Maximal Depth Property of Bigraded Modules

. MAXIMAL DEPTH PROPERTY OF BIGRADED MODULES arXiv:2007.05744v1 [math.AC] 11 Jul 2020 AHAD RAHIMI Abstract. Let S = K[x1 , . . . , xm , y1 , . . . , yn ] be the standard bigraded polynomial ring over a field K. Let M be a finitely generated bigraded S-module and Q = (y1 , . . . , yn ). We say M has maximal depth with respect to Q if there is an associated prime p of M such that grade(Q, M ) = cd(Q, S/p). In this paper, we study finitely generated bigraded modules with maximal depth with respect to Q. It is shown that sequentially Cohen–Macaulay modules with respect to Q have maximal depth with respect to Q. In fact, maximal depth property generalizes the concept of sequentially Cohen–Macaulayness. Next, we show that if M has maxgrade(Q,M) (M ) is not imal depth with respect to Q with grade(Q, M ) > 0, then HQ finitely generated. As a consequence, ”generalized Cohen–Macaulay modules with respect to Q” having ”maximal depth with respect to Q” are Cohen–Macaulay with respect to Q. All hypersurface rings that have maximal depth with respect to Q are classified. Introduction Let K be a field and S = K[x1 , . . . , xm , y1, . . . , yn ] be the standard bigraded polynomial ring over K. In other words, deg xi = (1, 0) and deg yj = (0, 1) for all i and j. We set the bigraded irrelevant ideals P = (x1 , . . . , xm ) and Q = (y1 , . . . , yn ). Let M be a finitely generated bigraded S-module. The author has been studying the algebraic properties of a finitely generated bigraded S-module M with respect to Q, see for instance [9], [13], [14]. We denote by cd(Q, M) the cohomological dimension of M with respect to Q which is the largest integer i for which HQi (M) 6= 0. A classical fact in commutative algebra ([9]) says that grade(Q, M) ≤ min{cd(Q, S/p) : p ∈ Ass(M)}. We set mgrade(Q, M) = min{cd(Q, S/p) : p ∈ Ass(M)}. We say M has maximal depth with respect to Q if the equality holds, i.e., grade(Q, M) = mgrade(Q, M). In other words, there is an associated prime p of M such that grade(Q, M) = cd(Q, R/p). Some examples of modules with maximal depth with respect to Q are given in Example 1.3. In this paper, the author studies depth property for finitely generated bigraded modules. We let P = 0 and consider R = K[y1 , . . . , yn ] as standard graded polynomial ring. Then, M as ordinary graded R-module has maximal depth if depth M = mdepth M where mdepth M = min{dim R/p : p ∈ Ass(M)}. This concept has already been working with several authors. Some known results in this regard are as follows: If I ⊆ R is a generic monomial ideal, then it has maximal depth, see [11, Theorem 2010 Mathematics Subject Classification. 