2
$\begingroup$

I am so sorry if you feel this kind of question is not appropriate for MS. But I hope you can sympathize with me, I tried to find the answer in all my books and even Google but I found nothing.

My question is : What is a standard graded algebra over a ring?

Please help me. Thanks.

Edit:

In the following paper On the asymtotic linearity of Castelnuovo-Mumford regularity, there is a definition of standard graded algebra. Suppose that $A$ is a ring, $R$ is a standard graded $A$ algebra if $R_{0}=A$ and $R$ is generated by the element of $R_1$. I did not fully understand this definition. Can anyone give here an example?

  • 0
    "Standa$r$d graded algebra" = "homogeneous algebra" in Bruns & Herzog terminology.2012-06-18

1 Answers 1

2

The definition of standard graded algebra is correct. You'll also see this condition stated as the graded algebra $A$ being finitely generated over $A_0$ by elements of degree $1$. There is a geometric reason why you would naturally restrict consideration to such rings, related to the Proj construction and the Verenose embedding. Briefly, the geometric space $\text{Proj} ~A$ attached to $A$ is isomorphic to $\text{Proj} ~A_{(d)}$, where $A_{(d)} = \oplus_{n \geq 0} A_{nd}$ is the graded subring of $A$ which keeps only the graded pieces whose degrees are multiples of $d$. Moreover, for any $A$ that is finitely generated over $A_0$ (but not necessarily in degree $1$), there always exists a $d > 0$ such that $A_{(d)}$ is finitely generated in degree $1$. See Exercises 7.4.F and 7.4.G and the discussion following them in the current (April 13, 2012) version of Ravi Vakil's book Foundations of Algebraic Geometry.