Local Cohomology Modules Of The Endomorphism,

The local cohomology module is an important tool for the study commutative algebra andalgebra geometry. It plays an important role in the characterizations of the properties of zero-divisors. For example, it can be used to characterize properties of Cohen-Macaulay mod-ules,Gorenstein rings and generalized Cohen-Macaulay modules. A local cohomology moduleis rarely finitely generated, the structure of it is still not clear, though much effort has been done.Many important results such as the injective dimension and Bass numbers have been estab-lished for the local cohomology modules over regular local rings containing a field. But for theunramified regular local rings containing no field, the known results about the injective dimen-sion of local cohomology modules are a little different from the case containing a field. It wasconjectured that there is no difference between them.In the proof of the results of local cohomology modules over a regular local ring R containinga field, there is an important result as follows. If x is an element of a regular system of parametersof a regular ring R containing a field, then the endomorphismH_I~i(R)→x H_I(i(R), i≥0are surjective for every ideal I(x∈I). We conjecture that for a unramified regular local ring(R,m) with p∈m,H_I~i(R)→x H_I(i(R), i≥0are surjective for every ideal I(p∈I).The purpose of the paper is to study the endomorphism (1). We will prove that (1) are surjec-tive for all ideal I(p∈I) and i≥0 if and only if (1) are surjective for all ideal I with p∈I andi = htI. Moreover, we have proved that (1) are surjective if dimR≤4.