Let $A$ be an $R$-algebra.
In the book "Noncommutative Geometry and Cayley-smooth Orders" by Le Bruyn one can find the notion of "Serre-smooth" in the introduction.
But no formal definition seems to be given in this (very long) introduction.
It is just mentioned that $A$ is Serre-smooth if $A$ has finite global dimension together with some extra features such as Auslander regularity or the Cohen-Macaulay property.


Then it is refered to the article <https://www.cambridge.org/core/journals/glasgow-mathematical-journal/article/some-properties-of-noncommutative-regular-graded-rings/244AE0A8E9C8AB6782515B504F1AC5C0> but there I can not find the word "Serre" (searching with Strg+F).



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> Question: Is there a reference for the complete definition of being Serre-smooth?
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What is a Serre-smooth algebra?
No, there is no such reference. The introduction to that book is based on a couple of lectures I gave in Luminy and there I had to distinguish between several notions of 'smoothness', formal smoothness a la Kontsevich-Rosenberg, Cayley-smoothness, and Artin-Schelter- or Auslander-Gorenstein-regularity as used by people working in NAG.
For the later category I then used the term 'Serre-smoothness' as it is the noncommutative equivalent of smoothness for commutative affine algebras (finite global dimension) to the noncommutative world.
If one only considers (maximal) orders in central simple algebras which are finite modules over their centers then one can do with less than the full repertoire of these homological conditions, I think.
Please read 'Serre-smoothness of A' as 'A is an Auslander-Gorenstein regular ring'. My apologies for the confusion this ad-hoc terminology may have caused.


