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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?

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    More context please. I would guess from the commutative algebra tag that the polynomial ring over that field would be a good guess though.2012-04-18
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    I am pretty awful at searching. But out of curiosity I typed "standard graded algebra" (with quotes) in Google. Quite a few hits, including definitions.2012-04-18
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    Alex's example is an example of what's describe in your new edit.2012-04-18
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    @AndréNicolas Heh. When I type in "standard graded algebra" into Google, this question is the first result.2012-04-18
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    I have just added some new informations. Please help me.2012-04-18
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    Yes, AlexYoucis gave an example you want. $R=A[x,y]$ with the natural grading.2012-04-18
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    "Standard graded algebra" = "homogeneous algebra" in Bruns & Herzog terminology.2012-06-18

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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.