Mitsuyasu Hashimoto

This book, first published in 2000, focuses on homological aspects of equivariant modules. It presents a homological approximation theory in the category of equivariant modules, unifying the Cohen-Macaulay approximations in commutative ring theory and Ringel's theory of delta-good approximations for quasi-hereditary algebras and reductive groups. The book provides a detailed introduction to homological algebra, commutative ring theory and homological theory of comodules of co-algebras over an arbitrary base. It aims to overcome the difficulty of generalising known homological results in representation theory. This book will be of interest to researchers and graduate students in algebra, specialising in commutative ring theory and representation theory.

We construct a quantized version of the theory of multilinear algebra, based on Jimbo's solution of Yang-Baxter equation of type #l 1 . Using this, we discuss the polynomial representations of quantum general linear groups.

Let R be a Noetherian commutative ring with, unit element, and X ij be variables with 1 ≤ i ≤ m and 1 ≤ j ≤ n . Let S = R[x ij ] be the polynomial ring over R , and I t be the ideal in S , generated by the t × t minors of the generic matrix (x ij ) ∈ M m, n (S) . For many years there has been considerable interest in finding a minimal free resolution of S/I t , over arbitrary base ring R . If we have a minimal free resolution P. over R = Z, the ring of integers, then R′ ⊗ z P . is a resolution of S/I t over the base ring R′ .