Graded algebra

A graded algebra over a field $k$ is a $k$-algebra $A={⨁}_{n\ge 0}{A}_n$ whose multiplication satisfies $A_i{A}_j\subseteq A_{i+j}$.

A standard graded algebra over a field $k$ has $A_0=k$ and is generated as a $k$-algebra by finitely many elements of degree one. Its Hilbert function agrees eventually with a polynomial.