Math 6310 • Algebra

Let \(G\) be a finite group, let \(H\) be a subgroup of \(G\), and let \(\mathbb {C}[G]\) (respectively, \(\mathbb {C}[H]\)) denote the complex group algebra of \(G\) (respectively, \(H\)). Let \(I_H^G\) denote the functor from the category of finite-dimensional \(\mathbb {C}[H]\) modules to the category of finite-dimensional \(\mathbb {C}[G]\) modules which, on the level of objects, is defined by \[ I_H^G(M)=\mathbb {C}[G]\otimes _{\mathbb {C}[H]}M. \] Prove or disprove: \(I_H^G\) is an exact functor.