The local cohomology functors

The main objective of this chapter is to introduce the a-torsion functor Γ a (throughout the book, a always denotes an ideal in a (non-trivial) commutative Noetherian ring R ) and its right derived functors, referred to as the local cohomology functors with respect to a . We shall see that Γ a is naturally equivalent to the functor and, indeed, that is naturally equivalent to the functor for each i ≥ 0; moreover, as Γ a turns out to be left exact, the functors Γ a and are naturally equivalent. This chapter also serves notice that our approach is based on fundamental techniques of homological commutative algebra, such as ones based on connected sequences of functors (see [52, pp. 212–214]): readers familiar with such ideas, and with the local cohomology functors, might like to just glance through this chapter and to move rapidly on to Chapter 2. Torsion functors Definition. For each R -module M , set, the set of elements of M which are annihilated by some power of a. Note that Γ a ( M ) is a submodule of M . For a homomorphism f : M → N of R -modules, we have f (Γ a ( M )) ⊆ Γ a ( N ), and so there is a mapping Γ a ( f ) : Γ a ( M )→ Γ a ( N ) which agrees with f on each element of Γ a ( M ). It is clear that, if g : M – N and h : N → L are further homomorphisms of R -modules and r ∈ R, then Γ a ( h o f ) = Γ a ( h ) o Γ a ( f ), Γ a ( f + g ) = Γ a ( f ) + Γ a ( g ), Γ a ( rf ) = r Γ a ( f ) and Γ a (Id M ) = Id Γa(M) .