Math. Proc. Camb. Phil. Soc. 121 (1997) 437–441 SHIFTS IN RESOLUTIONS OF MULTIGRADED MODULES
Srikanth Iyengar Abstract. Upper bounds are established on the shifts in a minimal resolution of a multigraded module. Similar bounds are given on the coefficients in the numerator of the BackelinLescot rational expression for multigraded Poincar´ e series.
Let K be a field and S = K[x1 , . . . , xn ] the polynomial ring with its natural n-grading. When I is an ideal generated by monomials in the variables x1 , . . . , xn , the ring R = S/I is n-graded for the induced grading. We denote by deg[y] the degree in Zn of a homogeneous element y in an n-graded R-module. 0 L An n-graded finite R-module M is known to have a free resolution with i th module j Rei,j , where the basis elements are n-homogeneous, and differentials ∂i of degree 0, given by matrices whose entries are monomials in the variables. Furthermore, the resolution may be chosen such that Ker ∂i−1 is minimally generated by {∂i (ei,j )}j . Such a minimal resolution is unique up to n-graded isomorphism. 1. Shifts For F a minimal resolution of M , the Betti numbers βi = rank(Fi ) and the shifts deg[ei,j ] = ai,j = (ai,j (1), . . . , ai,j (n)) ∈ Zn are invariants of M . We consider Zn with the product order: c < d if c(k) ≤ d(k) for all 1 ≤ k ≤ n and c 6= d. Every set {c1 , . . . , cm } ⊂ Zn has a least upper bound c = lub{c1 , . . . , cm }, where c(k) = max{c1 (k), . . . , cm (k)}. When a ∈ Zn , we denote the a(1) a(n) monomial x1 · · · xn by xa . We write buc for the greatest integer less than or equal to u and due for the least integer greater than or equal to u. In the recent paper [2], Bruns and Herzog have obtained upper bounds on the shifts in F. We present a new, more direct approach to this problem, which leads to sharper bounds in the case of infinite resolutions. Theorem 1. Let I be an ideal in the ring S = K[x1 , . . . , xn ], minimally generated by monomials xb1 , . . . , xbm and set R = S/I. For an n-graded finite R-module M with shifts ai,j , there are inequalities ai,j ≤ a +
i 2 b
for all
i, j ≥ 1,
1991 Mathematics Subject Classification. 13A02, 13D02. Typeset by AMS-TEX 1
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SRIKANTH IYENGAR
where b = lub{b1 , . . . , bm } and a = lub{a1,1 , . . . , a1,β1 }. If I 6= 0, then for each even i there is a k such that ai,j (k) < a(k) + 2i b(k).
The statement of Theorem 1 was suggested by the work of Bruns and Herzog [2]. As noted in [2, (3.2)], the case I = 0 of the theorem has the following consequence: Corollary. Let M be a finite n-graded module, over the polynomial ring S, with shifts ai,j . If cxa , where c ∈ K, is a non-zero entry of some ∂i in the multigraded resolution of M , then xa | lcm(xa1,1 , . . . , xa1,β1 ). In particular, the entries in the multigraded resolution of a monomial ideal divide the least common multiple of the generators of the ideal. The following example shows that, for each n, the bounds obtained in Theorem 1 are the best possible, in terms of a and b. Example 1. Let R = K[x1 , . . . , xn ]/(x1 · · · xn ) so that b = (1, . . . , 1). The R-module M = R/(x2n )R has a minimal resolution F:
x1 ···xn−1
x1 ···xn−1
x
x2
n n · · · −→ R −−−−−−→ R −→ R −−−−−−→ R −→ R −→ 0.
So a = (0, . . . , 0, 2) and there are inequalities ai,1 =
i b for odd i ≥ 1, 2i a + 2 b − (0, . . . , 0, 1) for even i ≥ 2. a+
Thus the shifts attain the upper bound of Theorem 1 when i is odd, and are equal to them in all but the last coordinate when i is even. Remark. Theorem 1 coincides with [2, (3.1)] when I = 0 and gives sharper bounds than [2, (3.4)] when i I 6= 0. Indeed, for M and R as in the example above, [2, (3.4)] yields ai,1 ≤ a + 2 b + (1, . . . , 1).
Proof of Theorem 1. We have to show that for each k there is an r such that ai,j (k) ≤ i a1,r (k) + 2 b(k). The inequality is clear for i = 1. For i ≥ 2 it is enough to prove:
(1) (2)
If i is even, then for each k there exists an s such that ai,j (k) ≤ ai−1,s (k) + b(k). If i is odd, then for each k there exists an r such that ai,j (k) ≤ ai−2,r (k) + b(k).
