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Cotangent functors and Herzog's last theorem

Antonino Ficarra

TL;DR

This work studies cotangent functors $\text{T}_i(S/R,S)$ through Palamodov's DG-algebra approach to relate them to deformations and differential data. By constructing resolvents and the cotangent complex, it shows $\text{T}_i(S/R,S)\cong H_i(L)$ for a suitable complex $L$ and connects these functors to Koszul homology and Tor, yielding exact sequences that link vanishing of higher cotangent modules to complete intersections. The central aim is to illuminate Herzog's Last Theorem, presenting partial results and a conditional proof under rational Poincaré-series assumptions, thereby situating Herzog's question within a homological and deformation-theoretic framework. The results expose deep interactions between conormal modules, Koszul homology, and the vanishing of cotangent functors, contributing to the broader understanding of when a quotient by an ideal is a complete intersection and how this is reflected in cotangent data.

Abstract

We present the theory of cotangent functors following the approach of Palamodov, and a conjecture of Herzog relating the vanishing of certain cotangent functors to the property of being a complete intersection.

Cotangent functors and Herzog's last theorem

TL;DR

This work studies cotangent functors through Palamodov's DG-algebra approach to relate them to deformations and differential data. By constructing resolvents and the cotangent complex, it shows for a suitable complex and connects these functors to Koszul homology and Tor, yielding exact sequences that link vanishing of higher cotangent modules to complete intersections. The central aim is to illuminate Herzog's Last Theorem, presenting partial results and a conditional proof under rational Poincaré-series assumptions, thereby situating Herzog's question within a homological and deformation-theoretic framework. The results expose deep interactions between conormal modules, Koszul homology, and the vanishing of cotangent functors, contributing to the broader understanding of when a quotient by an ideal is a complete intersection and how this is reflected in cotangent data.

Abstract

We present the theory of cotangent functors following the approach of Palamodov, and a conjecture of Herzog relating the vanishing of certain cotangent functors to the property of being a complete intersection.
Paper Structure (7 sections, 13 theorems, 46 equations)

This paper contains 7 sections, 13 theorems, 46 equations.

Table of Contents

  1. DG-algebras and the module of Kälher differentials
  2. A glimpse to DG-algebras
  3. A glimpse to the module of Kälher differentials
  4. The cotangent functors
  5. Cotangent functors of quotient rings
  6. Koszul homology, $\textup{Tor}(S,S)$ and the first Cotangent modules
  7. Herzog's Last Theorem

Key Result

Theorem 2.1

Let $\varphi: R\rightarrow S$ be a ring homomorphism, $M$ a $S$-module and $X$ a resolvent of $\varphi$. Then the homology of the complex $\Omega_{X/R}\otimes_XM$ is independent of the choices made in the construction of $X$.

Theorems & Definitions (24)

  • Conjecture A
  • Conjecture B
  • Theorem 2.1
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • proof
  • Lemma 3.3
  • ...and 14 more