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For my thesis, I'm defining affine $k$-algebras to be reduced, finitely generated $k$-algebras--each of which turns out to be isomorphic to the quotient of a polynomial ring by a radical ideal.

I'm looking for a similar characterisation of "projective $k$-algebras" (i.e. those which occur as coordinate rings of projective varieties). A first attempt was to use reduced, finitely generated, graded $k$-algebras. However, after some searching it seems that for example $k[X^2, X^3]$ is not isomorphic to the quotient of a graded polynomial ring by a homogeneous radical ideal, unless we change the grading: see here and here.

Is there an elegant way of tweaking this definition to enforce the correct behaviour?

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  • $\begingroup$ Do you want to say somehow "graded $k$-algebras"? AFAIK the coordinate rings of projective varieties are graded $k$-algebras. $\endgroup$
    – user26857
    May 14, 2015 at 20:29
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    $\begingroup$ Indeed they are graded $k$-algebras, but what I'm looking for is a characterisation, so that every $k$-algebra of such type arises as the coordinate ring of some projective variety. $\endgroup$
    – gerardlouw
    May 14, 2015 at 20:33
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    $\begingroup$ To start off with they clearly need to be finitely generated. Adding the restriction that they are reduced then takes care of the fact that we quotient by radical homogeneous ideals. The example I give of $k[X^2, X^3]$ seems to imply that we need a further restriction, unless you can provide a projective variety for which this is isomorphic to the coordinate ring. $\endgroup$
    – gerardlouw
    May 14, 2015 at 20:37
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    $\begingroup$ Well, in my definition graded $k$-algebras are finitely generated. One can also assume they are generated by degree one elements, and this is how they appear in algebraic geometry. But for some reasons it seems you want to let the generators having arbitrary degrees. (Or not?) In your example the ring is isomorphic to $k[u,v]/(u^2-v^3)$ and you can set $\deg u=3$ and $\deg v=2$. $\endgroup$
    – user26857
    May 14, 2015 at 20:41
  • $\begingroup$ Thank you, it seems that restricting the generators to degree one might provide the correct characterisation. $\endgroup$
    – gerardlouw
    May 14, 2015 at 20:46

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In other words, you want to know for which graded $k$-algebras $S$ the associated projective scheme $\operatorname{Proj} S$ is in fact a variety.

The usual scheme-theoretic definition of a (possibly reducible) variety over $k$ is a reduced, separated scheme of finite type over $k$. Since projective morphisms are automatically of finite type and separated (at least for noetherian schemes) [1], we just need $\operatorname{Proj S}$ to be reduced and projective over $k$.

To ensure that $S$ is reduced and projective over $k$, it's enough [2] that $S$ be reduced, $S_0 = k$, and $S$ finitely generated by $S_1$.

This condition isn't quite necessary for $\operatorname{Proj} S$ to be a projective scheme, I think -- for instance, I believe you could just take a graded ring and double all the gradings without affecting the associated scheme -- but I believe (someone should probably check this) that every projective variety arises from such a ring.

[1] Hartshorne, Theorem II.4.9

[2] Hartshorne, Example II.4.8.1

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