$E_n$-regularity implies $E_{n-1}$-regularity

Abstract

Vorst and Dayton-Weibel proved that $K_n$-regularity implies $K_{n-1}$-regularity. In this article 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 establish a Nisnevich descent result which is of independent interest.