The Algebraic Structure Of The Inertia Block

Let O be a complete discrete valuation ring with an algebraically closed residue field k of characteristic p.Let H be a finite group and let b be a block of H over O with a defect group Q. Let H’be a finite group and let b be a block of H over O.An indecomposable O(H×H’)-M induces a basic Morita equivalence between block algebra OHb and OH’b’, if M induces a Morita equivalence between OHb andOH’b’,and its source modules have their O-rank prime to p. Let H=NH(Q)andb be the Brauer corresponding of binH.LetB and B be respectively the block algebras of b and b; and if B and B are basically Morita equivalence,then B is an inertia block. If B is an inertia block,then we prove that there is a Q-interior algebra embedding betwwen the source algebras of B and B,which induces an isomorphism between the centers of source algebras, which is compatible with the isomorphism between the centers of block algebras.