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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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