Noether bound for invariants in relatively free algebras. (Q2630885)

Let \(V\) be a vector space of dimension \(d\geq 2\) over a fixed field \(K\) of characteristic 0. If \(X=\{x_1,\ldots,x_d\}\) is a basis of \(V\), denote by \(K[V]\) and by \(K\langle V\rangle\) the polynomial (commutative) algebra and the free unitary associative algebra freely generated by the set \(X\) over \(K\). The group \(GL(V)\) acts on \(V\) in a natural way and this action is extended to \(K[V]\) and to \(K\langle V\rangle\). Let \(R\) be a PI algebra and let \(I\) be its T-ideal (the ideal of all polynomial identities of \(R\)) inside \(K\langle V\rangle\), thus \(K\langle X\rangle/I\) will be the relatively free algebra of rank \(d\) in the variety of algebras \(\mathrm{var}(R)\) generated by \(R\). As T-ideals are invariant under the action of \(GL(V)\) the algebra \(F(R,V)=K\langle X\rangle/I\) is acted on by \(GL(V)\) as well. Assume \(G\) is a finite group and \(V\) is a \(G\)-module and denote by \(F(R,V)^G=\{f\in F(R,V)\mid gf=f\) for each \(g\in G\}\) the algebra of \(G\)-invariants. A classical theorem of E. Noether states that when \(G\) is finite then \(K[V]^G\) is finitely generated and moreover it is generated by the elements of degree \(\leq |G|\). One denotes by \(\beta(G,V)\) the least integer \(m\) such that the algebra of invariants \(K[V]^G\) is generated by elements of degree \(\leq m\), and by \(\beta(G)\) the largest \(\beta(G,V)\) where \(V\) runs over all \(G\)-modules. (Recall that \(\beta(G)\) is the Noether number of \(G\).) The paper under review studies natural analogues of \(\beta(G,V)\) and of \(\beta(G)\) for varieties of associative algebras. A theorem of Kharchenko states that in the case \(\dim V=2\) and \(F(R,V)^G\) is finitely generated for each finite subgroup \(G\) of \(GL(V)\) then the variety \(\mathrm{var}(R)\) is weakly Noetherian, that is each finitely generated algebra in this variety satisfies the ACC for two-sided ideals. Moreover Kharchenko proved that if \(\mathrm{var}(R)\) is weakly Noetherian then \(F(R,V)^G\) is finitely generated for each finite group \(G\) and each \(G\)-module \(V\) with \(\dim V<\infty\). Thus one is led to consider weakly Noetherian varieties of algebras. Let \(R\) be an associative algebra such that \(\mathrm{var}(R)\) is weakly Noetherian. The authors denote by \(\beta(G,R,V)\) the least \(m\) such that the algebra of invariants \(F(R,V)^G\) is generated by elements of degree \(\leq m\), and by \(\beta(G,R)\) the largest among all \(\beta(G,R,V)\) when \(V\) runs over the finite dimensional \(G\)-modules. The main theorem (Theorem 3.3) of the paper under review states that in case \(\mathrm{var}(R)\) contains properly the variety of all commutative algebras then \(\beta(G,R)\) is finite for each finite group \(G\). Moreover the authors produce an upper bound for \(\beta(G,R)\) as a function of \(|G|\) and several numerical invariants of the variety \(\mathrm{var}(R)\). These are the degree of a polynomial identity of a particular kind satisfied by \(R\) and the nilpotency index of the nil-algebras of nilindex \(\ell\) where \(\ell\) stands for the nilpotency index of the commutator ideal of the relatively free algebra \(F(R, V)\) where \(\dim V=|G|\). In case when \(G\) is abelian a better upper bound for \(\beta(G,R)\) is provided.