We generalize the notions of n-cluster tilting subcategories and τ-selfinjective algebras into n-precluster tilting subcategories and τn-selfinjective algebras, where we show that a subcategory naturally associated to n
Nearby in the stack
-precluster tilting subcategories has a higher Auslander--Reiten theory. Furthermore, we give a bijection between
n
-precluster tilting subcategories and
n
-minimal Auslander--Gorenstein algebras, which is a higher dimensional analog of Auslander--Solberg correspondence (Auslander--Solberg, 1993) as well as a Gorenstein analog of
n
-Auslander correspondence (Iyama, 2007). The Auslander--Reiten theory associated to an
n
-precluster tilting subcategory is used to classify the
n
-minimal Auslander--Gorenstein algebras into four disjoint classes. Our method is based on relative homological algebra due to Auslander--Solberg.