Smith reduction and sublattices of finite rank with an application to toric varietes

In this note I give a proof of the existence of a Smith normal form for matrices with integer entries. The existence of a good basis for a lattice with a finite index sublattice is a consequence of the Smith normal form. I conclude with an easy application to toric varieties. Theorem 1. Let A be a matrix with integer entries. Then there exist integer-values invertible matrices C and B such that A = CDB, where D is a diagonal integervalued matrix. Proof. We call an elementary operation on a matrix the addition of an integer multiple of one row (or column) to another row (or column). By means of a combination of elementary operations one can exchange two rows or columns, at the cost of a minus sign. An elementary operation on a matrix A corresponds to the multiplication from the left or the right of A with an elementary matrix, which is a matrix that has 1 along the diagonal and vanishing entries off the diagonal, except at one off-diagonal entry, where it is integer-valued. Thus elemetary matrices are invertible and have unit determinant; in particular, a product of a finite number of elementary matrices is an integer-valued invertible matrix. Let now A = (aij)1≤i,j≤n be an integer-valued matrix. By exchanging rows and or columns, we may assume that a11 6= 0. The greatest common divisor of the elements in the first row and the first column (i.e. all aij with i or j equal to 1) is an integer combination of the latter; hence we may arrive at the situation (by using the Euclidean algorithm) where a11 divides all elements of the first column and of the first row, by only using elemtary operations on A. Since now a11 divides all elements in the first row and first collumn, we can sweep the first row and column clean, and arrive at the following form of the matrix A: