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'' (and thus also an ''R''-module). 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 $M \otimes_R K$, 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 (with respect to a suitable basis) 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 $M_$ of ''M'' at $\mathfrak$. 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 $R_$-lattices $N_$ there exists an ''R''-lattice ''M'' satisfying $M_ = N_$. Over Dedekind domains a similar correspondence exists between ''R''-lattices and collections of lattices $N_\mathfrak$ over the completions of ''R'' with respect at primes $\mathfrak$.

A pair of lattices ''M'' and ''N'' over ''R'' admit a notion of relative index analogous to that of integer lattices in $\mathbb^n$. If ''M'' and ''N'' are projective (e.g. if ''R'' is a Dedekind domain), then ''M'' and ''N'' have trivial relative index if and only if ''M = N''.

Pure sublattices

An ''R''-submodule ''N'' of ''M'' that is itself a lattice is an ''R''-pure sublattice if ''M''/''N'' is ''R''-torsion-free. There is a one-to-one correspondence between ''R''-pure sublattices ''N'' of ''M'' and ''K''- subspaces ''W'' of ''V'', given byReiner (2003) p. 45
: $N \mapsto W = K \cdot N ; \quad W \mapsto N = W \cap M. \,$

See also

* Lattice (group), for the case where ''M'' is a Z-module embedded in a vector space ''V'' over the field of real numbers R, and the Euclidean metric is used to describe the lattice structure

References

*
* {{cite book , last=Reiner , first=I. , authorlink=Irving Reiner , title=Maximal Orders , series=London Mathematical Society Monographs. New Series , volume=28 , publisher=Oxford University Press , year=2003 , isbn=0-19-852673-3 , zbl=1024.16008

Module theory