Holmstrom 4 Memo notes Gillet

Gillet: K-theory and intersection theory

Have incorporated everything into the relevant CT pages.

Introduction

Brilliant historical background on Chow groups. Mention of Milnor K-theory and Rost’s cycle complexes, and Gersten’s conjecture.

Chow groups

Dimension and codimension

Cycles

Dimension relative to a base

Cartier divisors

Cap products with Cartier divisors and the divisor homomorphism

Rational equivalence

Basic properties of Chow groups

K-theory and intersection multiplicities

Serr’s Tor formula

$K_0$ with supports

The coniveau and the gamma filtations

Complexes computing Chow groups

Higher rational equivalence and Milnor K-theory

Rost’s axiomatics

Higher algebraic K-theory and Chow groups

Stable homotopy theory

Filtrations on cohomology of simplicial sheaves

Including various spectral sequences

Review of basic notions of K-theory

Quillen’s spectral sequence

K-theory and sheaf cohomology

Gersten’s conjecture, Bloch’s formula, and the Comparison of spectral sequences

The coniveau spectral sequence for other cohomology theories

It seems like Bloch-Ogus theories satisfy the Gersten conjecture, which has consequences for the coniveau filtration. Also identification of various spectral sequences (Leray, hypercohomology, coniveau).

Compatibility with products and localized intersection

Other cases of Gersten’s conjecture

Operations on the Quillen spectral sequence

The multiplicativity of the coniveau filtration: a proof using deformation to the normal cone

Bloch’s formula and singular varieties

Cohomology versus homology

Chow cohomology…

Local complete intersection subschemes and other cocycles

Chow groups of singular surfaces

Intersection theory of stacks

Chow groups for smooth Deligne-Mumford stacks. Discussion of weaker form of Bloch’s formula.

Deformation to the normal cone

Envelopes and hyperenvelopes

END

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