On log crystalline higher direct image

TL;DR

This paper develops the theory of big crystalline sites for log schemes, establishing key properties and the behavior of crystalline higher direct images, including Frobenius isogeny, advancing the understanding of log crystalline cohomology.

Contribution

It introduces the big crystalline site for log schemes and proves foundational theorems for crystalline higher direct images, including boundedness, base change, and perfectness.

Findings

Established the properties of the big crystalline site for log schemes.

Proved the boundedness, base change, and perfectness theorems for crystalline higher direct images.

Discussed the Frobenius isogeny property of the crystalline higher direct image of F-isocrystals.

Abstract

We define the big crystalline site for a log scheme and prove the basic properties. In particular, we show the boundedness, base change, and perfectness theorems for the crystalline higher direct image of quasi-coherent crystals between fine log schemes. We also introduce the big absolute crystalline sites and discuss the Frobenius isogeny property of the crystalline higher direct image of $F$-isocrystals.