Lattice in toric geometry ($N\cong {\mathbb{Z}}^n,M=\mathrm{Hom}(N,\mathbb{Z})$)

A toric lattice is a finite-rank free abelian group $N$, embedded as a discrete full-rank subgroup of $N_{\mathbb{R}}=N{\otimes }_{\mathbb{Z}}\mathbb{R}$. Its dual is $M=\mathrm{Hom}(N,\mathbb{Z})$. This lattice of integral vectors is distinct from an order-theoretic lattice. It supplies the integral structure for toric cones and their dual monomials.

A primitive lattice vector is a nonzero element $u$ of a finite-rank free abelian group $N$ that cannot be written $u=kv$ with $v\in N$ and an integer $|k|>1$. In an integral basis, this is equivalent to the greatest common divisor of its coordinates being one. The property is independent of the chosen lattice basis.

For example, $(2,3)$ is primitive although neither coordinate is one: the integer functional $\ell (x,y)=-x+y$ satisfies $\ell (2,3)=1$. Hence $N=\mathbb{Z}u\oplus \ker \ell$, showing that $u$ extends to a lattice basis. In contrast, $(2,4)=2(1,2)$ is not primitive.

Every rational ray of a fan contains a unique primitive generator pointing along it. This generator gives the integral normalization used in toric divisors and cocharacters; replacing it by a larger positive multiple leaves the geometric ray unchanged but changes its lattice data.

A lattice basis $v_1,\dots ,v_n$ of a toric lattice $N$ expresses each element uniquely as ${\sum }_i{a}_i{v}_i$ with $a_i\in \mathbb{Z}$. Relative to a fixed integral basis, the new vectors form a lattice basis exactly when the matrix of their coordinates has determinant $\pm 1$.

Being a real basis is weaker. The vectors $(1,0),(1,2)$ are linearly independent over the real numbers, but their integer combinations have even second coordinate and do not fill ${\mathbb{Z}}^2$. Their determinant is two.

A primitive lattice vector is nonzero and generates a saturated sublattice. In coordinates its entries have greatest common divisor one. The Euclidean algorithm completes it to a lattice basis. The nonzero condition excludes the zero vector, whose zero subgroup is saturated but which cannot be a basis vector.^[ref]