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Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring

Henry Potts-Rubin

TL;DR

This work determines the minimal graded free resolution of the $k$-linear derivations $ ext{Der}_{R|k}$ on a determinantal ring $R=Q/I_n$, where $I_n$ is generated by the maximal minors of an $n\times(n+1)$ generic matrix. The authors construct a relative bar resolution via a differential graded (dg) Hilbert-Burch algebra $\\mathcal{A}$ and a dg module $\\mathcal{U}$ resolving $ ext{coker}(J^T_R)$, establish a dg action of $\\mathcal{A}$ on $\\mathcal{U}$, and show that both $\,R o Q\,$ and $\, ext{coker}(J^T_R)\$ are Golod. Using Iyengar's relative bar construction and truncation, they produce a minimal $R$-free resolution of $ ext{Der}_{R|k}$ with explicit generators and maps, and compute its Poincaré series as $P^R_{ ext{Der}_{R|k}}(t)=\dfrac{n(n+1)(2+nt)}{1-t-nt^2}$. Notably, the $n=2$ case yields a linear resolution, while the construction provides a characteristic-free, algebraic route to the derivation module’s resolution and generators, aligning with and complementing known results obtained by other methods.

Abstract

We use the construction of the relative bar resolution via differential graded structures to obtain the minimal graded free resolution of $\text{Der}_{R \mid k}$, where $R$ is a determinantal ring defined by the maximal minors of an $n \times (n+1)$ generic matrix and $k$ is its coefficient field. Along the way, we compute an explicit action of the Hilbert-Burch differential graded algebra on a differential graded module resolving the cokernel of the Jacobian matrix whose kernel is $\text{Der}_{R \mid k}$. As a consequence of the minimality of the resulting relative bar resolution, we get a minimal generating set for $\text{Der}_{R \mid k}$ as an $R$-module, which, while already known, has not been obtained via our methods.

Resolving the Module of Derivations on an $n \times (n+1)$ Determinantal Ring

TL;DR

This work determines the minimal graded free resolution of the -linear derivations on a determinantal ring , where is generated by the maximal minors of an generic matrix. The authors construct a relative bar resolution via a differential graded (dg) Hilbert-Burch algebra and a dg module resolving , establish a dg action of on , and show that both and are Golod. Using Iyengar's relative bar construction and truncation, they produce a minimal -free resolution of with explicit generators and maps, and compute its Poincaré series as . Notably, the case yields a linear resolution, while the construction provides a characteristic-free, algebraic route to the derivation module’s resolution and generators, aligning with and complementing known results obtained by other methods.

Abstract

We use the construction of the relative bar resolution via differential graded structures to obtain the minimal graded free resolution of , where is a determinantal ring defined by the maximal minors of an generic matrix and is its coefficient field. Along the way, we compute an explicit action of the Hilbert-Burch differential graded algebra on a differential graded module resolving the cokernel of the Jacobian matrix whose kernel is . As a consequence of the minimality of the resulting relative bar resolution, we get a minimal generating set for as an -module, which, while already known, has not been obtained via our methods.
Paper Structure (6 sections, 19 theorems, 119 equations)

This paper contains 6 sections, 19 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Useful Lemmas
  4. Resolving the Cokernel over $Q$
  5. Differential Graded Structure
  6. The Resolution of $\mathop{\mathrm{Der}}\nolimits_{R \mid k}$

Key Result

Theorem 1

Let $\mathcal{A}$ be the minimal graded $Q$-free resolution of $R$ and $\mathcal{U}$ be the minimal graded $Q$-free resolution of $\mathop{\mathrm{coker}}\nolimits J^T_R$. The minimal graded $R$-free resolution of $\mathop{\mathrm{Der}}\nolimits_{R \mid k}$ is the truncated relative bar resolution where $M_{i,j}$ comes from the action $\mathcal{A}_i\cdot \mathcal{U}_j$ of the dg algebra $\mathcal

Theorems & Definitions (56)

  • Theorem : Theorem \ref{['mainresult']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Proposition 2.7: Proposition 5.2.4, IFR
  • Corollary 2.8
  • proof
  • ...and 46 more