Let R be a non-negatively graded Cohen-Macaulay ring with R_0 a Cohen-Macaulay factor ring of a local Gorenstein ring. Let d be the dimension of R, m be the maximal homogeneous ideal of R, and M be a finitely generated graded R-module. It has long been known how to read information about the socle degrees of the local cohomology module H_m^0(M) from the twists in position d in a resolution of M by free R-modules. It has also long been known how to use local cohomology to read valuable information from complexes which approximate resolutions in the sense that they have positive homology of small Krull dimension. The present paper reads information about the maximal generator degree (rather than the socle degree) of H_m^0M from the twists in position d-1 (rather than position d) in an approximate resolution of M. We apply the local cohomology results to draw conclusions about the maximum generator degree of the second symbolic power of the prime ideal defining a monomial curve and the second symbolic power of the ideal defining a finite set of points in projective space. There is an application to general hyperplane sections of subschemes of projective space over an infinite field. There is an application of the local cohomology techniques to partial Castelnuovo-Mumford regularity. An application to the ideals generated by the lower order Pfaffians of an alternating matrix will appear in a future paper. One additional application to the study of blow-up algebras appears in a separate paper.

中文翻译：

---

局部同调的度界

设R为非负渐变的Cohen-Macaulay环，R_0为局部Gorenstein环的Cohen-Macaulay因子环。设d为R的维数，m为R的最大均匀理想值，M为有限生成的渐变R模。早就知道如何通过自由的R模块以M的分辨率从位置d的扭曲中读取有关局部同调模块H_m ^ 0（M）的初阶的信息。长期以来，人们还知道如何使用局部同调来从复杂的化合物中读取有价值的信息，这些化合物在分辨率方面具有小Krull维的正同源性。本文从位置d-1（而不是位置d）的扭曲中以大约M的分辨率读取有关H_m ^ 0M的最大生成度（而不是鞋底度）的信息。我们应用局部同调结果得出关于定义理想多项式的主要理想的第二符号幂的最大生成度和定义投影空间中有限点集的理想的第二符号幂的结论。在无限域上有一个到投影空间子计划的一般超平面部分的应用。局部同调技术在部分Castelnuovo-Mumford正则性中的应用。在未来的论文中将出现对由交替矩阵的低阶Pfaffian生成的理想的应用。另一种论文是爆破代数研究的另一项应用。