H(t) = + t + t + t t + t ( t) Since this is not corner-free, Lemma 38 implies Hreg H = > = δ ( Q) In particular, the Hilbert regularity of Q(t)/( t) d depends on d: for H (t) = t + t t 3 + t 4 ( t) d with d 4, we have Hreg H = by Theorem 47 Example 48 also explains that non-negativity of Q(t)/( t) k for some k N does not ensure Hreg H k The decomposition Q(t) k ( t) k = Q i (t) ( t) i with nonnegative Q i turned into one of Z[t] according to [, Theorem ] can be Q(t) ( t) max{deg Q i } by exchanging t and t, but, if d < max{deg Q i }, this does not yield a decomposition of Q(t)/( t) d Due to the difficulty illustrated by the previous example the general description of Hilbert regularity is less straightforward than that of Hilbert depth In the remaining case of deg Q > d, δ( Q), the (, deg Q)- boundary presentation is nonnegative, and hence, Hreg H deg Q If Hreg H < deg Q, then the (, deg Q)-boundary presentation can be reduced to a nonnegative (, k)-boundary presentation with smaller k Such a reduction may be performed in steps; therefore, we investigate whether a reduction from k to k is possible in what follows

20 76 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA Proposition 49 Let H(t) = k f i t i ( t) n + ctk ( t) n + g j t k ( t) d j with nonnegative coefficients Then, c = Hreg H k f k g g j+ g j for j =,, d n Proof Let Hreg H k Then there exists a boundary presentation (4) H(t) = k f i ti ( t) n + c t k ( t) n + g j tk ( t) d j with nonnegative coefficients By Lemma 35, this presentation may be transformed into k f i t i H(t) = ( t) n + (c + g j )tk ( j ( t) n + g i )tk ( t) d j, and, by uniqueness of the (n, k)-boundary presentation, we have f k = c + g j g j = g Necessity of the other conditions was already noted in Corollary 4 (b) and Proposition 45 If the conditions on the right are satisfied then Corollary 36 yields a nonnegative (, k )-boundary presentation (4) The (, Hreg H)-boundary presentation can be achieved by iterated reduction steps starting from the (, deg Q)-boundary presentation The reduction continues as long as the conditions of Proposition 49 remain valid Hence, it ends in one of the three cases illustrated by the diagrams in Figure 3

21 HILBERT REGULARITY OF Z-GRADED MODULES 77 b d b d a a k a < a k δ d δ d b j b j+ < b d a a k δ d Figure 3 Construction of the (, Hreg H)-boundary presentation may be described as follows Beginning with k = deg Q, we consider the (, k)- boundary presentation As long as k > δ d ( Q) and f k = g (k), there is also a nonnegative and corner free (, k )-boundary presentation; thus, we continue with k instead of k As soon as k = δ d ( Q) or f k g (k), we have reached the minimal k for which a nonnegative and corner-free (, k)-boundary presentation exists If k = δ d ( Q) or f k < g (k), no further reduction is possible; hence, Hreg H = k However, if k > δ d ( Q) and f k g (k), one last reduction step, leading to a non corner-free boundary presentation, may be performed;

22 78 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA thus, Hreg H = k in this case Note that, here, Lemma 38 implies Hreg H = deg H Theorem 4 Let H(t) = Q(t)/( t) d = n a nt n be a nonnegative series with d >, and let Q(t)/( t) j j N (i) If deg Q d or δ d ( Q) deg Q, then Hreg H = δ d ( Q) (ii) Otherwise, with we have k := min{i δ d ( Q) i deg Q Hreg H = a j = b (j+) d { k k and for all j = i,, deg Q}, if k = δ d ( Q) a k < b (k) d if k > δ d ( Q) a k > b (k) d = n b(j) n t n for Proof The cases in (i) were already treated in Theorem 47 Part (ii) follows from the discussion preceding this theorem; the number k, which is well defined by Proposition 39, is merely the width of the minimal nonnegative and corner-free boundary presentation The closing result of this section is the analogue of Proposition 46 for δ(q) Lemma 4 Let H(t) = Q(t)/( t) d be nonnegative and Then, δ(q) = δ e (Q) e := max{δ d ( Q), deg (Q) + } Proof The (d δ e (Q), δ e (Q))-boundary presentation of the series H is nonnegative by Lemma 33, and the definition of δ d ( Q) and δ e (Q) Hence the (d δ e (Q), δ e+m (Q))-boundary presentation with m is nonnegative as well, but this implies that δ e+m (Q) δ e (Q) for all m N Therefore, δ(q) = δ e (Q)

