Uniqueness of Exact Borel Subalgebras and Bocses

In the article by König, Külshammer, and Ovsienko (Adv. Math <bold>262</bold> (2014), 546–592), together with Koenig and Ovsienko, the first author showed that every quasi-hereditary algebra can be obtained as the (left or right) dual of a directed bocs. In this monograph, we prove that if one additionally assumes that the bocs is basic, a notion we define, then this bocs is unique up to isomorphism. This should be seen as a generalisation of the statement that the basic algebra of an arbitrary associative algebra is unique up to isomorphism. The proof associates to a given presentation of the bocs an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Subscript normal infinity"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">A_\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -structure on the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E x t"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mi>x</mml:mi> <mml:mi>t</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Ext</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -algebra of the standard modules of the corresponding quasi-hereditary algebra. Uniqueness then follows from an application of Kadeishvili’s theorem.