13C14, 13C15, 16W50, 13F20, 13D45. Key words and phrases. Maximal depth, Sequentially Cohen–Macaulay, Generalized Cohen– Macaulay, Local cohomology, Monomial ideal, Hypersurface ring. 1 2.2]. If a monomial ideal I has maximal depth, then so does its polarization, see [6]. Edge ideals of cycle graphs, transversal polymatroidal ideals and high powers of connected bipartite graph with maximal depth property are classified in [16]. The paper is organized as follows: In the preliminary section, we give some facts about mgrade(Q, M). Let F : 0 = M0 M1 ··· Md = M be the dimension filtration of M with respect to Q. We observe that all the Mi have the same mgrade with respect to Q, namely this number is cd(Q, M1 ). From this fact, we deduce that if M is sequentially Cohen–Macaulay with respect to Q, then M has maximal depth with respect to Q, see Proposition 1.6. Of course this class is rather large. In Example 1.7, the ring R is not sequentially Cohen–Macaulay with respect to Q and has maximal depth with respect to Q. If P = 0, we deduce that the ordinary sequentially Cohen–Macaulay modules have maximal depth, see [15]. Let (R, m) be a Noetherian local ring and M a finitely generated R-module. This is a known fact that if there exists p ∈ Ass(M) such that dim R/p = j > 0, then Hmj (M) is not finitely generated. Inspired by this fact, we may ask the following question: Question 0.1. Assume that there exists p ∈ Ass(M) such that cd(Q, S/p) = j > 0. Does it follow that HQj (M) is not finitely generated? In Section 2, we give a positive answer to this question in a particular case. We first show that if M has maximal depth with respect to Q with grade(Q, M) > 0 and |K| = ∞, then there exists a bihomogeneous M-regular element y ∈ Q of degree (0, 1) such that M/yM has maximal depth with respect to Q. An example is given to show that the converse is not true in general. This fact is used to answer the above question in the following case: if M has maximal depth with respect to Q with grade(Q,M ) grade(Q, M) > 0, then HQ (M) is not finitely generated, see Theorem 2.4. As a consequence, ”generalized Cohen–Macaulay modules with respect to Q” with ”maximal depth with respect to Q” are Cohen–Macaulay with respect to Q. In fact, we show: If M is generalized Cohen–Macaulay with respect to Q with grade(Q, M) > 0, then M has maximal depth with respect to Q is equivalent to say that ”M is sequentially Cohen–Macaulay with respect to Q” and this is the same as ”M is Cohen–Macaulay with respect to Q”. If P = 0, we deduce that ordinary generalized Cohen–Macaulay modules with maximal depth are Cohen–Macaulay, see [15]. In the following section, we let I ⊆ S be a monomial ideal. It is shown that mgrade(Q, S/I) = n − d where d is the maximal height of an associated prime of I in Q. Moreover, if S/I is Cohen–Macaulay, then S/I has maximal depth with respect to P and Q. We also show that the maximal depth property is preserved under tensor product and direct sum. In the final section, we classify all hypersurface rings that have maximal depth with respect to Q. 1. Preliminaries Let K be a field and S = K[x1 , . . . , xm , y1, . . . , yn ] be the standard bigraded polynomial ring over K. In other words, deg xi = (1, 0) and deg yj = (0, 1) for all i 2 and j. We set the bigraded irrelevant ideals P = (x1 , . . . , xm ) and Q = (y1 , . . . , yn ). Let M be a finitely generated bigraded S-module. We denote by cd(Q, M) the cohomological dimension of M with respect to Q which is the largest integer i for which HQi (M) 6= 0. Let |K| = ∞. In [9, Proposition 1.7] it is