On a construction of modules over a polynomial ring in the case of an arbitrary field. (English. Russian original) Zbl 1084.16012

Popov, O. N.

Russ. Math. Surv. 59, No. 3, 583-584 (2004); translation from Usp. Mat. Nauk 59, No. 3, 175-176 (2004).

The author reminds us of a construction of modules over a polynomial ring, introduced in [Mat. Sb. 193, No. 3, 115-134 (2002; Zbl 1019.16005)], namely: Let \(A\) be a finite-dimensional associative algebra with \(1\) over a field \(k\) with basis \(e_1,\dots,e_d\) and let \(\varrho\colon A\to M_n(k)\) be its matrix representation corresponding to an \(A\)-module \(M\) with \(\dim_kM=n\). For a natural number \(l\) we consider the ring \(R=k[x_1,\dots,x_{dl}]\) and denote by \(F_l(M)\) the factor-module of the free \(R\)-module \(R^n\) by its submodule generated by the columns of the matrix \(\text{Id}_j=\sum_i\varrho(e_i)x_{ij}\), \(j=1,\dots,l\).
He states that “If \(A\) is a maximally central algebra, then for any \(l>1\) all indecomposable \(A\)-modules turn into Cohen-Macaulay \(R\)-modules, and \(F_l\) is an exact fully faithful functor into the category of graded \(R\)-modules. If, in addition \(A\) is equidimensional, then all \(A\)-modules turn into Cohen-Macaulay modules”, and further states that the reverse implications also hold.

Reviewer: Sophie Frisch (Graz)

MSC:

16G50	Cohen-Macaulay modules in associative algebras
16D90	Module categories in associative algebras
13F20	Polynomial rings and ideals; rings of integer-valued polynomials

Keywords:

finite-dimensional algebras; matrix representations; categories of finite-dimensional modules; Cohen-Macaulay modules; exact functors