Coalgebra actions on Azumaya algebras (Q1173880)

Let \(k\) be a commutative ring, \(C\) a \(k\)-coalgebra and \(A\) an Azumaya \(k\)-algebra. The author proves that \(\text{Meas}(C,\text{End }A)\), the set of \(C\)-measurings on \(A\), is in 1-1 correspondence with \(\mathbf{I}(C^* \otimes A)\), the set of right \(C^*\) submodules \(I\) of \(C^*\otimes A\) such that \(\kappa: I\otimes A\to C^*\otimes A\), \(\kappa(x\otimes a) = x(1\otimes a)\), is an isomorphism. If \(C\) is cocommutative, \(\text{Meas}(C,\text{End }A)\) is a group under convolution, and there is an exact sequence of groups \(1\to\text{Inn}(C,\text{End }A)\to\text{Meas}(C,\text{End }A)\to\text{Pic}(C^*)\), where \(\text{Inn}(C,\text{End }A)\) is the subgroup of inner measurings. The author then proves a Noether-Skolem type theorem for measurings. If \(C\) is cocommutative and \(\text{Pic}(C^*)\) is trivial, or if \(k\) is artinian and \(C\) is finitely generated, or if \(k\) is a field (\(C\) arbitrary), every \(C\) measuring on \(A\) is inner.