23 HILBERT REGULARITY OF Z-GRADED MODULES 79 5 Computation of Hilbert depth and Hilbert regularity The aim of this section is an algorithm for computing the Hilbert depth and Hilbert regularity of a module with given Hilbert series H(t) = Q(t)/( t) d, see Algorithm 5 An algorithm solely for the Hilbert depth was given by Popescu [] The correctness of Algorithm 5 follows immediately from the previous results The output could easily be extended by boundary presentations realizing Hdepth or Hreg since the required coefficients are computed in the course; for example, a nonnegative boundary presentation of the minimal height Hdepth H is given by d h a e a (h) i t (j+) e te i ( t) d j for e = deg Q > δ d ( Q) H(t) = ( t) h + d h b (δ d( Q)) j t e ( t) d j for e = δ d ( Q) deg Q, with a and b used as in the description of the algorithm, and h := Hprojdim H For completeness, we give an upper bound for the number of repetitions of the loop in the second step of Algorithm 5 The idea is to replace Q(t) = i q it i with a polynomial q + rt such that, for all n, i N, the coefficient c (k) n of ( q + rt)/( t) k is not greater than the coefficient b (k) n of Q(t)/( t) k Such a polynomial may be obtained by repeated application of the map f = m h i t i m h i t i + min{h m, h m + h m }t m to the polynomial Q Since q + rt ( t) k = [ ( ) ( n + k n + k q + r n n n = n )] t n [ k (n + j) ( q (n + k ) + r(n )) k! ] t n,

24 8 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA Algorithm 5: Computing Hilbert depth and Hilbert regularity Input: Q Z[t], d Z with H(t) = Q(t)/( t) d nonnegative Q(t) := Q( t); - - Determine δ d ( Q): k := ; repeat k := k + ; Compute the first d coefficients b (k),, b(k) d of Q(t)/( t) k ; until b (k),, b(k) d nonnegative; δ d ( Q) = k; Determine Hprojdim H: e := max{δ d ( Q), deg(q) + }; k := ; repeat k := k + ; Compute the first e coefficients a (k),, a(k) e of Q(t)/( t)k ; until a (k),, a(k) e nonnegative; Hprojdim H = k; 4 Hdepth H = d Hprojdim H; Determine Hreg H: if deg Q d or δ d ( Q) deg Q then Hreg H = δ d ( Q); else Compute the ith coefficient a i of H for i =,, deg Q; Compute the (d )th coefficient b (j) d of Q(t) ( t) j for j = δ d ( Q),, deg Q; k := min{i δ d ( Q) i deg Q and a j = b (j+) d for all j = = i,, deg Q}; if a k b (k) d and k > δ d( Q) then Hreg H = k ; else Hreg H = k; end end Output: Hdepth H, Hreg H

25 HILBERT REGULARITY OF Z-GRADED MODULES 8 we want to determine the least k such that (5) q (n + k ) + r(n ) = ( q + r)(n ) + k q holds for n d Without loss of generality, we may assume that q + r < Then, (5) is equivalent to n q k q + r This inequality must be valid, in particular, for n = d, and thus, for k ( d)( q + r) q, the first d coefficients of ( q + rt)/( t) k Q(t)/( t) k are nonnegative and a fortiori those of Example 5 Let H(t) = 5t + t + 4t 3 ( t) 7 Then Q(t) = Q( t) = 9t + 3t 4t 3, and we find δ 7 ( Q) = 7 since Q(t) ( t) 5 = + t t 4t 3 + t 4 + 7t t 6 + Q(t) ( t) 6 = + 3t + t 3t 3 3t 4 + 4t 5 + 7t 6 + Q(t) ( t) 7 = + 5t + 6t + 3t 3 + t 4 + 4t t 6 + In order to determine the Hilbert depth we first compute the δ 7 ( Q) = 7 coefficients of Q(t)/( t) k for k Since Q(t) ( t) 5 = + 5t + 6t + 4t 3 + t 4 3t 5 + t 6 + Q(t) ( t) 6 = + 7t + 3t + 7t 3 + 7t 4 + 4t 5 + 4t 6 +, we have Hdepth H = 7 6 = The Hilbert regularity requires no further computations since deg Q = 3 < 7 = d, and thus, Hreg H =

