  Motivated by the goal of studying cluster algebras in infinite type, we study
the stability domains of modules for the preprojective algebra in the
corresponding infinite types. Specifically, we study real bricks: those modules
whose endomorphism algebra is a division ring and which have no
self-extensions. We define "shard modules" to be those real bricks whose
stability domain is as large as possible (meaning, of dimension one less than
the rank of the preprojective algebra). We show that all real bricks are
obtained by applying the Baumann-Kamnitzer reflection functors to simple
modules, and we give a recursive formula for the stability domain of a real
brick. We show that shard modules are in bijection with Nathan Reading's
"shards", and that their stability domains are the shards; we also establish
many foundational results about shards in infinite type which have not
previously appeared in print. With an eye toward applications to cluster
algebras, our paper is written to handle skew-symmetrizable as well as
skew-symmetric exchange matrices, and we therefore discuss the basics of the
theory of species for preprojective algebras. We also give some counterexamples
to show ways in which infinite type is more subtle than the well-studied finite
type cases.
