Lattice (module)
In mathematics, particularly in the field of ring theory, a lattice is an algebraic structure which, informally, provides a general framework for taking a sparse set of points in a larger space. Lattices generalize several more specific notions, including integer lattices in real vector spaces, orders in algebraic number fields, and fractional ideals in integral domains. Formally, a lattice is a kind of module over a ring that is embedded in a vector space over a field.
Formal definition
Let R be an integral domain with field of fractions K, and let V be a vector space over K. An R-submodule M of a V is called a lattice if M is finitely generated over R. It is called full if, i.e. if M contains a K-basis of V. Some authors require lattices to be full, but we do not adopt this convention in this article.Any finitely-generated torsion-free module M over R can be considered as a full R-lattice by taking as the ambient space, the extension of scalars of M to K. To avoid this ambiguity, lattices are usually studied in the context of a fixed ambient space.
Properties
The behavior of the base ring R of a lattice M strongly influences the behavior of M. If R is a Dedekind domain, M is completely decomposable as a direct sum of fractional ideals. Every lattice over a Dedekind domain is projective.Lattices are well-behaved under localization and completion: A lattice M is equal to the intersection of all the localizations of M at. Further, two lattices are equal if and only if their localizations are equal at all primes. Over a Dedekind domain, the local-global-dictionary is even more robust: any two full R-lattices are equal all all but finitely many localizations, and for any choice of -lattices there exists an R-lattice M satisfying. Over Dedekind domains a similar correspondence exists between R-lattices and collections of lattices over the completions of R with respect at primes.
A pair of lattices M and N over R admit a notion of relative index analogous to that of integer lattices in. If M and N are projective, then M and N have trivial relative index if and only if M = N.