Bounds on cohomology and Castelnuovo-Mumford regularity.

*(English)*Zbl 0912.13007Let \(X\subseteq \mathbb{P}^N_K\) be a projective scheme over an algebraically closed field \(K\). We denote by \({\mathcal I}_X\) the ideal sheaf of \(X\). Then \(X\) is said to be \(m\)-regular if \(H^i (\mathbb{P}^N_K, {\mathcal I}_X(m-i)) =0\) for all \(i\geq 1\). The Castelnuovo-Mumford regularity \(\text{reg} (X)\) of \(X \subseteq \mathbb{P}^N_K\), first introduced by Mumford by generalizing ideas of Castelnuovo, is the least such integer \(m\). The interest in this concept stems partly from the well-known fact that \(X\) is \(m\)-regular if and only if for every \(p\geq 0\) the minimal generators of the \(p\)-th syzygy module of the defining ideal \(I\) of \(X \subseteq \mathbb{P}^N_K\) occur in degree \(\leq m+p\). There are good bounds in some cases if \(X\) is assumed to be smooth.

Our interest is to consider the case where \(X\) is locally Cohen-Macaulay and equidimensional. Under this assumption there is a nonnegative integer \(k\) such that \[ (X_0, \dots, X_N)^k\oplus\Bigl[ {\underset l\in\mathbb{Z} \bigoplus} H^i\bigl( \mathbb{P}_K^N, {\mathcal I}_X(l) \bigr) \Bigr]= 0\quad \text{for }1\leq i\leq \dim X, \] where \(\mathbb{P}^N_K =\text{Proj} K[X_0, \dots, X_N]\). In this case \(X \subseteq \mathbb{P}_K^N\) is called a \(k\)-Buchsbaum scheme. A refined version is a \((k,r)\)-Buchsbaum scheme: Let \(k\) and \(r\) be integers with \(k\geq 0\) and \(1\leq r\leq\dim X\). Then we call \(X \subseteq \mathbb{P}^N_K\) a \((k,r)\)-Buchsbaum scheme if, for all \(j=0, \dots, r-1\), \(X \cap V\) is a \(k\)-Buchsbaum scheme for every \((N-j)\)-dimensional complete intersection \(V\) in \(\mathbb{P}^N_K\) with \(\dim (X\cap V) =\dim(X) -j\).

In recent years upper bounds on the Castelnuovo-Mumford regularity of such a variety \(X\subseteq\mathbb{P}^N_K\) have been given by several authors in terms of \(\dim(X)\), \(\deg(X)\), \(k\), and \(r\). These bounds are stated as follows; \[ \text{reg}(X) \leq\left \lceil {\deg(X)- 1\over \text{codim} (X)}\right \rceil + C(k,r, d), \] where \(d=\dim X\), \(C(k,r,d)\) is a constant depending on \(k, r\), and \(d\), and \(\lceil n\rceil\) is the smallest integer \(l\geq n\) for a rational number \(n\). In case \(X\) is arithmetically Cohen-Macaulay, that is, \(k=0\), it is well known that \(C(k,r,d)\leq 1\). We assume \(k\geq 1\). In case \(r=1\) it was shown that \(C(k,1,d) \leq {d+1 \choose 2} k-d+1\) and improved to \(C(k,1,d) \leq(2d-1) k-d+1\). Further, a better bound \(C(k,1,d) \leq dk\) was obtained U. Nagel and P. Schenzel [“Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity”, MPI/94-31, preprint]. The general case \(r\geq 1\) was first studied in Le Tuan Hoa, R. M. Mirò-Roig, M. Rosa and W. Vogel [Hiroshima Math. J. 24, No. 2, 299-316 (1994; Zbl 0822.14023)] which was improved by Lê Tuân Hoa and W. Vogel [J. Algebra 163, No. 2, 348-365 (1994; Zbl 0816.14023)] by showing that \(C(k,r,d)\leq(r-1)k+{d+2-r\choose 2}k-d+1\).

The purpose of this paper is to give bounds on \(\text{reg}(X)\) in terms of \(\dim(X)\), \(\deg(X)\), \(k\), and \(r\), which improve some of the previous results. In the general case we show that \(C(k,r,d) \leq dk-r+1\). Moreover, in case \(k=1\), we show that \(C(1,r,d) \leq \left \lceil {d\over r} \right \rceil\).

