At least (algebraic) geometers denote inverse image functor/extension of scalars by $f^*$, direct image functor/restriction of scalars by $f_*$ and its right adjoint by $f^!$. Because one wants to think of $f$ being in a geometric direction, that is opposite to the morphism of rings. Geometrically minded algebraists (and sheaf theorists) follow this convention as well.

]]>Small correction to the left action on $f_*(M)$. If S is not commutative then the multiplcation needs to be on the right. otherwise you get a right action instead of a left action

Justin Desrochers

]]>Gave concrete formula for coextension of scalars and a case where extension and coextension agree.

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