Indeed, if i is odd, then repeated application of (2) yields the desired inequality. If i even, then using (1) find an s such that ai,j (k) ≤ ai−1,s (k) + b(k). When i = 2 we are done, else (i − 1) ≥ 3 is odd, so there is an r with ai−1,s (k) ≤ a1,r (k) + i−1 2 b(k). Thus, ai,j (k) ≤ a1,r (k) +
i−1 b(k) + b(k) = a1,r (k) + 2i b(k). 2
The proof of (1) and (2) uses the following observation which is verified directly: (3)
b(k)+1
If F is a free R-module and u ∈ F , then xk
b(k)
u = 0 implies xk
u = 0.
SHIFTS IN RESOLUTIONS OF MULTIGRADED MODULES µ
ν
Fix a segment FP i −→Fi−1 −→Fi−2 of the minimal resolution F. Let ν(ei,j ) = and µ(ei−1,j ) = s µsj ei−2,s . Note that (4)
ai,j = deg[ν(ei,j )] = deg[νsj ] + ai−1,s
for all s
3
P
s
νsj ei−1,s
with νsj 6= 0.
If i ≥ 2 is even and (1) fails, then choose k such that for all s, 1 ≤ s ≤ βi−1 , we have ai,j (k) > ai−1,s (k) + b(k). Together with (4) this implies that, for d = b(k), the monomial 0 νsj is of the form xd+1 νsj for all s. Set k w=
X
0 νsj ei−1,s ∈ Fi−1
and
y = xdk w ∈ Fi−1 .
s
Note that xd+1 µ(w) = µ(ν(ei,j )) = 0. In view of (3), this shows that xdk µ(w) = 0. Therek fore, µ(y) = 0, that is, y ∈ Kerµ. But now ν(ei,j ) = xk y ∈ m Kerµ, where m = (x1 , . . . , xn ) is the homogeneous maximal ideal of R. As F is minimal, this is a contradiction. This completes the proof of (1). Assuming i ≥ 3 is odd and (2) fails, choose k such that (5)
ai,j (k) > ai−2,r (k) + b(k) for all 1 ≤ r ≤ βi−2 .
If ai−1,s (k) ≥ ai,j (k) for some s, then from (5) we get ai−1,s (k) > ai−2,r (k) + b(k), for all r. Since (i − 1) ≥ 2 is even, this contradicts (1). Thus, for each s we have, 0 ai,j (k) > ai−1,s (k). Hence, using (4), we write each monomial νsj in the form xk νsj . Two applications of (4) yield ai,j = deg[νsj ] + deg[µrs ] + ai−2,r whenever νsj 6= 0 and µrs 6= 0. With (5) this shows that, for d = b(k), the monomial νsj µrs is of the form xd+1 λsr k for all r and s. Consider the elements w=
X r,s
λsr ei−2,r ∈ Fi−2
and y =
X
0 νsj ei−1,s ∈ Fi−1 .
s
A straightforward computation shows that xdk w = µ(y). As in the proof of (1), we see that xd+1 w = µ(ν(ei,j )) = 0. By (3), this implies µ(y) = xdk w = 0. Thus y ∈ Kerµ and k hence ν(ei,j ) = xk y ∈ m Kerµ. Again, we get a contradiction to the minimality of F. Thus (2) holds. To complete the proof of Theorem 1 we need to prove that when I 6= 0 i and i is even, ai,j < a + 2 b. Assume to the contrary. Choose j such that ai,j = a + 2i b. As F is minimal, there is a νs,j 6= 0. From (4) we get ai,j = deg[νs,j ] + ai−1,s . Hence a+
i i−1 i b = deg[ν ] + a ≤ deg[ν ] + a + b = deg[ν ] + a + s,j i−1,s s,j s,j 2 2 2 b−b
Therefore deg[νs,j ] ≥ b and so νs,j = 0. This contradicts the choice of νs,j . Thus Theorem 1 is proved.
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SRIKANTH IYENGAR
2. Poincar´ e Series For each i ≥ 0, the K-vector space TorR i (M, K) is multigraded with homogeneous basis elements of degrees ai,j for j = 1, . . . , βi . The multigraded Poincar´e series of M in the P R variables t and s1 , . . . , sn is defined by PM (s, t) = ri (s)ti , where i>0
ri (s) =
X
d dimK TorR i (M, K)d s
d∈Zn ±1 is a Laurent polynomial in Z[s±1 1 , . . . , sn ]. Note that by setting s1 = · · · = sn = u R one obtains the usual Poincar´e series PM (u, t) of the graded R-module associated to the ±1 n-graded R-module M . We denote by degk (f ) the degree of f ∈ Z[s±1 1 , . . . , sn ] in the variable sk . R Theorem of Backelin and Lescot [1,5]. If M is finitely generated, then PM (s, t) is a rational function of the form Pp M i R i=0 fi (s)t P PM (s, t) = , q 1 + j=1 gjR (s)tj
R with fiM ∈ Z[s±1 . . . , s±1 n ] and gj ∈ Z[s1 , . . . , sn ]. The denominator is independent of M 1 ,P and satisfies q ≤ k b(k) and degk (gjR ) ≤ b(k) for k = 1, . . . , n where b = lub{b1 , . . . , bm }.
