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Two-sided homological properties of special and one-relator monoids

Robert D. Gray, Benjamin Steinberg

Abstract

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi-$\mathrm{FP}_n$ if its group of units is of type $\mathrm{FP}_n$. We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form $\langle A \mid r=1 \rangle$. In particular, we show that all such monoids are of type bi-$\mathrm{FP}_\infty$. Moreover, we show that if $r$ is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most $2$, while if $r$ is a proper power then it has infinite Hochschild cohomological dimension. For any non-special one-relator monoid $M$ with defining relation $u=v$ we show that if there is no nonempty word $w$ such that $u,v \in A^*w \cap w A^*$ then $M$ is of type bi-$\mathrm{FP}_\infty$ and has Hochschild cohomological dimension at most $2$.

Two-sided homological properties of special and one-relator monoids

Abstract

A monoid presentation is called special if the right-hand side of each defining relation is equal to 1. We prove results which relate the two-sided homological finiteness properties of a monoid defined by a special presentation with those of its group of units. Specifically we show that the monoid enjoys the homological finiteness property bi- if its group of units is of type . We also obtain results which relate the Hochschild cohomological dimension of the monoid to the cohomological dimension of its group of units. In particular we show that the Hochschild cohomological dimension of the monoid is bounded above by the maximum of 2 and the cohomological dimension of its group of units. We apply these results to prove a Lyndon's Identity type theorem for the two-sided homology of one-relator monoids of the form . In particular, we show that all such monoids are of type bi-. Moreover, we show that if is not a proper power then the one-relator monoid has Hochschild cohomological dimension at most , while if is a proper power then it has infinite Hochschild cohomological dimension. For any non-special one-relator monoid with defining relation we show that if there is no nonempty word such that then is of type bi- and has Hochschild cohomological dimension at most .

Paper Structure

This paper contains 9 sections, 18 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Finitely presented special monoids
  4. Proving that $N \cong G \ast C^*$ for some finite set $C$.
  5. Proving $M \times M$ is a free $N$-set
  6. Proving the set of collapsed edges is a finitely generated free $M \times M^{\mathrm{op}}$-set
  7. Two-sided homological finiteness properties of special monoids
  8. Applying the general result to special one-relator monoids
  9. Other one-relator monoids

Key Result

Lemma 2.1

Let $R$ be a ring. If $0\to A\to B\to C\to 0$ is an exact sequence of left $R$-modules such that $A$ is of type $\mathrm{FP}_{n-1}$ and $B$ is of type $\mathrm{FP}_n$, then $C$ is of type $\mathrm{FP}_n$.

Figures (1)

  • Figure 1: Homological finiteness properties satisfied by monoids that admit finite complete rewriting systems, and all implications between them. The proofs that all these implications hold, and that none of the implications in the diagram are reversible, can be found in the papers cremanns1994finitelafont1995newPride1995wang2000secondKobayashiOtto2003Kobayashi2005Pride2006Kobayashi2010.

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Proposition 4.1
  • proof
  • ...and 26 more