Knowlpedia

Definition

Full lattice over an integral domain

A finitely generated submodule spanning a finite-dimensional vector space over the fraction field.

Let RR be an integral domain with KK, and let VV be a finite-dimensional KK-. A full RR-lattice in VV is an RR- L⊆VL\subseteq V that is and satisfies KL=VKL=V.

Equivalently, the scalar-extension map K⊗RL→VK\otimes_R L\to V, a⊗ℓ↦aℓa\otimes\ell\mapsto a\ell, is an isomorphism.

Integral coordinates

For R=ZR=\mathbb Z and K=QK=\mathbb Q, a full lattice is free of rank dim⁡QV\dim_{\mathbb Q}V. For example, Ze1⊕12Ze2\mathbb Z e_1\oplus\tfrac12\mathbb Z e_2 is a full lattice in Q2\mathbb Q^2.

Over a general , a full lattice is finitely generated and projective, but need not be free. An integral basis is therefore not part of the definition.

Related meanings

An is a full lattice also closed under multiplication and containing the identity. A is closed under the Lie bracket instead.

An additionally lives in a real inner-product space; no topology or inner product is required here. This is unrelated to an order-theoretic .

References
  1. John Voight, Quaternion Algebras, author's text, Definition 9.3.1, Remark 9.3.4, and Theorem 9.3.6.