Titlebook: Relative Nonhomogeneous Koszul Duality; Leonid Positselski Book 2021 The Editor(s) (if applicable) and The Author(s), under exclusive lice

方便

Homogeneous Quadratic Duality over a Base Ring,All the . in this book are unital. We will always presume unitality without mentioning it, so all the left and ring . over associative rings are unital, all the . take the unit to the unit, all the . contain the unit, and all the . and . are such that the unit element belongs to the degree-zero grading/filtration component.

大气层

Flat and Finitely Projective Koszulity,So far in this book, we denoted a graded ring by . (and for the most part we will continue to do so in the sequel), but this is a colloquial abuse of notation. A graded abelian group . is properly thought of as a collection of abelian groups .. Then there are several ways to produce an ungraded group from a graded one.

排斥

Relative Nonhomogeneous Quadratic Duality,Nonhomogeneous quadratic rings . can be informally described as rings defined by nonhomogeneous quadratic relations over a fixed base ring ..

袭击

notice

Comodules and Contramodules Over Graded Rings,Let . be a nonnegatively graded ring. We denote the underlying ungraded ring of . by the same letter . =  Σ. (see Sect. .) and consider ungraded right .-modules.

导师

Relative Nonhomogeneous Derived Koszul Duality: The Comodule Side,Let (., ., .) be a curved DG-ring, as defined in Sect. ., and let . be the related quasi-differential graded ring constructed in Theorem ..

hedonic

Delirium

The Co-Contra Correspondence,Let . be a nonnegatively graded ring, and let . be a ring endowed with a ring homomorphism .→... We start from the case of ungraded comodules and contramodules before passing to the graded ones.

充满装饰

sclera

Examples,In this section we recall the main points of the discussion of the tensor rings and, more importantly, the symmetric and exterior algebras, which was presented in the examples given throughout Chaps. .–.; see, in particular, Examples ., ., and ..