Let
be a graph whose edges are labeled by ideals of a commutative ring.
We introduce a
generalized spline, which is a vertex labeling of
by
elements of the ring so that the difference between the labels of any two adjacent
vertices lies in the corresponding edge ideal. Generalized splines arise naturally in
combinatorics (algebraic splines of Billera and others) and in algebraic topology
(certain equivariant cohomology rings, described by Goresky, Kottwitz, and
MacPherson, among others). The central question of this paper asks when an
arbitrary edge-labeled graph has nontrivial generalized splines. The answer is
“always”, and we prove the stronger result that the module of generalized
splines contains a free submodule whose rank is the number of vertices in
.
We describe the module of generalized splines when
is a tree, and give several ways to describe the ring of generalized
splines as an intersection of generalized splines for simpler subgraphs of
. We
also present a new tool which we call the
GKM matrix, an analogue of the incidence
matrix of a graph, and end with open questions.
Keywords
splines, GKM theory, equivariant cohomology, algebraic
graph theory