It should be noted that the assumption that M is n-graded is essential. Indeed, Jacobsson [3, Corollary 2.3] constructs a graded module M over an n-graded ring R such that R PM (u, t) is not a rational function. Using Theorem 1, we establish upper bounds on degk (fi ) and simplify a crucial step in the proof the Backelin and Lescot’s result. Theorem 2. If M has no free direct summand, then there are inequalities: degk (fiM ) ≤ a(k) + d 2i eb(k)
for
1≤k≤n
0 ≤ i ≤ p.
Proof. We keep the notation from the proof of Theorem 1, and write fi and gi for fiM and giR respectively. Thus, a1,j = deg[µsj ] + a0,s for all s with µsj 6= 0. As M has no free summands, for each s there is a j with νsj 6= 0 and hence, a0,s ≤ a1,j . Coupled with Theorem 1, this yields degk (ri ) ≤ a(k) + 2i b(k) for all i ≥ 0. By Backelin’s result degk (gi ) ≤ b(k) for all i ≥ 1. In view of these inequalities we get X i−1 degk rj gi−j ≤ max{degk (rj gi−j )} ≤ max{a(k)+ 2j b(k)+b(k)} ≤ a(k)+ i−1 b(k). 2 j=0
j m. Set Rk = K[x1 , . . . , x ˆk , . . . , xn ]/(xb1 , . . . , xbm )
and
A = Rk [xk ].
Then A is an n-graded ring and R ∼ = A/xk J as n-graded rings. If {ci,j } are the shifts of M considered as an A-module, Theorem 1 yields ci,j (k) ≤ c(k) + 2i d(k), where d = lub{b1 , . . . , bm } and c = lub{c1,1 , . . . , c1,β1 }. Clearly d(k) = 0 and ci,j (k) ≤ c(k) for all i, j ≥ 1. Thus, with r 0 = c(k), we get (6)
0
A PM (s, t) = g0 + g1 sk + · · · + gr0 srk
with
gi ∈ Z[[t, s1 , · · · , sˆk , · · · , sn ]]
The remainder of the proof is as in [5]. As A = Rk [xk ] and Rk is an (n − 1)-graded ring (7)
Rk A PK (s, t) = (1 + sk t)PK
with
Rk PK ∈ Z[[s1 , . . . , sˆk , . . . , sn , t]]
A As Ker(TorA → TorR ∗ (M, K) − ∗ (M, K)) is an n-graded subspace of Tor∗ (M, K), its ntuple Hilbert series B(s, t) is in Z[[s1 , . . . , sˆk , . . . , sn , t]][sk ]. This series appears in the following expression
(8)
R R A A PM (s, t)/PK (s, t) = [PM (s, t) − (1 + t)B(s, t)]/PK (s, t)
which is the n-tuple version of [6, Chapter 4, (4.1)], and is due to the fact that A − → R is a Golod homomorphism [6 , Theorem 2.3]. The proposition follows by combining relations (6), (7) and (8). Acknowledgement. I am indebted to my thesis advisor Luchezar Avramov for the many helpful discussions concerning this paper.
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SRIKANTH IYENGAR
References 1. J. Backelin, Les anneaux locaux ` a relations monomiales ont des s´ eries de Poincar´ e-Betti rationelles, C. R. Acad. Sci. Paris, S´ er. I Math. 295 (1982), 607–610. 2. W. Bruns and J. Herzog, On multigraded resolutions, Math. Proc. Cambridge Phil. Soc. 118 (1995), 245–257. 3. C. Jacobsson, Finitely presented graded Lie algebras and homomorphism of local rings, J. Pure Appl. Algebra 38 (1985), 243–253. 4. D. Eisenbud, Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980), 35–64. 5. J. Lescot, S´ eries de Poincar´ e des modules multi-gradu´ es sur les anneaux monomiaux, Algebraic Topology. Rational Homotopy (Y. F´ elix, ed.), Lecture Notes Math. 1318, Springer-Verlag, 1986, pp. 155–161. 6. G. Levin, Lectures on Golod homorphisms, Reports 15, Matematiska Instit., Stockholms Univ., Stockholm, Sweden, 1976. 7. J. Tate, Homology of noetherian rings and local rings, Illinois J. Math. 1 (1957), 14-27. Deparment of Mathematics, Purdue University, West Lafayette, Indiana 47907. E-mail address:
[email protected] 