Let $\Delta =\left(\begin{smallmatrix} A & {_AN_B}\\ {_BM_A} B \\\end{smallmatrix}\right)$ be a Morita ring with $M\otimes_{A}N=0=N\otimes_{B}M$.We first study how to construct (complete) duality pairs of $\Delta$-modules using $A$-modules and $B$-modules, generalizing the result Mao (Comm. Algebra, 2020, 12: 5296--5310) about over triangular matrix ring. Moreover, we Gorenstein projective modu...

The theory of local homology modules was initiated by Matlis in 1974. It is a dual version of the theory of local cohomology modules. Mohammadi and Divaani-Aazar (2012) studied the connection between local homology and Gorenstein flat modules by using Gorenstein flat resolutions. In this paper, we introduce generalized local homology modules for complexes and we give several ways for computing ...

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...

Abstract We provide a semiorthogonal decomposition for the derived category of fibrations quintic del Pezzo surfaces with rational Gorenstein singularities. There are three components, two which equivalent to categories base and remaining non-trivial component is flat finite degree 5 scheme over base. introduce methods construction decomposition. One moduli space approach following work Kuznets...

We consider the question of simplicity a ring $R$ under action its differential operators $D_R$. give examples to show that even when is Gorenstein and has rational singularities need not be simple $D_R$-module; for example, this case homogeneous coordinate smooth cubic surface. Our are rings Fano varieties, our proof proceeds by showing tangent bundle such variety big. also partial converse pr...

The Auslander correspondence is a fundamental result in Auslander-Reiten theory. In this paper we introduce the category modadm(E) of admissibly finitely presented functors and use it to give version for any exact E. An important ingredient proof localization theory categories. We also investigate how properties E are reflected modadm(E), example being (weakly) idempotent complete or having eno...

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...

Abstract We show that, for any fixed weight, there is a natural system of Hodge sheaves, whose Higgs field has no poles, arising from flat projective family varieties parametrized by regular complex base scheme, extending the analogous classical result smooth families due to Griffiths. As an application, based on positivity direct image we establish criterion spaces rational Gorenstein be gener...