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Betti numbers for connected sums of graded Gorenstein artinian algebras

Nasrin Altafi, Roberta Di Gennaro, Federico Galetto, Sean Grate, Rosa M. Miro-Roig, Uwe Nagel, Alexandra Seceleanu, Junzo Watanabe

Abstract

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring.

Betti numbers for connected sums of graded Gorenstein artinian algebras

Abstract

The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that a connected sum of doublings is the doubling of a fiber product ring.
Paper Structure (12 sections, 21 theorems, 116 equations, 3 tables)

This paper contains 12 sections, 21 theorems, 116 equations, 3 tables.

Table of Contents

  1. Introduction
  2. background
  3. Oriented AG algebras
  4. Macaulay dual generators
  5. Fiber product
  6. Connected sum
  7. Doubling
  8. Graded Betti numbers of the Connected Sum
  9. Two Summands
  10. Arbitrary many summands
  11. Connected Sum as a Doubling
  12. Motivating examples

Key Result

Lemma 2.5

Let $R = K[x_1,\ldots,x_m]$ and $S = K[y_1,\ldots,y_n]$ be polynomial rings over $K$ with homogeneous maximal ideals $\mathbf{x} = (x_1,\ldots,x_m)$ and $\mathbf{y} = (y_1,\ldots,y_n)$, respectively. Let $Q = R \otimes_K S = K[x_1,\ldots,x_m, y_1,\ldots,y_n]$. If $A = R/\mathfrak a$ and $B = S/\math where in eq:fiber product presentation$\mathfrak a, \mathfrak b, \mathbf{x},\mathbf{y}$ denote exte

Theorems & Definitions (56)

  • Definition 2.1: IMS
  • Remark 2.2
  • Example 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Example 2.6
  • Definition 2.7
  • Remark 2.8
  • Lemma 2.9
  • ...and 46 more