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Point varieties and point-exactness of Koszul algebras

Haigang Hu, Wenchao Wu, Yu Ye

Abstract

In this paper, we introduce the point-exact condition for a Koszul algebra $A$, which is useful for characterizing the (G1) condition of $A$ in the sense of Mori. Let $B = A/(f)$, where $f \in A_2$ is a regular normal element. We show that if $A$ satisfies the (G1) condition and is point-exact up to degree $\ell \geq 2$, then $B$ also satisfies the (G1) condition and is point-exact up to degree $\ell$. Moreover, we show that skew polynomial algebras satisfy the point-exact condition.

Point varieties and point-exactness of Koszul algebras

Abstract

In this paper, we introduce the point-exact condition for a Koszul algebra , which is useful for characterizing the (G1) condition of in the sense of Mori. Let , where is a regular normal element. We show that if satisfies the (G1) condition and is point-exact up to degree , then also satisfies the (G1) condition and is point-exact up to degree . Moreover, we show that skew polynomial algebras satisfy the point-exact condition.
Paper Structure (10 sections, 21 theorems, 80 equations)

This paper contains 10 sections, 21 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Matrices over a vector space
  4. Koszul algebras
  5. Point varieties
  6. Point-exact conditions
  7. Minimal resolutions
  8. Main results
  9. Skew polynomial algebras
  10. Appendix: Smooth $\pm 1$-skew quadric surfaces

Key Result

Theorem 1.2

Let $A$ be a connected graded algebra, $f \in A_m$ a regular normal element with $m \geq 1$, and $B = A/(f)$. Let $P_\bullet \xrightarrow{\sim} \Bbbk_A$ be a minimal resolution. The sequence $T_\bullet$ defined in (miniB), which is constructed from $P_\bullet$, is a free resolution of $\Bbbk_B$. If

Theorems & Definitions (51)

  • Theorem 1.2: Theorem \ref{['T1']}
  • Theorem 1.3: Theorem \ref{['thm-main']}
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6: AS87
  • Definition 2.7: ATV90
  • Definition 2.8: Mo06
  • ...and 41 more