Comparison of Dualizing Complexes

TL;DR

This paper establishes a deep connection between various complexes in algebraic geometry, showing their quasi-isomorphisms and equivalences, thus advancing the understanding of duality and cycle complexes.

Contribution

It proves the existence of a map inducing a quasi-isomorphism between Bloch's cycle complex and Kato's Milnor K-theory complex, and relates Bloch's complex to Spiess' dualizing complex.

Findings

Map from Bloch's cycle complex to Kato's complex induces a quasi-isomorphism.

Truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex.

Establishes equivalences between étale sheafified cycle complex and Gersten complex.

Abstract

We prove that there is a map from Bloch's cycle complex to Kato's complex of Milnor K-theory, which induces a quasi-isomorphism from \'{e}tale sheafified cycle complex to the Gersten complex of logarithmic de Rham--Witt sheaves. Next we show that the truncation of Bloch's cycle complex at -3 is quasi-isomorphic to Spiess' dualizing complex.