More about a construction of modules over a polynomial ring. (English. Russian original) Zbl 1081.16023

Popov, O. N.

Russ. Math. Surv. 58, No. 2, 386-387 (2003); translation from Usp. Mat. Nauk 58, No. 2, 173-174 (2003).

The author considers the following construction for modules over a polynomial ring. Let \(A\) be a finite-dimensional associative algebra over a field \(k\) with basis \(e_1,\dots,e_d\), and let \(\rho\colon A\to M_n(k)\) be the matrix representation of \(A\) corresponding to an \(A\)-module \(M\) of dimension \(n\) over \(k\). For a fixed integer \(l>1\) one considers the polynomial ring \(R=k[X_{11},\dots,X_{dl}]\) and the \(R\)-module \(F_l(M)\) defined as the factor module of the free \(R\)-module \(R^n\) by the submodule generated by the columns of the matrices \(\text{Id}_j=\sum_{i=1}^d\rho(e_i)X_{ij}\), \(j=1,\dots,l\).
Related to this construction, the author proved in an earlier paper [Mat. Sb. 193, No. 3, 115-134 (2002; Zbl 1019.16005)] that for algebras \(A\) which can be represented as a direct sum of tensor products of some simple \(k\)-algebras \(S_i\) by finite-dimensional commutative algebras \(K_i\) over \(C(S_i)\), the center of \(S_i\), all indecomposable \(A\)-modules of the form \(F_l(M)\) are Cohen-Macaulay. In the paper under review he proves that the converse of this result also holds, provided that the ground field \(k\) is perfect.

Reviewer: Cornel Baetica (Bucureşti)

MSC:

16G50	Cohen-Macaulay modules in associative algebras
16G30	Representations of orders, lattices, algebras over commutative rings
13C14	Cohen-Macaulay modules

Keywords:

Cohen-Macaulay modules; indecomposable modules; perfect fields; simple modules; finite-dimensional algebras; matrix representations; tensor products of simple algebras