Definitions

Let \( M \) be a finitely-generated, graded module over \( R = \mathbb{K}[x_1,\dots,x_n] \).

• Definition. A (finite) free resolution \( F_\bullet \) of \( M \) is an exact sequence of finitely-generated, free modules \[ F_\bullet \, \colon \quad 0 \longrightarrow F_n \longrightarrow F_{n-1} \longrightarrow \dots \longrightarrow F_1 \longrightarrow F_0 \longrightarrow M \longrightarrow 0. \] If the \( F_i \) are graded, then we say that \( F_\bullet \) is a graded free resolution of \( M \). We also say \( F_\bullet \) is minimal if there are no constant terms in any of the maps.
• Definition. Suppose \( F_\bullet \) is a graded free resolution of \( M \). The Betti table (or Betti diagram) of \( M \) is a diagram where the entry \( b_{i,j} \) in column \( i \) and row \( j \) is the rank of the free module \( F_i \) in the \( j \)-th degree, i.e., \( b_{i,j} \) indicates that \( F_i \) has a summand of the form \( R(-i-j)^{b_{i,j}} \).
• Definition. The Castelnuovo-Mumford regularity (or simply regularity) of \( M \), denoted \( \text{reg}(M) \), is the smallest integer \( r \) such that \( b_{i,r} \neq 0 \) for all \( i \), i.e., the index of the bottom-most nonzero row in the Betti table of \( M \).

Motivations

The regularity of a module is a fundamental invariant that measures the complexity of the module's generators and relations. While it has many equivalent formulations, the definition above is very useful for intuition as it comes directly from the minimal free resolution. Of importance, the regularity gives a bound on the maximal degree of the minimal generators of the module. With the regularity, we can also deduce bounds relating to syzygy computations and describe the complexity of Gröbner basis algorithms in terms of the regularity.

The projective dimension (the length of the minimal free resolution, or the index of the last nonzero column of the Betti table) of \( M \) is bounded by the number of variables; this is Hilbert's syzygy theorem. Unlike the projective dimension, the regularity does not enjoy such a simple bound, and in fact is known to be much more difficult to compute. For this reason, much work has gone into understanding the regularity of various classes of modules, with much motivation coming from algebraic geometry. Bounding the regularity is a central problem in commutative algebra.

Toric varieties

Toric varieties are a class of varieties that also have a combinatorial description, opening the door to many computational techniques. Many classical examples in algebraic geometry are toric varieties, such as projective spaces, rational normal curves, etc. Because of their combinatorial data, toric varieties provide provide a rich source of examples to experiment with. In this context, we are interested in understanding the regularity of the coordinate ring of a toric variety.

When the toric variety is a monomial curve, the combinatorial data is encoded as a numerical semigroup: integer points on a line. In 1996, L'vovsky showed that the regularity is bounded by the sum of the two largest "gaps" in the semigroup, a combinatorial bound for a notoriously hard-to-compute algebraic invariant. So, it should be expected that a similar combinatorial bound exists for the next case: toric surfaces.