shown that (1) grade(Q, M) ≤ min{cd(Q, S/p) : p ∈ Ass(M)}. We set mgradeS (Q, M) = min{cd(Q, S/p) : p ∈ Ass(M)}. For simplicity, we write mgrade(Q, M) instead of mgradeS (Q, M). We recall the following facts which will be used in the sequel. Fact 1.1. The following statements hold. (a) cd(Q, M) = max{cd(Q, S/p) : p ∈ Ass(M)}, see [3, Corollary 4.6]. (b) cd(P, M) = dim M/QM and cd(Q, M) = dim M/P M, see [13, Formula 3]. (c) grade(Q, M) ≤ dim M − cd(P, M), and the equality holds if M is Cohen– Macaulay, see [13, Formula 5]. (d) grade(Q, M) = 0 if and only if there exists p ∈ Ass(M) such that Q ⊆ p. Observe that grade(Q, M) ≤ mgrade(Q, M) ≤ cd(Q, M) ≤ dim M. Fact 1.1(a) provides the second inequality. Note that grade(Q, M) = 0 if and only if mgrade(Q, M) = 0. Thus, if mgrade(Q, M) = 1, then grade(Q, M) = 1. Definition 1.2. We say M has maximal depth with respect to Q if the equality (1) holds, i.e., grade(Q, M) = mgrade(Q, M). In other words, there is an associated prime p of M such that grade(Q, M) = cd(Q, S/p). Example 1.3. Some examples of modules with maximal depth property are as follows: • Let q ∈ Z. In [13], we say M is Cohen–Macaulay with respect to Q if we have only one non-vanishing local cohomology. In other words, grade(Q, M) = cd(Q, M) = q. Cohen–Macaulay modules with respect to Q have maximal depth with respect to Q because grade(Q, M) = cd(Q, S/p) for every associated prime p of M. • If cd(Q, M) ≤ 1, then M has maximal depth with respect to Q. • If grade(Q, M) = 0, then M has maximal depth with respect to Q. In fact, Fact 1.1(d) provides an associated prime p of M such that Q ⊆ p. Hence cd(Q, S/p) = dim S/(P + p) = dim S/(P + Q) = 0. The first equality follows from Fact 1.1(b). Therefore, M has maximal depth with respect to Q. A finite filtration D: 0 = D0 D1 · · · Dr = M of bigraded submodules of M is the dimension filtration of M with respect to Q if Di−1 is the largest bigraded submodule of Di for which cd(Q, Di−1 ) < cd(Q, Di ) for all i = 1, . . . , r. Fact 1.4. let D be the dimension filtration of M with respect to Q. Then (a) Ass(Di ) = {p ∈ Ass(M) : cd(Q, S/p) ≤ cd(Q, Di )}, see [10, Lemma 1.7]. 3 (b) Ass(M/Di ) = Ass(M) − Ass(Di ), see [10, Corollary 1.10]. Lemma 1.5. Let D be the dimension filtration of M with respect to Q. Then Ass(Di /Di−1 ) = {p ∈ Ass(M) : cd(Q, S/p) = cd(Q, Di )}. In particular, (2) Ass(M) = r [ Ass(Di /Di−1 ). i=1 Proof. We set A = {p ∈ Ass(M) : cd(Q, S/p) = cd(Q, Di )}. Let p ∈ Ass(Di /Di−1 ). By Fact 1.1(a) we have cd(Q, S/p) ≤ cd(Q, Di /Di−1 ) = cd(Q, Di ). The embedding 0 → Di /Di−1 → M/Di−1 also yields p ∈ Ass(M/Di−1 ). Hence p ∈ Ass(M) and p 6∈ Ass(Di−1 ) by Fact 1.4(b). Consequently, cd(Q, Di−1 ) < cd(Q, S/p) ≤ cd(Q, Di ). Fact 1.4(a) provides the first inequality. It follows that cd(Q, S/p) = cd(Q, Di ) and hence p ∈ A. Now let p ∈ A. Thus cd(Q, S/p) = cd(Q, Di ) and so p ∈ Ass(Di ) by Fact 1.4(a). As p 6∈ Ass(Di−1 ), the containment Ass(Di ) ⊆ Ass(Di−1 ) ∪ Ass(Di /Di−1 ) implies p ∈ Ass(Di /Di−1 ).  