Quasilength, latent regular sequences, and content of local cohomology

The quasilength of a finitely generated module that is killed by a power of a finitely generated ideal I is introduced: it is the length of a shortest filtration of the module with factors that are cyclic modules killed by I. This notion is then used to define a notion of content for the dth local cohomology module of a ring or module with support in an ideal generated by d elements. In the case of a ring, which is central, this content is a real number between 0 and 1. It is not known whether it is independent of the choice of the d generators nor whether it can change if the generators are replaced by powers. In positive prime characteristic it is shown that the content is always 0 or 1. This is an open question in equal characteristic 0 and in mixed characteristic. It is conjectured that if the elements form a system of parameters in a local ring of dimension d, then the content is 1. This is proved if the ring contains a field. In mixed characteristic it is an open question in dimension 3 or more, and implies the direct summand conjecture. The relationship between the notions of quasilength and content and the property that a sequence of elements can become a regular sequence on a ring or module after base change is explored.