| Summary: | We already know that the noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noetherian algebras resemble commutative local Noetherian rings in many respects. We also know that commutative rings have the important property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic, and we want to generalize such properties to noncommutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">N</mi></semantics></math></inline-formula>-graded Noetherian algebra. By generalizing the conclusions about commutative rings and combining what we already know about noncommutative graded algebras, we identify a class of noncommutative graded algebras with the property that every minimal acyclic complex of finitely generated graded free modules is totally acyclic. We also discuss how the relationship between AS–Gorenstein algebras and AS–Cohen–Macaulay algebras admits a balanced dualizing complex. We show that AS–Gorenstein algebras and AS–Cohen–Macaulay algebras with a balanced dualizing complex belong to this algebra.
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