A finite filtration F : 0 = M0 M1 ··· Mr = M of M by bigraded submodules M is called a Cohen–Macaulay filtration with respect to Q if each quotient Mi /Mi−1 is Cohen–Macaulay with respect to Q and 0 ≤ cd(Q, M1 /M0 ) < cd(Q, M2 /M1 ) < · · · < cd(Q, Mr /Mr−1 ). If M admits a Cohen–Macaulay filtration with respect to Q, then we say M is sequentially Cohen–Macaulay with respect to Q. Note that if M is sequentially Cohen–Macaulay with respect to Q, then the filtration F is uniquely determined and it is just the dimension filtration of M with respect to Q, that is, F = D, see [14]. Proposition 1.6. Let F : 0 = M0 M1 ··· Md = M be the dimension filtration of M with respect to Q. Then, mgrade(Q, Mi ) = cd(Q, M1 ) for i = 1, . . . , d. Moreover, if M is sequentially Cohen–Macaulay with respect to Q, then M has maximal depth with respect to Q. Proof. We first show that mgrade(Q, M) = cd(Q, M1 ). We set mgrade(Q, M) = t. Thus there exists p ∈ Ass(M) such that cd(Q, S/p) = t. Hence p ∈ Ass(Mi /Mi−1 ) for some i by (2). Thus cd(Q, S/p) = cd(Q, Mi ) = t again by (2). Note that t = mgrade(Q, M) ≤ cd(Q, M1 ). If t < cd(Q, M1 ), then cd(Q, Mi ) < cd(Q, M1 ) for some i, a contradiction. Therefore, mgrade(Q, M) = cd(Q, M1 ). Now we observe that t = mgrade(Q, M) ≤ mgrade(Q, Md−1 ) ≤ · · · ≤ mgrade(Q, M1 ) ≤ cd(Q, M1 ) = t. Consequently, mgrade(Q, Mi ) = t for i = 1, . . . , d. To show the second part, let M be sequentially Cohen–Macaulay with respect to Q. Thus the dimension filtration of F with respect to Q is the Cohen–Macaulay filtration with respect to Q. As M1 is Cohen–Macaulay with respect to Q, it has maximal depth with respect to Q and so grade(Q, M1 ) = mgrade(Q, M1 ). Since M 4 is sequentially Cohen–Macaulay with respect to Q, it follows that grade(Q, Mi ) = grade(Q, M) for all i, see [14, Fact 2.3]. Using the first part, we have grade(Q, M) = grade(Q, M1 ) = mgrade(Q, M1 ) = mgrade(Q, M), as desired.  In the following, we give an example which is not sequentially Cohen–Macaulay with respect to Q and has maximal depth with respect to Q. Example 1.7. Let S = K[x1 , x2 , y1 , y2, y3 , y4 ] be the standard bigraded polynomial ring. We set R = S/I where I = (x1 x2 , x1 y3 , x1 y4 , x2 y1 , y1y3 , y1 y4 , y2 y4 , y2y3 ) and Q = (y1 , y2 , y3 , y4). By using CoCoA([4]), the ideal I has the minimal priT mary decomposition I = 3i=1 pi where p1 = (x1 , y1 , y2 ), p2 = (x2 , y3 , y4) and p3 = (x1 , y1 , y3, y4 ). Fact 1.1(a) provides mgrade(Q, R) = 1. Thus grade(Q, R) = 1 and so R has maximal depth with respect to Q. On the other hand, R is not sequentially Cohen–Macaulay with respect to Q, see [12, Example 2.15]. 2. Not finitely generated local cohomology modules Let (R, m) be a Noetherian local ring and M a finitely generated R-module. This is a known fact that if there exists p ∈ Ass(M) such that dim R/p = j > 0, then Hmj (M) is not finitely generated, see [1, Corollary 11.3.3] and [1, Exercise 11.3.9]. Inspired by this fact, we may ask the following question: Question 2.1. Assume that there exists p ∈ Ass(M) such that cd(Q, S/p) = j > 0. Does it follow that HQj (M) is not finitely generated? In this section, we have a positive answer for this question in a particular case. First, we prove the following crucial lemma: Lemma 2.2. Suppose