Our method here is to use a spectral sequence theory for graded modules developed by C. Miyazaki [Tokyo J. Math. 12, No. 1, 1-20 (1989; Zbl 0696.13016); J. Pure Appl. Algebra 85, No. 2, 143-161 (1993; Zbl 0773.13001); and in “Commutative Algebra 1992 ICTP, Trieste, 164-176 (1994)] in order to get bounds on the local cohomology and the Castelnuovo-Mumford regularity.

Our interest is to consider the case where \(X\) is locally Cohen-Macaulay and equidimensional. Under this assumption there is a nonnegative integer \(k\) such that \[ (X_0, \dots, X_N)^k\oplus\Bigl[ {\underset l\in\mathbb{Z} \bigoplus} H^i\bigl( \mathbb{P}_K^N, {\mathcal I}_X(l) \bigr) \Bigr]= 0\quad \text{for }1\leq i\leq \dim X, \] where \(\mathbb{P}^N_K =\text{Proj} K[X_0, \dots, X_N]\). In this case \(X \subseteq \mathbb{P}_K^N\) is called a \(k\)-Buchsbaum scheme. A refined version is a \((k,r)\)-Buchsbaum scheme: Let \(k\) and \(r\) be integers with \(k\geq 0\) and \(1\leq r\leq\dim X\). Then we call \(X \subseteq \mathbb{P}^N_K\) a \((k,r)\)-Buchsbaum scheme if, for all \(j=0, \dots, r-1\), \(X \cap V\) is a \(k\)-Buchsbaum scheme for every \((N-j)\)-dimensional complete intersection \(V\) in \(\mathbb{P}^N_K\) with \(\dim (X\cap V) =\dim(X) -j\).

In recent years upper bounds on the Castelnuovo-Mumford regularity of such a variety \(X\subseteq\mathbb{P}^N_K\) have been given by several authors in terms of \(\dim(X)\), \(\deg(X)\), \(k\), and \(r\). These bounds are stated as follows; \[ \text{reg}(X) \leq\left \lceil {\deg(X)- 1\over \text{codim} (X)}\right \rceil + C(k,r, d), \] where \(d=\dim X\), \(C(k,r,d)\) is a constant depending on \(k, r\), and \(d\), and \(\lceil n\rceil\) is the smallest integer \(l\geq n\) for a rational number \(n\). In case \(X\) is arithmetically Cohen-Macaulay, that is, \(k=0\), it is well known that \(C(k,r,d)\leq 1\). We assume \(k\geq 1\). In case \(r=1\) it was shown that \(C(k,1,d) \leq {d+1 \choose 2} k-d+1\) and improved to \(C(k,1,d) \leq(2d-1) k-d+1\). Further, a better bound \(C(k,1,d) \leq dk\) was obtained U. Nagel and P. Schenzel [“Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity”, MPI/94-31, preprint]. The general case \(r\geq 1\) was first studied in Le Tuan Hoa, R. M. Mirò-Roig, M. Rosa and W. Vogel [Hiroshima Math. J. 24, No. 2, 299-316 (1994; Zbl 0822.14023)] which was improved by Lê Tuân Hoa and W. Vogel [J. Algebra 163, No. 2, 348-365 (1994; Zbl 0816.14023)] by showing that \(C(k,r,d)\leq(r-1)k+{d+2-r\choose 2}k-d+1\).

The purpose of this paper is to give bounds on \(\text{reg}(X)\) in terms of \(\dim(X)\), \(\deg(X)\), \(k\), and \(r\), which improve some of the previous results. In the general case we show that \(C(k,r,d) \leq dk-r+1\). Moreover, in case \(k=1\), we show that \(C(1,r,d) \leq \left \lceil {d\over r} \right \rceil\).

Our method here is to use a spectral sequence theory for graded modules developed by C. Miyazaki [Tokyo J. Math. 12, No. 1, 1-20 (1989; Zbl 0696.13016); J. Pure Appl. Algebra 85, No. 2, 143-161 (1993; Zbl 0773.13001); and in “Commutative Algebra 1992 ICTP, Trieste, 164-176 (1994)] in order to get bounds on the local cohomology and the Castelnuovo-Mumford regularity.

##### MSC:

13D45 | Local cohomology and commutative rings |

13C14 | Cohen-Macaulay modules |

14F17 | Vanishing theorems in algebraic geometry |

14M05 | Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) |