E_n-regularity implies E_(n-1)-regularity

TL;DR

This paper generalizes a known regularity implication from rings to differential graded categories and schemes, extending algebraic K-theory results to a broader, more abstract setting with applications to descent and fundamental theorems.

Contribution

It extends the implication of E_n-regularity to E_(n-1)-regularity from rings to differential graded categories and schemes, broadening the scope of algebraic K-theory.

Findings

Regularity is preserved under desuspensions, fibers, direct factors, and sums.

The implication holds for schemes.

Bass' fundamental theorem is extended to this setting.

Abstract

Vorst and latter Dayton-Weibel proved that K_n-regularity implies K_(n-1)-regularity. In this note we generalize this result from (commutative) rings to differential graded categories and from algebraic K-theory to any functor which is Morita invariant, continuous, and localizing. Moreover, we show that regularity is preserved under taking desuspensions, fibers of morphisms, direct factors, and arbitrary direct sums. As an application, we prove that the above implication also holds for schemes. Along the way, we extend Bass' fundamental theorem to this broader setting and prove a Nisnevich descent result which is of independent interest