grade(Q, M) > 0 and |K| = ∞. If M has maximal depth with respect to Q, then there exists a bihomogeneous M-regular element y ∈ Q of degree (0, 1) such that M/yM has maximal depth with respect to Q. Proof. Here we follow the proof of [9, Proposition 1.7]. By our assumption, there exists p ∈ Ass(M) such that grade(Q, M) = cd(Q, S/p). As grade(Q, M) > 0, there exists a bihomogeneous M-regular element y ∈ Q such that grade(Q, M/yM) = grade(Q, M) − 1. The element p ∈ Ass(M) is properly contain in an element q ∈ Ass(M/yM). The element y may be chosen to avoid all the minimal prime ideal of Supp(S/(P + p)), too. Observe that grade(Q, M) − 1 = ≤ = < = = 5 grade(Q, M/yM) cd(Q, S/q) dim S/(P + q) dim S/(P + p) cd(Q, S/p) grade(Q, M). Consequently, (3) grade(Q, M/yM) = cd(Q, S/q) where q ∈ Ass(M/yM). Therefore, M/yM has maximal depth with respect to Q.  The following example shows that the converse of Lemma 2.2 is not true in general. Example 2.3. Let S = K[x1 , x2 , y1, y2 ] be the standard bigraded polynomial ring. We set R = S/(f ) where f = x1 y1 + x2 y2 and Q = (y1 , y2). The ring R is Cohen– Macaulay of dimension 3 and grade(Q, R) = dim R − cd(P, R) = 3 − 2 = 1 > 0 by Fact 1.1(c). The bihomogenous element y = y1 + y2 ∈ S is R-regular. Consider the following isomorphism R/yR ∼ = S/(f, y). Using Macaulay2 ([5]) gives us the associated primes of T = S/(f, y) that is {(y1 , y2 ), (y1 + y2 , x1 − x2 )}. Since T has an associated prime contains Q, it follows from Fact 1.1(d) that grade(Q, T ) = grade(Q, R/yR) = 0. Therefore, R/yR has maximal depth with respect to Q. On the other hand, as Ass(R) = {(f )} we have mgrade(Q, R) = cd(Q, R) = dim R/P R = dim S/P = 2. Fact 1.1(b) explains the second equality. As grade(Q, R) = 1, the ring R has no maximal depth with respect to Q. Theorem 2.4. Assume M has maximal depth with respect to Q with grade(Q, M) > grade(Q,M ) 0 and |K| = ∞. Then HQ (M) is not finitely generated. Proof. We set grade(Q, M) = r. We proceed by induction on r. Our assumption says that there exists p ∈ Ass(M) such that grade(Q, M) = cd(Q, S/p). Suppose r = 1. The exact sequence 0 → S/p → M → U → 0 yields the exact sequence 0 → HQ0 (U) → HQ1 (S/p) → HQ1 (M) → HQ1 (U) → 0. This is a known fact that for cd(Q,N ) every finitely generated S-module N, the local cohomology module HQ (N) is not finitely generated. Now suppose HQ1 (M) is finitely generated. Then the exact sequence 0 → HQ0 (U) → HQ1 (S/p) → K → 0 where K is a submodule of HQ1 (M) provides HQ1 (S/p) is finitely generated, a contradiction. Therefore, HQ1 (M) is not finitely generated. Now suppose r ≥ 2 and that the statement holds for all modules N which has maximal depth with respect to Q and grade(Q, N) < r. We want to prove it for M which has maximal depth with respect to Q with grade(Q, M) = r. By Lemma 2.2, there exists a bihomogeneous M-regular element y ∈ Q of degree (0, 1) such that M/yM has maximal depth with respect to Q. In fact, by (3) we have, grade(Q, M/yM) = cd(Q, S/q) for some q ∈ Ass(M/yM). y The exact sequence 0 → M → M → M/yM → 0 yields the exact sequence y 0 → HQr−1 (M/yM) → HQr (M) → HQr (M). Our induction hypothesis provides 6 HQr−1 (M/yM) is not finitely generated. Therefore, HQr (M) is not finitely generated too, as desired.  