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 V = K · M, i.e. if M contains a K-basis of V. ^[1] Some authors require lattices to be full, but we do not adopt this convention in this article. ^[2]

Any finitely-generated torsion-free module M over R can be considered as a full R-lattice by taking as the ambient space ${\displaystyle M\otimes _{R}K}$, 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. ^[3]

Lattices are well-behaved under localization and completion: A lattice M is equal to the intersection of all the localizations ${\displaystyle M_{({\mathfrak {p}})}}$ of M at ${\displaystyle {\mathfrak {p}}}$. 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 ^[4] of ${\displaystyle R_{({\mathfrak {p}})}}$-lattices ${\displaystyle N_{({\mathfrak {p}})}}$ there exists an R-lattice M satisfying ${\displaystyle M_{({\mathfrak {p}})}=N_{({\mathfrak {p}})}}$. Over Dedekind domains a similar correspondence exists between R-lattices and collections of lattices ${\displaystyle N_{\mathfrak {p}}}$ over the completions of R with respect at primes ${\displaystyle {\mathfrak {p}}}$. ^[5]

A pair of lattices M and N over R admit a notion of relative index analogous to that of integer lattices in ${\displaystyle \mathbb {R} ^{n}}$. 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. ^[6]

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 by ^[7]

${\displaystyle N\mapsto W=K\cdot N;\quad W\mapsto N=W\cap M.\,}$

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

1. Reiner (2003) pp. 44, 108
2. Voight (2021) p. 141
3. Voight (2021) p. 142
4. We require that the local lattices be consistent, in the sense that there exists some R-lattice P with ${\displaystyle P_{({\mathfrak {p}})}=N_{({\mathfrak {p}})}}$ for all but finitely many primes.
5. Voight (2021), pp. 143–146
6. Voight 2021, p. 147 f.
7. Reiner (2003) p. 45

• Voight, John (2021). Quaternion Algebras. Graduate Texts in Mathematics. Vol. 288. Springer. ISBN   978-3-030-56692-0.
• Reiner, I. (2003). Maximal Orders. London Mathematical Society Monographs. New Series. Vol. 28. Oxford University Press. ISBN   0-19-852673-3. Zbl   1024.16008.