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Modules over a polynomial ring obtained from representations of finite-dimensional associative algebras. (English. Russian original) Zbl 1019.16005
Sb. Math. 193, No. 3, 423-443 (2002); translation from Mat. Sb. 193, No. 3, 115-134 (2002).
Let $$A$$ be a finite-dimensional associative algebra with identity over a field $$k$$ with the basis $$e_1,\dots,e_d$$ and let $$\varrho\colon A\to M_m(k)$$ be its matrix representation corresponding to the $$A$$-module $$M$$ with $$\dim_kM=m$$. For a natural integer $$l$$ we shall consider the ring $$R=k[x_1,\dots,x_{dl}]$$ of polynomials and we denote by $$F_l(M)$$ the factor-module of the free $$R$$-module $$R^m$$ by its submodule generated by the columns of the matrix $$\text{Id}_j=\sum_i\varrho(e_i)x_{ij}$$, $$j=1,\dots,l$$. If either $$l=1$$ or $$A=\bigoplus_iS_i\otimes_{Z(S_i)}K_i$$, where $$S_i$$ is a simple algebra over $$k$$, $$Z(S_i)$$ is its center and $$K_i$$ is a finite-dimensional commutative algebra over $$Z(S_i)$$ with $$\dim_{Z(S_i)}S_i=n_i^2$$, then (1) $$F_l(\cdot)$$ is an exact completely strict functor from the category of finite-dimensional $$A$$-modules into the category of graded $$R$$-modules and homogeneous homomorphisms of degree 0; (2) if either $$l=1$$ or $$n_i=n$$ for each $$i$$, then the functor $$F_l(\cdot)$$ maps the finite dimensional $$A$$-modules into Cohen-Macaulay $$R$$-modules of projective dimension $$(l-1)n+1$$; (3) if $$l=1$$ then for each $$M$$ the annihilator of $$F_l(M)$$ is a principal ideal (Theorem 1). Recall that a finite-dimensional associative algebra with identity element over the field $$k$$ is called: $$K$$-algebra, if there is an $$l>1$$ such that the functor $$F_l(\cdot)$$ transforms the finite-dimensional irreducible $$A$$-modules into Cohen-Macaulay ones; $$O$$-algebra, if the functor $$F_l(\cdot)$$ is exact for some $$l>0$$; and $$T$$-algebra if there is an $$l>1$$ such that the module $$F_l(M)$$ is Cohen-Macaulay whenever $$M$$ is a finite-dimensional $$A$$-module. The $$O$$-algebras and $$T$$-algebras over a perfect field $$k$$ are just the direct sums presented in Theorem 1 above. Moreover, for $$T$$-algebras the integers $$n_i$$ are all equal (Theorem 2). In the last Theorem 3 several invariants of $$A$$-modules are presented under the hypothesis that all the integers $$n_i$$ are equal.

##### MSC:
 16G50 Cohen-Macaulay modules in associative algebras 16D90 Module categories in associative algebras 16S36 Ordinary and skew polynomial rings and semigroup rings
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