Let M a finitely generated bigraded S-module. We say M is generalized Cohen– Macaulay with respect to Q if the local cohomology module HQi (M) is finitely generated for all i < cd(Q, M). As a consequence, we have the following Corollary 2.5. Suppose M is generalized Cohen–Macaulay with respect to Q with grade(Q, M) > 0 and |K| = ∞. Then the following statements are equivalent: (a) M has maximal depth with respect to Q, (b) M is sequentially Cohen–Macaulay with respect to Q, (c) M is Cohen–Macaulay with respect to Q. Proof. The implication (c) ⇒ (b) is obvious. The implication (b) ⇒ (a) follows from Proposition 1.6. For (a) ⇒ (c), we need to show grade(Q, M) = cd(Q, M). Suppose grade(Q, M) < cd(Q, M). As M is generalized Cohen–Macaulay with respect to Q, grade(Q,M ) we have that HQ (M) is finitely generated. This contradicts with Theorem 2.4. Therefore, grade(Q, M) = cd(Q, M), as desired.  Example 2.6. Let S = K[x1 , x2 , y1 , y2 , y3 , y4] be the standard bigraded polynomial ring and set Q = (y1 , y2 , y3 , y4). We set R = S/(p1 ∩ p2 ) where p1 = (x1 , y1 , y2) and p2 = (x2 , y3 , y4). One has cd(Q, R) = 2. The exact sequence 0 → R → S/p1 ⊕ S/p2 → S/m → 0 yields HQ0 (R) = 0 and HQ1 (R) ∼ = S/m. Here = HQ0 (S/m) ∼ 1 m is the unique maximal ideal of S. Hence HQ (R) is finitely generated. Therefore, R is generalized Cohen–Macaulay with respect to Q. As grade(Q, R) = 1 and mgrade(Q, R) = 2, the ring R has no maximal depth with respect to Q. 3. Monomial ideal, tensor product and direct sum In the following, we discuss the maximal depth property of monomial ideals. Proposition 3.1. Let I ⊂ S = K[x1 , . . . , xm , y1 , . . . , yn ] be a monomial ideal. The following statements hold: (a) If the maximal height of an associated prime of I in Q is d, then mgrade(Q, S/I) = n − d. (b) If S/I is Cohen–Macaulay, then S/I has maximal depth with respect to P and Q. T Proof. (a): Let I = ri=1 qi be an irredundant irreducible decomposition of I where qi are pi -primary monomial ideals, see [8]. We may write qi = qxi + qyi where qxi = (xαi11 , . . . , xαikk ) and qyi = (yiβ11 . . . , yiβss ) are the monomial ideals in K[x] and √ √ K[y],prespectively. We set qi = pi = pxi + pyi for all i where pxi = qxi and pyi = qyi . The ideal I has the irredundant irreducible decomposition where I = (q1 ∩ · · · ∩ qa1 ) ∩ · · · ∩ (qar−1 +1 ∩ · · · ∩ qat ) height pyai−1 +1 = · · · = height pyai = dyi 7 for i ∈ {1, . . . , t}; assuming a0 = 0 and dy1 < dy2 < · · · < dyt . Observe that mgrade(Q, S/I) = = = = = min{cd(Q, S/p) : p ∈ Ass(S/I)} min{dim S/(P + p) : p ∈ Ass(S/I)} min{dim S/(P + py ) : p ∈ Ass(S/I)} min{dim K[y]/py : p ∈ Ass(S/I)} n − dyt . Fact 1.1(b) provides the second step in this sequence and the remaining steps are standard. (b): We show that S/I has maximal depth with respect to Q. The argument for P is similar. Since S/I is Cohen–Macaulay, it follows that dxt < · · · < dx2 < dx1 where height pxai−1 +1 = · · · = height pxai = dxi and dxi + dyi = height pi . Observe that grade(Q, S/I) = = = = for i ∈ {1, . . . , t}; dim S/I − cd(P, S/I) m + n − (dxt + dyt ) − (m − dxt ) n − dyt mgrade(Q, S/I). By Fact 1.1(c) explains the first step and the forth step follows from Part(a). The remaining steps are obvious.  