Definition 1.1. A curved L∞ algebra over A consists of a locally free finitely generated graded A]-module V, together with a cohomological degree 1 and square zero derivation d : Ŝym(V∨[−1])→ Ŝym(V∨[−1]) where V∨ is the A]-linear dual and the completed symmetric algebra is also over A]. There are two additional requirements on the derivation d: (1) The derivation d makes Ŝym(V∨[−1]) into a dga ...

This paper develops a duality theory for connected cochain DG algebras, with particular emphasis on the non-commutative aspects. One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology. As an application, it is proved that if the canonical module A/A has a semi-free resolutio...

Let A be a noetherian local commutative ring and let M be a suitable complex of A-modules. This paper proves that M is a dualizing complex for A if and only if the trivial extension A ⋉M is a Gorenstein Differential Graded Algebra. As a corollary follows that A has a dualizing complex if and only if it is a quotient of a Gorenstein local Differential Graded Algebra. Let A be a noetherian local ...

Given an additive equational category with a closed symmetric monoidal structure and a potential dualizing object, we find sufficient conditions that the category of topological objects over that category has a good notion of full subcategories of strong and weakly topologized objects and show that each is equivalent to the chu category of the original category with respect to the dualizing obj...

This paper describes the historical background and motivation involved in the discovery (or invention) of Chu categories. In 1975, I began a sabbatical leave at the ETH in Zürich, with the idea of studying duality in categories in some depth. By this, I meant not such things as the duality between Boolean algebras and Stone spaces, nor between compact and discrete abelian groups, but rather sel...

This paper is devoted to some fundumental calculation on 2-dimensional smoothable semi-log-terminal singularities. If we study minimal or canonical models of one parameter degeneration of algebraic surfaces, we must treat singularities that appear in the central fiber. Smoothable semi-log-terminal singularities are the singularities of the central fiber of the minimal model of degeneration, and...

the concepts of fuzzy semi-ideals of r with respect to h≤r and generalized fuzzy quotient rings are introduced. some properties of fuzzy semiideals are discussed. finally, several isomorphism theorems for generalized fuzzy quotient rings are established.