Gorenstein dimensions for stable Auslander algebras of Nakayama algebras

Let A be a Nakayama algebra and Aus(A) its stable Auslander algebra. We introduce a new homological invariant, called the finitistic Gorenstein dimension, which coincides with the classical Gorenstein dimension whenever the latter is finite, and compute this invariant for Aus(A) when A has infinite global dimension. Combined with an accompanying criterion for the finiteness of the Gorenstein dimension of Aus(A), our results provide a complete description of this dimension and verify the Gorenstein Symmetry Conjecture for this class of algebras. Furthermore, we show that the finitistic Gorenstein dimension of Aus(A) is also left-right symmetric, is explicitly related to the finitistic dimension of A, and can be attained among the injective dimensions of the projective Aus(A)-modules corresponding to simple A-modules. Our proofs rely on computing injective dimensions in terms of distances in the coresolution forest of A, a combinatorial structure closely related to Ringel's coresolution quiver.