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More about a construction of modules over a polynomial ring. (English. Russian original) Zbl 1081.16023
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.
##### MSC:
 16G50 Cohen-Macaulay modules in associative algebras 16G30 Representations of orders, lattices, algebras over commutative rings 13C14 Cohen-Macaulay modules
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