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A remark on 0-cycles as modules over algebras of finite correspondences

M. Rovinsky

Abstract

Given a smooth projective variety $X$ over a field, consider the $\mathbb Q$-vector space $Z_0(X)$ of 0-cycles (i.e. formal finite $\mathbb Q$-linear combinations of the closed points of $X$) as a module over the algebra of finite correspondences. Then the rationally trivial 0-cycles on $X$ form an absolutely simple and essential submodule of $Z_0(X)$.

A remark on 0-cycles as modules over algebras of finite correspondences

Abstract

Given a smooth projective variety over a field, consider the -vector space of 0-cycles (i.e. formal finite -linear combinations of the closed points of ) as a module over the algebra of finite correspondences. Then the rationally trivial 0-cycles on form an absolutely simple and essential submodule of .
Paper Structure (8 sections, 7 theorems, 8 equations)

This paper contains 8 sections, 7 theorems, 8 equations.

Table of Contents

  1. Category algebras and non-degenerate modules
  2. Algebras of finite correspondences and their modules
  3. The socle of $Z_0(S)$
  4. Motivic $A_S$-modules
  5. Loewy filtrations on $Z_0(S)$
  6. The case of curves
  7. Consequences of the filtration conjecture
  8. Correspondences on non-proper varieties?

Key Result

Lemma 1.1

If $\mathcal{C}$ is a small preadditive category then $\mathcal{C}^{\vee}$ and $\mathrm{Mod}_{\mathcal{C}}$ are equivalent abelian categories. In particular, if two small preadditive categories $\mathcal{C}$ and $\mathcal{C}'$ are equivalent then the categories $\mathrm{Mod}_{\mathcal{C}}$ and $\mat

Theorems & Definitions (18)

  • Lemma 1.1: Morita equivalence
  • proof
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Example 2.4: P.Samuel, §2
  • Theorem 2.5
  • proof
  • ...and 8 more