Extension of scalars in the corestriction of an algebra (Q1077505)

For algebras over fields the corestriction over finite separable field extensions is a very useful technique. Let A be an algebra over a field L, L/K a finite separable extension and E/K an arbitrary extension. Thus \(E\otimes_ KL=L_ 1\otimes...\otimes L_ s\). The author shows that \[ E\otimes_ KCor_{L/K}A=(Cor_{L_ 1/E}A_{L_ 1})\otimes_ E...\otimes_ E(Cor_{L_ s/E}A_{L_ s}) \] where \(A_{L_ i}=L_ i\otimes_ LA\). The special case where L/K is an extension of global fields, while E is the completion of K at a prime, yields that the \(L_ i\) are the completions of L at the primes lying over the prime of K. In a final section the author studies the integral analogue of these results and shows that when certain primes ramify the results do not carry over.