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Homological properties of rings defined by $n+1$ general quadrics in $n$ variables

Rachel Diethorn, Sema Güntürkün, Alexis Hardesty, Pinar Mete, Liana Şega, Aleksandra Sobieska, Oana Veliche

TL;DR

This work analyzes rings defined by $n+1$ general quadrics in $n$ variables via their almost complete intersection structure and a linked Gorenstein partner. By constructing Golod maps from a complete intersection $P$ to $R$ and $A$, the authors derive explicit rational Poincaré series for the residue field and establish that $A$ has minimal Backelin rate while $R$ does not, with $A$’s Yoneda algebra generated in degrees $1$ and $2$. They show that all finitely generated modules over $R$ and $A$ have rational Poincaré series, and that $R$ is level; for odd $n$ they provide bounds on Betti numbers in terms of auxiliary rings, plus exact values for certain Betti numbers, consistent with conjectures in the literature. The appendix furnishes general Golod criteria via trivial Massey operations, underpinning the Golod proofs in the main text and enabling the broad generalization of Koszul-like properties to these almost complete intersection and linked-Gorenstein settings.

Abstract

We study the almost complete intersection ring $R$ defined by $n+1$ general quadrics in a polynomial ring in $n$ variables over a field $\sf{k}$ and a corresponding linked Gorenstein ring $A$. The overarching theme is that, while not Koszul (except for some small values of $n$), these rings have homological properties that extend those of Koszul rings. We establish that finitely generated modules over these rings have rational Poincaré series and we give concrete formulas for the Poincaré series of $\sf{k}$ over both $A$ and $R$. We also show that $A$ has minimal rate and its Yoneda algebra $\text{Ext}_A(\sf{k},\sf{k})$ is generated by its elements of degrees $1$ and $2$. While the graded Betti numbers of $R$ and $A$ over the polynomial ring are not known when $n$ is odd, our approach provides bounds and yields values for two of these Betti numbers, showing in particular that $R$ is level.

Homological properties of rings defined by $n+1$ general quadrics in $n$ variables

TL;DR

This work analyzes rings defined by general quadrics in variables via their almost complete intersection structure and a linked Gorenstein partner. By constructing Golod maps from a complete intersection to and , the authors derive explicit rational Poincaré series for the residue field and establish that has minimal Backelin rate while does not, with ’s Yoneda algebra generated in degrees and . They show that all finitely generated modules over and have rational Poincaré series, and that is level; for odd they provide bounds on Betti numbers in terms of auxiliary rings, plus exact values for certain Betti numbers, consistent with conjectures in the literature. The appendix furnishes general Golod criteria via trivial Massey operations, underpinning the Golod proofs in the main text and enabling the broad generalization of Koszul-like properties to these almost complete intersection and linked-Gorenstein settings.

Abstract

We study the almost complete intersection ring defined by general quadrics in a polynomial ring in variables over a field and a corresponding linked Gorenstein ring . The overarching theme is that, while not Koszul (except for some small values of ), these rings have homological properties that extend those of Koszul rings. We establish that finitely generated modules over these rings have rational Poincaré series and we give concrete formulas for the Poincaré series of over both and . We also show that has minimal rate and its Yoneda algebra is generated by its elements of degrees and . While the graded Betti numbers of and over the polynomial ring are not known when is odd, our approach provides bounds and yields values for two of these Betti numbers, showing in particular that is level.
Paper Structure (9 sections, 23 theorems, 135 equations)

This paper contains 9 sections, 23 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Regularity over the complete intersection ring $P$
  4. Regularity in the general case
  5. Computation of Poincaré series
  6. Rate computations
  7. The Yoneda Ext algebra of $A$
  8. Bounds on Betti numbers over $Q$ when $n$ is odd
  9. Appendix

Key Result

Theorem 1

Let $n\ge 2$ be an integer and set $\ell=\left\lfloor\frac{n-2}{2}\right\rfloor$. Let ${\sf{k}}$ be an infinite field of characteristic zero or greater than $n$ and consider the polynomial ring $Q={\sf{k}}[x_1, \dots, x_n]$. A general quotient $R$ of $Q$ defined by $n+1$ quadrics $f_1, \dots, f_{n+1 The Gorenstein ring $A=Q/G$ with $G=(f_1, \dots, f_n)\colon (f_{n+1})$ has the following properties

Theorems & Definitions (59)

  • Theorem 1
  • Remark 2.8
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Remark 3.6
  • Proposition 3.8
  • proof
  • Definition 4.2
  • ...and 49 more