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On a construction of modules over a polynomial ring in the case of an arbitrary field. (English. Russian original) Zbl 1084.16012
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.
##### MSC:
 16G50 Cohen-Macaulay modules in associative algebras 16D90 Module categories in associative algebras 13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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