26 8 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA δ 7 ( Q) = 7 Moreover, in this case, the boundary presentation H(t) = + 7t + 3t + 7t 3 + 7t 4 + 4t 5 + 4t 6 t + 4t7 ( t) + 3t7 ( t) 4 + 6t7 ( t) 5 + 5t7 ( t) 6 + t7 ( t) 7 simultaneously has the minimal height Hdepth H and the minimal width Hreg H Finally, we give two examples illustrating the remaining cases occurring in Algorithm 5 Example 5 Let H(t) = t + t3 ( t) Then, Q(t) = t + 3t t 3 and δ ( Q) = Since deg Q exceeds δ d ( Q) as well as d, the final loop of our algorithm applies By Q(t) ( t) = + t + t + t 3 + Q(t) ( t) = + t + Q(t) ( t) 3 = + t +, we find k = = δ ( Q) Hence, Hreg H = This example confirms that Hreg H = δ d ( Q) may also occur if deg Q > d, δ d ( Q) Example 53 For H(t) = t + t t 3 ( t), we have δ ( Q) =, and the calculations may be summarized by The third subcase of Hreg H > δ d ( Q), which leads to a non cornerfree (, Hreg H)-boundary presentation, was already shown in Example 48

27 HILBERT REGULARITY OF Z-GRADED MODULES Figure 4 H(t) = ( t + t 3 )/( t) Figure 5 H(t) = ( t + t t 3 )/( t) Acknowledgements The first author thanks the Mathematical Sciences Research Institute, Berkeley, CA, for support and hospitality during Fall when this work was started

28 84 W BRUNS, JJ MOYANO-FERNÁNDEZ AND J ULICZKA REFERENCES AM Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm Algebra (993), W Bruns and J Herzog, Cohen-Macaulay rings, Stud Adv Math 39, Cambridge University Press, Cambridge, W Bruns, C Krattenthaler and J Uliczka, Stanley decompositions and Hilbert depth in the Koszul complex, J Commutative Algebra (), J Elias, L Robbiano and G Valla, Number of generators of ideals, Nagoya Math J 3 (99), J Herzog, A survey on Stanley depth, in Monomial ideals, Computations and applications, A Bigatti, P Giménez and E Sáenz-de-Cabezón, eds, Lect Notes Math 83, Springer, New York, 3 6 HA Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm Algebra (993), B Ichim and JJ Moyano-Fernández, How to compute the multigraded Hilbert depth of a module, Math Nachr 87 (4), AS Jahan, Prime filtrations and Stanley decompositions of squarefree modules and Alexander duality, Manuscr Math 3 (9), JJ Moyano-Fernández and J Uliczka, Hilbert depth of graded modules over polynomial rings in two variables, J Algebra 373 (3), 3 5 A Popescu, An algorithm to compute the Hilbert depth, J Symb Comp 66 (5), 7 J Uliczka, Remarks on Hilbert series of graded modules over polynomial rings, Manuscr Math 3 (), Universität Osnabrück, Institut für Mathematik, 4969 Osnabrück, Germany address: wbruns@uosde Universitat Jaume I, Departamento de Matemáticas & Institut Universitari de Matemàtiques i Aplicacions de Castelló, 7 Castellón de la Plana, Spain address: moyano@ujies Universität Osnabrück, Institut für Mathematik, 4969 Osnabrück, Germany address: juliczka@uosde