Remark 3.2. The following example shows that Proposition 3.1(b) is no longer true if I is not a monomial ideal. Consider the hypersurface ring R = K[x1 , x2 , y1, y2 ]/(f ) where f = x1 y1 + x2 y2 . The ring R is Cohen–Macaulay of dimension 3. By Fact 1.1(c) we have grade(Q, R) = dim R − cd(P, R) = 3 − 2 = 1. As Ass(R) = {(f )}, then mgrade(Q, R) = cd(Q, R) = 2. Thus R has no maximal depth with respect to Q. The converse Proposition 3.1(b) does not hold in general. We set R = S/(p1 ∩ p2 ) where S = K[x1 , x2 , y1 , y2 ], p1 = (x1 , y1 ), p2 = (x2 , y2 ) and Q = (y1 , y2 ). One has grade(Q, R) = mgrade(Q, R) = 1 and grade(P, R) = mgrade(P, R) = 1. Thus R has maximal depth with respect to P and Q. The ring R is not Cohen–Macaulay. In fact, dim R = 2 and depth R = 1. Note also that the ring R has maximal depth with respect to P and Q, but not maximal depth with respect to P + Q = m. In the following, we show that the maximal depth property with respect to Q is preserved under tensor product and direct sum. We first recall the following fact. We set K[x] = K[x1 , . . . , xm ] and K[y] = K[y1 , . . . , yn ]. Fact 3.3. Let K be an algebraically closed field. Let L and N be two non-zero finitely generated graded modules over K[x] and K[y], respectively. We set M = L ⊗K N and consider M as an S-module. Then AssS (M) = {p1 + p2 : p1 ∈ AssK[x](L) and p2 ∈ AssK[y](N)}, see [7, Corollary 2.8]. 8 Proposition 3.4. Continue with the notation and assumptions as above. Then, M has maximal depth with respect to Q if and only if N has maximal depth. Proof. Note that grade(Q, M) = depthK[y] N. Let p ∈ Ass(M). By Fact 3.3, there exist p1 ∈ AssK[x](L) and p2 ∈ AssK[y] (N) such that p = p1 + p2 . Observe that mgrade(Q, M) = = = = = min{cd(Q, S/p) : p ∈ Ass(M)} min{dim S/(P + p) : p ∈ Ass(M)} min{dim S/(P + p2 ) : p2 ∈ Ass(N)} min{dim K[y]/p2 ) : p2 ∈ Ass(N)} mdepthK[y] N. Fact 1.1(b) explains the second step in this sequence and the remaining steps are standard. Therefore, the assertion follows.  Proposition 3.5. Let M1 , . . . , Mn be finitely generated bigraded S-modules. Then L n i=1 Mi has maximal depth with respect to Q if and only if there exists j such that grade(Q, Mj ) ≤ grade(Q, Mk ) for all k and Mj has maximal depth with respect to Q. L has maximal depth with respect Proof. Suppose ni=1 MiL
Thus there exists Ln to Q. n an associated prime p of
i=1 Mi such that grade Q, i=1 Mi = cd(Q, S/p). Note L that grade Q, ni=1 Mi = grade(Q,
