Correction to on the Brauer Group of Algebras Having a Grading and an Action

Extract

Let R be a commutative ring, G a finite abelian group. Let A be an R-algebra which is graded by G (i.e. A = Σ⊕_σ∈GA_σ, where A_σA_τ ⊂ A_στ for σ, τ in G) and for which A₁ is an R-module of finite type. In Remark 4.1 (a) of [1] we asserted that under these hypotheses if u is in A and u + pA is homogeneous in A/pA for each maximal ideal p of R then u is homogeneous in A. We used this assertion for u a unit in A such that a → uau^–1 is a grading-preserving homomorphism. K. Ulbrich has kindly pointed out a counterexample to the assertion: R = Z/4Z, G = {1, σ};, u = 2σ + 1, p = 2R. Proposition 4.2 of [1] uses the erroneous result and is in turn invoked later in the paper.