We prove that any faithful Frobenius functor between abelian categories preserves the Gorenstein projective dimension of objects. Consequently, it and reflects give conditions on when a stable objects, singularity defect categories, respectively. In appendix, we direct proof following known result: for an category with enough projectives injectives, its global coincides injective dimension.

We study a pair of conjectures on better behaved GKZ hypergeometric systems PDEs inspired by Homological mirror symmetry for crepant resolutions Gorenstein toric singularities. prove the in case dimension two.

Let φ : (R, m)→ (S, n) be a local homomorphism of commutative noetherian local rings. Suppose that M is a finitely generated S-module. A generalization of Grothendieck’s non-vanishing theorem is proved for M (i.e. the Krull dimension of M over R is the greatest integer i for which the ith local cohomology module of M with respect to m, Hi m(M), is non-zero). It is also proved that the Gorenstei...

Let Z be the center of a nonnoetherian dimer algebra on torus. Although itself is also nonnoetherian, we show that it has Krull dimension 3, and locally noetherian an open dense set Max . Furthermore, reduced / nil depicted by Gorenstein singularity, contains precisely one closed point positive geometric dimension.

We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.

We put cluster tilting in a general framework by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal one-orthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.

We introduce a refinement of the Gorenstein flat dimension for complexes over an associative ring—the flat-cotorsion —and prove that it, unlike dimension, behaves as one expects homological without extra assumptions on ring. Crucially, we show it coincides with where latter is finite, and right coherent rings—the setting known to behave expected.

Let $R$ be a Noetherian local ring. We prove that is regular of dimension at most four if, and only every prime ideal, defining Gorenstein quotient ring, syzygetic. deduce characterization these rings in terms the Andr\'e-Quillen homology.