Gillespie posed two questions in [Front. Math. China 12 (2017) 97-115], one of which states that “for what rings R do we have K(AC)=K(R-Inj)?”. We give an answer to such a question. As applications, obtain new homological approach unifies some well-known conditions Krause’s recollement holds, and example show there exists Gorenstein injective module is not AC-injective. also improve Neeman’s an...

In this article, we prove that any Q-factorial weak Fano 3-fold with only terminal singularities has a smoothing. 0. Introduction Definition 0.1. Let X be a normal Gorenstein projective variety of dimension 3 over C which has only terminal singularities. (1) If −KX is ample, we call X a Fano 3-fold. (2) If −KX is nef and big, we call X a weak Fano 3-fold. Definition 0.2. Let X be a normal Goren...

We classify the smooth projective symmetric G-varieties with Picard number one (and G semisimple). Moreover we prove a criterion for the smoothness of the simple (normal) symmetric varieties whose closed orbit is complete. In particular we prove that, given a such variety X which is not exceptional, then X is smooth if and only if an appropriate toric variety contained in X is smooth. keywords:...

in this paper, we study projective randers change and c-conformal change of p-reduciblemetrics. then we show that every p-reducible generalized landsberg metric of dimension n  2 must be alandsberg metric. this implies that on randers manifolds the notions of generalized landsberg metric andberwald metric are equivalent.

We describe how simple machine learning methods successfully predict geometric properties from Hilbert series (HS). Regressors embedding weights in projective space to ${\sim}1$ mean absolute error, whilst classifiers dimension and Gorenstein index $>90\%$ accuracy with ${\sim}0.5\%$ standard error. Binary random forest managed distinguish whether the underlying HS describes a complete intersec...

Let G be a finite subgroup of SL(r,C). In dimensions r = 2 and r = 3, McKay correspondence provides a natural bijection between the set of irreducible representations of G and a cohomology-ring basis of the overlying space of a projective, crepant desingularization of C/G. For r = 2 this desingularization is unique and is known to be determined by the Hilbert scheme of the Gorbits. Similar stat...

The classical global and weak dimensions of rings play an important role in the theory of rings and have a great impact on homological and commutative algebra. In this paper, we define and study the Gorenstein homological dimensions of commutative rings (Gorenstein projective, injective, and flat dimensions of rings) which introduce a new theory similar to the one of the classical homological d...