The Endomorphism Ring Theorem for Galois and D2 extensions

Abstract

Let $S$ be the left bialgebroid $\End {}_BA_B$ over the centralizer $R$ of a right D2 algebra extension $A\Vert B$, which is to say that its tensor-square is isomorphic as $A$-$B$-bimodules to a direct summand of a finite direct sum of $A$ with itself. We prove that its left endomorphism algebra is a left $S$-Galois extension of $A^{\mathrm{op}}$. As a corollary, endomorphism ring theorems for D2 and Galois extensions are derived from the D2 characterization of Galois extension (cf. math.QA/0502188 and math.QA/0409589). We note the converse that a Frobenius extension satisfying a generator condition is D2 if its endomorphism algebra extension is D2.

20 pp, some additional material including a converse endomorphism ring theorem for certain Frobenius extensions, which yields a complete answer to question 1 in math.RA/0107064

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