Motives over S

This chapter shows that the operation φ‎ 𝑉 of the definition introduced in the previous chapter extends to a cohomology operation over 𝑘, and that it satisfies the recognition criterion of a theorem, so that φ‎ 𝑉 must be β‎𝑃𝑏. This construction of the cohomology operation utilizes the machinery of motives over a simplicial noetherian scheme. The chapter first presents this scheme in three parts, initially summarizing the basic theory of motives over a scheme 𝑆 before discussing motives over a simplicial scheme and over a smooth simplicial scheme. It then presents the slice filtration and generalizes from simplicial scheme 𝔛 to embedded schemes. Finally, this chapter defines the operations φ‎ 𝑖 and φ‎ 𝑉.