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Closed projective embedding from a radical homogeneous ideal

Statement

Assume the Axiom of Choice. If $J$ is homogeneous radical and $V_{+} (J) \neq \emptyset$ , then $I_{+} (V_{+} (J)) = J$ ; its inclusion in $P_{k}^{n}$ is a closed projective embedding and its homogeneous coordinate ring is $k [x_{0}, \dots, x_{n}] / J$ .

Facts & Assumptions

Given: The Axiom of Choice, the algebraically closed base field $k$ fixed by projective space points, and a homogeneous radical ideal $J \subseteq k [x_{0}, \dots, x_{n}]$ with nonempty $X = V_{+} (J)$ .

[F1]

The projective space $P_{k}^{n}$ is defined over a fixed algebraically closed field $k$ (projective space points); its projective algebraic sets are the homogeneous zero loci $V_{+} (J)$ (projective algebraic set).

[F2]

Over this algebraically closed field, under AC the projective radical-ideal correspondence gives $I_{+} (V_{+} (J)) = J$ for homogeneous radical $J$ with nonempty zero locus (projective irreducibility homogeneous prime).

[F3]

The homogeneous coordinate ring of a projective algebraic set $X$ is $S (X) = k [x_{0}, \dots, x_{n}] / I_{+} (X)$ , with its induced grading (homogeneous coordinate ring).

[A1]

Assume the Axiom of Choice. Its use is the radical-ideal correspondence in [F2] (The Axiom of Choice).

Proof

technique · direct

1.1A1F1F2given

The base field is algebraically closed by the standing projective-space convention in [F1]. Since $J$ is homogeneous and radical and $X = V_{+} (J)$ is nonempty, all hypotheses of [F2] hold; under [A1] it gives $I_{+} (X) = J$ .

2.1F1step 1.1

By [F1], $X = V_{+} (J)$ is projectively closed. Its inclusion identifies $X$ with this closed algebraic subset and has the identity as inverse onto its image, so it is a closed projective embedding in the classical coordinate sense used here.

3.1F3step 1.1step 2.1algebra∎

Substituting $I_{+} (X) = J$ from step 1.1 into [F3] gives $S (X) = k [x_{0}, \dots, x_{n}] / J$ . The quotient carries the induced grading because $J$ is homogeneous. This proves the stated coordinate-ring formula together with the closed embedding in step 2.1.

Depends on

Used by

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Sources

J. S. Milne, Algebraic Geometry, Chapter 6 (standard reference, not scraped)
Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)