Commutative algebra and the linear diophantine problem of Frobenius

Let $A$ be a finite set of relatively prime positive integers, and let $S(A)$ be the set of all nonnegative integral linear combinations of elements of $A$. The set $S(A)$ is a semigroup that contains all sufficiently large integers. The largest integer not in $S(A)$ is the Frobenius number of $A$, and the number of positive integers not in $S(A)$ is the genus of $A$. Sharp and Sylvester proved in 1884 that the Frobenius number of the set $A = \{a,b\}$ is $ab-a-b$, and that the genus of $A$ is $(a-1)(b-1)/2$. Graded rings and a simple form of Hilbert's syzygy theorem are used to give a commutative algebra proof of this result.