Let L denote a finite lattice with at least two points and let A denote the incidence algebra of L. We prove that L is distributive if and only if A is an Auslander regular ring, which gives a homological characterisation of distributive lattices. In this case, A has an explicit minimal injective coresolution, whose i-th term is given by the elements of L covered by precisely i elements. We give a combinatorial formula of the Bass numbers of A. We apply our results to show that the order dimension of a distributive lattice L coincides with the global dimension of the incidence algebra of L. Also we categorify the rowmotion bijection for distributive lattices using higher Auslander-Reiten translates of the simple modules.

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Optimal upper bounds are provided for the dominant dimensions of Nakayama algebras and more generally algebras A with an idempotent e such that there is a minimal faithful injective-projective module eA and such that eAe is a Nakayama algebra. This answers a question of Abrar and proves a conjecture of Yamagata for monomial algebras.

Gendo-symmetric algebras were introduced by Fang and Koenig in [FanKoe2] as a generalisation of symmetric algebras. Namely, they are endomorphism rings of generators over a symmetric algebra. This article studies various algebraic and homological properties of representation-finite gendosymmetric biserial algebras. We show that the associated symmetric algebras for these gendo-symmetric algebras are Brauer tree algebras, and classify the generators involved using Brauer tree combinatorics. We also study almost ν-stable derived equivalences, introduced in [HuXi1] between representation-finite gendo-symmetric biserial algebras. We classify these algebras up to almost ν-stable derived equivalence by showing that the representative of each equivalence class can be chosen as a Brauer star with some additional combinatorics. We also calculate the dominant, global, and Gorenstein dimensions of these algebras. In particular, we found that representation-finite gendo-symmetric biserial algebras are always Iwanaga-Gorenstein algebras.Recall that symmetric Nakayama algebra can also be called Brauer star algebras. We can now summarise our results using the following table, which compares the gendo-symmetric generalisation with the classical situation: Symmetric algebras Gendo-symmetric algebras Brauer tree special gendo-Brauer tree || || Representation-finite biserial Representation-finite biserial || || Derived equivalent to Almost ν-stable derived equivalent to Brauer star algebras special gendo-Brauer star algebras ON REPRESENTATION-FINITE GENDO-SYMMETRIC BISERIAL ALGEBRAS 3Note that if we take away "special" from the first row or the last row, then the corresponding equality will not hold any more. For example, the Auslander algebra of the local Brauer tree algebra K[X]/(X 4 ) is a gendo-Brauer tree algebra which is neither representation-finite, nor biserial. If we replace "special gendo-Brauer star algebras" by "gendo-symmetric Nakayama algebra", then the second equality also will not hold; see Example 4.9. An advantage of focusing on special gendo-Brauer tree algebras is that we can employ combinatorics of Brauer tree algebras to help investigate various properties of those algebras. Indeed, we will use the Green's walk around a Brauer tree [Gre] to obtain formulae for calculating various homological dimensions of special gendo-Brauer tree algebras.Proposition. (Proposition 5.6) Every representation-finite gendo-symmetric biserial algebra is Iwanaga-Gorenstein. Moreover, both Gorenstein dimension and dominant dimension are independent of the exceptional multiplicity of the associated Brauer tree.Note that while all special gendo-Brauer tree algebras are Iwanaga-Gorenstein, this is not true for general gendo-Brauer tree algebras. An explicit example of a non-Iwanaga-Gorenstein gendo-Brauer tree algebra is detailed in [Ma4].In addition to the above proposition, we can also determine special gendo-Brauer trees which have finite global dimension. In particular, we can classify the special gendo-Brauer tree algebras which ...

Gendo-symmetric algebras were recently introduced by Fang and König in [FanKoe]. An algebra is called gendo-symmetric in case it is isomorphic to the endomorphism ring of a generator over a finite dimensional symmetric algebra. We show that a finite dimensional algebra A over a field K is gendo-symmetric if and only if there is a bocs-structure on (A, D(A)), where D = Hom K (−, K) is the natural duality. Assuming that A is gendosymmetric, we show that the module category of the bocs (A, D(A)) is isomorphic to the module category of the algebra eAe, when e is an idempotent such that eA is the unique minimal faithful projective-injective right A-module. We also prove some new results about gendo-symmetric algebras using the theory of bocses.2010 Mathematics Subject Classification. Primary 16G10, 16E10.