In this paper, the principal tool to describe transversal polymatroids with Gorenstein base ring is polyhedral geometry, especially the Danilov−Stanley theorem for the characterization of canonical module. Also, we compute the a − invariant and the Hilbert series of base ring associated to this class of transversal polymatroids.

In this note, it is proved that over a commutative noetherian henselian non-Gorenstein local ring there are infinitely many isomorphism classes of indecomposable totally reflexive modules, if there is a nonfree cyclic totally reflexive module.

Gross and Hopkins have proved that in chromatic stable homotopy, Spanier-Whitehead duality nearly coincides with Brown-Comenetz duality. We give a conceptual interpretation of this phenomenon in terms of a Gorenstein condition [8] for maps of ring spectra.

We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, Gwdim (R) ≤ 1 does not imply that R is an arithmetical ring.

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.

In this thesis, we focus on the study of the base rings associated to some transversal polymatroids. A transversal polymatroid is a special kind of discrete polymatroid. Discrete polymatroids were introduced by Herzog and Hibi [16] in 2002. The thesis is structured in four chapters. Chapter 1 starts with a short excursion into convex geometry. Next, we look at some properties of affine semigrou...

Given a one-dimensional Cohen-Macaulay local ring $$(R,{\mathfrak {m}},k)$$ , we prove that it is almost Gorenstein if and only $${\mathfrak {m}}$$ canonical module of the {m}}:{\mathfrak . Then, generalize this result by introducing notions ideal gAGL proving R an We use fact to characterize when Gorenstein, provided has minimal multiplicity. This generalization proved Chau, Goto, Kumashiro, M...

If k is a field and R commutative k-algebra, we explore the question of when ring DR|k k-linear differential operators on isomorphic to its opposite ring. Under mild hypotheses, prove this case whenever Gorenstein local or invariants. As key step in proof show that many cases interest canonical modules admit right D-module structures.

We develop in this paper a stable theory for projective complexes, by which we mean to consider chain complex of finitely generated modules as an object the factor category homotopy modulo split complexes. As result are able prove that over generically Gorenstein ring is exact if and only its dual exact. This shows dependence total reflexivity conditions ring.