Determinantal rank and flat modules

Periodic Flat Modules, and Flat Modules for Finite Groups

If R is a ring of coefficients and G a finite group, then a flat RG-module which is projective as an R-module is necessarily projective as an RG-module. More generally, if H is a subgroup of finite index in an arbitrary group Γ, then a flat RΓmodule which is projective as an RH-module is necessarily projective as an RΓ-module. This follows from a generalization of the first theorem to modules o...

Bi-polynomial rank and determinantal complexity

The permanent vs. determinant problem is one of the most important problems in theoretical computer science, and is the main target of geometric complexity theory proposed by Mulmuley and Sohoni. The current best lower bound for the determinantal complexity of the d by d permanent polynomial is d/2, due to Mignon and Ressayre in 2004. Inspired by their proof method, we introduce a natural rank ...

Relative Cotorsion Modules and Relative Flat Modules

Let R be a ring, M a right R-module, and n a fixed non-negative integer. M is called n-cotorsion if Extn+1 R N M = 0 for any flat right R-module N . M is said to be n-flat if ExtR M N = 0 for any n-cotorsion right R-module N . We prove that ( n n is a complete hereditary cotorsion theory, where n (resp. n) denotes the class of all n-flat (resp. n-cotorsion) right R-modules. Several applications...

The Determinantal Ideals of Link Modules. Ii

The Determinantal Ideals of Link Modules. I

of ZfiΓ-modules and ZH-homomorphisms will be called an augmentation sequence if A is a finitely generated Ziϊ-module. Since ZH is a noetherian ring, A is then a finitely presented Zff-module; also, the module B in an augmentation sequence is finitely presented, since it is isomorphic to a submodule of A. For instance, if L £ S is a tame link of m components, G = π^S — L), and G is the commutato...