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Enumerative and planar combinatorics of trivariate monomial resolutions

Erika Ordog

Abstract

The canonical sylvan resolution is a resolution of an arbitrary monomial ideal over a polynomial ring that is minimal and has an explicit combinatorial formula for the differential. The differential is a weighted sum over lattice paths of weights of chain-link fences, which are sequences of faces that are linked to each other via higher-dimensional analogues of spanning trees. Along a lattice path in the three-variable case, these weights can be condensed to a single weight contributing to the combinatorial formula for the differential that bypasses any computation of chain-link fences. The main results in this paper express the sylvan matrix entries for monomial ideals in three variables as a sum over lattice paths of simpler weights that depend only on the number of specific Koszul simplicial complexes that lie along the corresponding lattice path. Certain entries have numerators equal to the number of lattice paths in $\mathbb{N}^2$ that follow specific restrictions.

Enumerative and planar combinatorics of trivariate monomial resolutions

Abstract

The canonical sylvan resolution is a resolution of an arbitrary monomial ideal over a polynomial ring that is minimal and has an explicit combinatorial formula for the differential. The differential is a weighted sum over lattice paths of weights of chain-link fences, which are sequences of faces that are linked to each other via higher-dimensional analogues of spanning trees. Along a lattice path in the three-variable case, these weights can be condensed to a single weight contributing to the combinatorial formula for the differential that bypasses any computation of chain-link fences. The main results in this paper express the sylvan matrix entries for monomial ideals in three variables as a sum over lattice paths of simpler weights that depend only on the number of specific Koszul simplicial complexes that lie along the corresponding lattice path. Certain entries have numerators equal to the number of lattice paths in that follow specific restrictions.

Paper Structure

This paper contains 8 sections, 23 theorems, 34 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sylvan matrix entries for the map $F_0 \leftarrow F_1$
  4. Coefficients in the maps from $F_2$ to $F_1$
  5. Lattice path cases
  6. Sylvan matrix entries $D_{ve}^{{\mathbf a} \mathbf{b}}$ when ${\mathbf a} = \mathbf{b} - ne_i$
  7. Sylvan matrix entries $D_{ve}^{{\mathbf a} \mathbf{b}, \lambda}$ along a single, non-unique lattice path
  8. Examples

Key Result

Theorem 2.15

The canonical sylvan homomorphism $D^{{\mathbf a} \mathbf{b}}: \widetilde{C}_i K^{\mathbf{b}}I \rightarrow \widetilde{C}_{i-1} K^{{\mathbf a}} I$ is given by its sylvan matrix, with entries

Figures (1)

  • Figure 1: All cases for Koszul complexes along lattice paths that begin by moving back in the $i$-direction contributing to the maps $F_1 \leftarrow F_2$

Theorems & Definitions (64)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 54 more