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A freeness criterion for complexes with derived actions

Sylvain Brochard, Srikanth B. Iyengar, Chandrashekhar B. Khare

TL;DR

This work develops a freeness criterion for the zeroth homology of finite free complexes with derived actions over local rings, seeking to avoid full patching. It identifies a precise defect bound $p=\operatorname{edim}A-\operatorname{edim}B$ that governs freeness and provides several cases where $\operatorname{H}_0(F)$ is free: (i) the regular case under derived-action bounds, (ii) the case where $F$ is a complex of $B$-modules with finite flat dimension, and (iii) the complete intersection case yielding detailed Betti-number relations and a Koszul-model conclusion. It also discusses consequences for patching in number theory, showing that derived-action freeness can obstruct strengthening patching data and lifting Hecke actions, while highlighting favorable outcomes when $B/\mathfrak{m}_A B$ is a complete intersection. Overall, the results support the plausibility of a non-patching freeness criterion and clarify the interplay between derived actions, depth, dimension, and complete intersection structure in various regimes.

Abstract

Inspired by the patching method of Calegari and Geraghty, and a conjecture of de Smit that has been proved by the first author, we present a conjectural freeness criterion without patching for complexes over commutative noetherian local rings with derived actions, and verify it in several cases.

A freeness criterion for complexes with derived actions

TL;DR

This work develops a freeness criterion for the zeroth homology of finite free complexes with derived actions over local rings, seeking to avoid full patching. It identifies a precise defect bound that governs freeness and provides several cases where is free: (i) the regular case under derived-action bounds, (ii) the case where is a complex of -modules with finite flat dimension, and (iii) the complete intersection case yielding detailed Betti-number relations and a Koszul-model conclusion. It also discusses consequences for patching in number theory, showing that derived-action freeness can obstruct strengthening patching data and lifting Hecke actions, while highlighting favorable outcomes when is a complete intersection. Overall, the results support the plausibility of a non-patching freeness criterion and clarify the interplay between derived actions, depth, dimension, and complete intersection structure in various regimes.

Abstract

Inspired by the patching method of Calegari and Geraghty, and a conjecture of de Smit that has been proved by the first author, we present a conjectural freeness criterion without patching for complexes over commutative noetherian local rings with derived actions, and verify it in several cases.

Paper Structure

This paper contains 7 sections, 13 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The regular case
  4. The case of a complex of B-modules
  5. A consequence for patching
  6. The complete intersection case
  7. Graded modules over a non-commutative algebra

Key Result

Theorem 1.2

In the context of Question qu:Shekhar assume furthermore that $F$ is quasi-isomorphic in ${\mathsf{D}}_{{}}(A)$ to a complex of $B$-modules. Then $\operatorname{H}_i(F)=0$ for $i\ne 0$, the $B$-module $\operatorname{H}_0(F)$ is free, and the map $A\to B$ is an exceptional complete intersection.

Theorems & Definitions (31)

  • Theorem 1.2
  • Theorem 1.3
  • Example 2.2
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['th:regular_case']}
  • Theorem 4.1
  • Example 4.2
  • Lemma 4.4
  • ...and 21 more