The Stacks project

62.1 Introduction

A foundational reference is [SV].

In this chapter we only define what are called the universally integral relative cycles in [SV]. This choice makes the theory somewhat simpler to develop than in the original, but of course we also lose something.

Fix a morphism $X \to S$ of finite type between Noetherian schemes. A family $\alpha $ of $r$-cycles on fibres of $X/S$ is simply a collection $\alpha = (\alpha _ s)_{s \in S}$ where $\alpha _ s \in Z_ r(X_ s)$. It is immediately clear how to base change $g^*\alpha $ of $\alpha $ along any morphism $g : S' \to S$. Then we say $\alpha $ is a relative $r$-cycle on $X/S$ if $\alpha $ is compatible with specializations, i.e., for any morphism $g : S' \to S$ where $S'$ is the spectrum of a discrete valuation ring, we require the generic fibre of $g^*\alpha $ to specialize to the closed fibre of $g^*\alpha $. See Section 62.6.