Counterexamples to the tilting and (p,r)-filtration conjectures

In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the {(p,r)}-Filtration Conjecture.

The finiteness of the Gorenstein dimension for Artin algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

Simple objects in equivariantized module categories

Let [Formula: see text] be a finite group acting on a fusion category [Formula: see text] and let [Formula: see text] be a subgroup of [Formula: see text]. Let [Formula: see text] be a semisimple indecomposable module category over [Formula: see text]. Considering [Formula: see text] a simple object in [Formula: see text] and [Formula: see text] the stabilizer (or inertia) subgroup of [Formula: see text], we establish a bijective correspondence between isomorphism classes of simple objects in equivariantizations [Formula: see text] and [Formula: see text], where such simple objects contain [Formula: see text] as a direct summand. Also, as an application to projective representations of [Formula: see text], we relate isomorphism classes of simple objects in [Formula: see text] with irreducible projective representations of [Formula: see text].

Ziegler's Indecomposability Criterion

Ziegler’s Indecomposability Criterion is used to prove that a totally transcendental, i.e., ∑-pure injective, indecomposable left module over a left noetherian ring is a directed union of finitely generated indecomposable modules. The same criterion is also used to give a sufficient condition for a pure injective indecomposable module RU to have an indecomposable local dual

TRIVIAL MAXIMAL 1-ORTHOGONAL SUBCATEGORIES FOR AUSLANDER 1-GORENSTEIN ALGEBRAS

Let $\Lambda $ be an Auslander 1-Gorenstein Artinian algebra with global dimension two. If $\Lambda $ admits a trivial maximal 1-orthogonal subcategory of $\text{mod } \Lambda $, then, for any indecomposable module $M\in \text{mod } \Lambda $, the projective dimension of $M$ is equal to one if and only if its injective dimension is also equal to one, and $M$ is injective if the projective dimension of $M$ is equal to two. In this case, we further get that $\Lambda $ is a tilted algebra.

CLASSIFICATION OF GRADED LEFT-SYMMETRIC ALGEBRAIC STRUCTURES ON WITT AND VIRASORO ALGEBRAS

We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra32 (2004) 243–251; J. Nonlinear Math. Phys.6 (1999) 222–245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys.6 (1999) 222–245].

KERNELS OF MORPHISMS BETWEEN INDECOMPOSABLE INJECTIVE MODULES

We show that the endomorphism rings of kernels ker ϕ of non-injective morphisms ϕ between indecomposable injective modules are either local or have two maximal ideals, the module ker ϕ is determined up to isomorphism by two invariants called monogeny class and upper part, and a weak form of the Krull–Schmidt theorem holds for direct sums of these kernels. We prove with an example that our pathological decompositions actually take place. We show that a direct sum ofnkernels of morphisms between injective indecomposable modules can have exactlyn! pairwise non-isomorphic direct-sum decompositions into kernels of morphisms of the same type. IfERis an injective indecomposable module andSis its endomorphism ring, the duality Hom(−,ER) transforms kernels of morphismsER→ERinto cyclically presented left modules over the local ringS, sending the monogeny class into the epigeny class and the upper part into the lower part.

THE LOEWY SERIES OF THE STEINBERG-PIM OF FINITE GENERAL LINEAR GROUPS

In this paper we calculate the Loewy series of the projective indecomposable module of the unipotent block contained in the Gelfand–Graev module of the finite general linear group in the case of non-describing characteristic and Abelian defect group.