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Affine Semigroup Algebras And Their Fibered Sums

C-Y. Jean Chan, I-Chiau Huang, Jung-Chen Liu

Abstract

We study affine semigroup rings as algebras over subsemigroup rings. From this relative viewpoint with respect to a given subsemigroup ring, the fibered sum of two affine semigroup algebras is constructed. Such a construction is compared to the tensor product and to the classical gluings of affine semigroup rings as defined in Rosales (1997). While fibered sum can always be achieved, gluings of affine semigroup rings do not always exist. Therefore, we further investigate when the fibered sum of affine semigroup algebras gives rise to a gluing. A criterion is recovered in terms of the defining semigroups under which the gluing may take place.

Affine Semigroup Algebras And Their Fibered Sums

Abstract

We study affine semigroup rings as algebras over subsemigroup rings. From this relative viewpoint with respect to a given subsemigroup ring, the fibered sum of two affine semigroup algebras is constructed. Such a construction is compared to the tensor product and to the classical gluings of affine semigroup rings as defined in Rosales (1997). While fibered sum can always be achieved, gluings of affine semigroup rings do not always exist. Therefore, we further investigate when the fibered sum of affine semigroup algebras gives rise to a gluing. A criterion is recovered in terms of the defining semigroups under which the gluing may take place.
Paper Structure (5 sections, 21 theorems, 46 equations)

This paper contains 5 sections, 21 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Fibered Sums
  3. Apéry elements
  4. Algebras
  5. Fibered Sum versus Gluing

Key Result

Lemma 2.3

In the category of cancellative monoids, given homomorphisms $S\to S_1$ and $S\to S_2$, the monoid $S_1 \oplus_{S} S_2$ together with the canonical homomorphisms $S_1\to S_1 \oplus_{S} S_2$ and $S_2\to S_1 \oplus_{S} S_2$ is the fibered sum of $S_1$ and $S_2$ over $S$.

Theorems & Definitions (52)

  • Example 2.1
  • Lemma 2.3
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Example 2.7
  • Proposition 2.8
  • proof
  • Corollary 2.9
  • ...and 42 more