FSF-Modules And Rational Invariants Of Special Subgroups Of Classical Groups

Local cohomology is an effective tool in studying commutative algebra and algebric geometry. Many scholars have been absorbed in studying it and have made some effort to develop it. In 1974, J. Herzog introduced the notation of generalized local cohomology. Then in 2009, R. Takahashi and others extended it to local cohomology with respect to a pair of ideals. In this paper, we study FSF modules and their properties firstly. In view of these, we abtain some good conclusions:If M is a FSF module and t a non-negative integer such that HIi,J(M) is FSF module for all i< t, then the R-module HomR(R/I,J(M)) is FSF, as a consequence, the associated prime of HIt,J(M) is finite; Let M be a finitely generated projective R-module, N be a R module and t a non-negative integer such that ExtRtM/IM,N) is FSF, then for any FSF submodule U of the first non I - FSF finite module HIt(M,N), the R-module HomR(M/IM,HIt(M,N)/U) is FSF, as a consequence, the set of associated prime of HIt(M, N)/U is finite.All the time, the rational invariant theory absorbed many mathematicians. It affects a lot of mathematical branches and physical domains widely. In 1911. L. E. Dickson gave the rational invariants of GLn (Fq) and SLn(Fq). Recently, the invariants of other classical groups also gained good results. The second part of this paper gives some important conclusions: If G1,G2 are subgroups of classical groups, then rational invariants of their internal direct product are the intersection of their rational invariants; If G1, G2 are subgroups of classical groups, and Fq(T1), Fq(T2) are rational invariants of G1,G2 respectively, then the rational invariants of G1∩G2 are Fq(T1)(T2). Besides,we study the rational invariants of the group∑and SOn(Fq).