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Hereditary algebras are quasi-hereditary with respect to any adapted partial order on the indexing set of the isomorphism classes of their simple modules. For any adapted partial order on {1,... ,n}, we compute the quiver and relations for the Ext-algebra of standard modules over the path algebra of a uniformly oriented linear quiver with n vertices. Such a path algebra always admits a regular exact Borel subalgebra in the sense of König and we show that there is always a regular exact Borel subalgebra containg the idempotents e 1 ,... ,e n and find a minimal generating set for it. For a quiver Q and a deconcatenation Q = Q 1 ⊔Q 2 of Q at a sink or source v , we describe the Ext-algebra of standard modules over K Q, up to an isomorphism of associative algebras, in terms of that over K Q 1 and K Q 2 . Moreover, we determine necessary and sufficient conditions for K Q to admit a regular exact Borel subalgebra, provided that K Q 1 and K Q 2 do. We use these results to obtain sufficient and necessary conditions for a path algebra of a linear quiver with arbitrary orientation to admit a regular exact Borel subalgebra.

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We provide a classification of generalized tilting modules and full exceptional sequences for a certain family of quasi-hereditary algebras, namely dual extension algebras of path algebras of uniformly oriented linear quivers, modulo the ideal generated by paths of length two, with their opposite algebra. An important step in the classification is to prove that all indecomposable self-orthogonal modules (with respect to extensions of positive degree) admit a filtration with standard subquotients or a filtration with costandard subquotients. Furthermore, we prove that that every generalized tilting module, not equal to the characteristic tilting modules, admits either a filtration with standard subquotients or a filtration with costandard subquotients. Since the algebras studied in this article admit a simple-preserving duality, combining these two statements reduces the problem to classifying generalized tilting modules admitting a filtration with standard subquotients. To finalize the classification of generalized tilting modules we develop a combinatorial model for the poset of indecomposable self-orthogonal modules with standard filtration with respect to the relation arising from higher extensions. This model is also used for the classification of full exceptional sequences.

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