Straightening Laws on Modules and Their Symmetric Algebras

Several modules M over algebras with straightening law A have a structure which is similar to the structure of A itself: M has a system of generators endowed with a natural partial order, a standard basis over the ring B of coefficients, and the multiplication A × M → A satisfies a “straightening law”. We call them modules with straightening law, briefly MSLs. In section 1 we recall the notion of an algebra with straightening law together with those examples which will be important in the sequel. Section 2 contains the basic results on MSLs, whereas section 3 is devoted to examples: (i) powers of certain ideals and residue class rings with respect to them, (ii) “generic” modules defined by generic, alternating or symmetric matrices of indeterminates, (iii) certain modules related to differentials and derivations of determinantal rings. The essential homological invariant of a module is its depth. We discuss how to compute the depth of an MSL in section 4. The main tool are filtrations related to the MSL structure. The last section contains a natural strengthening of the MSL axioms which under certain circumstances leads to a straightening law on the symmetric algebra. The main examples of such modules are the “generic” modules defined by generic and alternating matrices. The notion of an MSL was introduced by the author in [Br.3] and discussed extensively during the workshop. The main differences of this survey to [Br.3] are the more detailed study of examples and the treatment of the depth of MSLs which is almost entirely missing in [Br.3]