Sn Ms ) where grade(Q, Ms ) ≤ grade(Q, Mk ) for Ln all k. Since p ∈ Ass i=1 Mi = i=1 Ass(Mi ), it follows that p ∈ Ass(Ms ) where grade(Q, Ms ) ≤ grade(Q, Mk ) for all k. Indeed, otherwise p ∈ Ass(Ml ) for some l and there exists h such that grade(Q, Mh ) < grade(Q, Ml ). Hence grade(Q, Ms ) ≤ < ≤ = grade(Q, Mh ) grade(Q, Ml ) cd(Q, S/p) grade(Q, Ms ), a contradiction. The third inequality follows from (1). Therefore, the conclusion follows. The other implications are obvious.  4. Hypersurface rings with maximal depth In the following, we classify all hypersurface rings that have maximal depth with respect to Q. Let f ∈ S be a bihomogeneous element of degree (a, b) and consider the hypersurface ring R = S/f S. We may write X cαβ xα y β where cαβ ∈ K. f= |α|=a |β|=b Note that R is a Cohen–Macaulay S-module of dimension m + n − 1. Theorem 4.1. Let f ∈ S be a bihomogeneous element of degree (a, b) and R = S/f S be the hypersurface ring. Then R has maximal depth with respect to Q if and only if one of the following conditions holds true 9 (a) f = h1 h2 h3 where deg h1 = (α1 , 0) with α1 > 0, deg h2 = (α2 , β1 ) with α2 , β1 > 0, deg h3 = (0, β2 ) with β2 > 0 and α1 + α2 = a and β1 + β2 = b; (b) f = h2 h3 where deg h2 = (α2 , β1 ) with a = α2 , β1 > 0, deg h3 = (0, β2 ) with β2 > 0 and β1 + β2 = b; (c) f = h1 h3 where deg h1 = (a, 0) with a ≥ 0 and deg h3 = (0, b) with b ≥ 0. Q Proof. Let f = ri=1 fi be the unique factorization of f into bihomogeneous irQs−1 reducible factors fi . We may write h1 = i=1 fi where deg fi = (ai , 0) with Qt ai ≥ 0 for i = 1, . . . , s − 1, h2 =Q i=s fi where deg fi = (ai , bi ) with ai , bi > 0 for i = s, s + 1, . . . , t and h3 = ri=t+1 fi where deg fi = (0, bi ) with bi ≥ 0 for P P Ps−1 i = t + 1, . . . , r. Note that ri=1 ai = a and ri=1 bi = b. We set i=1 ai = α1 , Pr Pt Pt b = β . Thus α + α = a and β + β2 = b. b = β and a = α , 2 1 2 1 1 2 i=t+1 i i=s i i=s i We consider several cases: Case 1: Suppose that α1 , α2 , β1 , β2 > 0. Then cd(P, R) = = = = = max{cd(P, S/p) : p ∈ Ass(R)} max{dim S/(Q + (fi )) : i = 1, . . . , r} max{dim S/(Q + (fi )) : i = s, . . . , r} dim S/Q m. Fact 1.1(b) explains the second step. Hence by Fact 1.1(c) we have grade(Q, R) = dim R − cd(P, R) = m + n − 1 − m = n − 1. On the other hand, mgrade(Q, R) = = = = min{cd(Q, S/p) : p ∈ Ass(R)} min{dim S/(P + (fi )) : i = 1, . . . , r} min{dim S/(P + (fi )) : i = t, . . . , r} n − 1. Therefore, R has maximal depth with respect to Q. Case 2: Suppose that α1 = 0 and β1 , β2 , α2 = a > 0. Then cd(P, R) = = = = max{cd(P, S/p) : p ∈ Ass(R)} max{dim S/(Q + (fi )) : i = s, . . . , r} dim S/Q m. Fact 1.1(c) provides grade(Q, R) = dim R − cd(P, R) = m + n − 1 − m = n − 1. 10 On the other hand, mgrade(Q, R) = = = = min{cd(Q, S/p) : p ∈ Ass(R)} min{dim S/(P + (fi )) : i = s, . . . , r} min{dim S/(P + (fi )) : i = t, . . . , r} n − 1. Thus, R has maximal depth with respect to Q. Case 3: If α2 = β1 = 0 and α1 , β2 > 0, then grade(Q, R) = mgrade(Q, R) = n − 1, and so R has maximal depth with respect to Q, too. Case 4: Suppose that β2 = 0 and α1 , α2 , β1 = b > 0. A similar arguments as above shows that n − 1 = grade(Q, R) 6= mgrade(Q, R) = n. Thus , R has no maximal depth with respect to Q in this case. Case 5: Suppose α1 = β2 = 0 and α2 = a > 0, β1 = b > 0. Then grade(Q, R) = dim R − cd(P, R) = m + n − 1 − m = n − 1, and mgrade(Q, R) = n. Hence R has no maximal depth with respect to Q. Now the desired conclusion follows from the above observations. References [1] M. Brodmann and R.Y. 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