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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A nilpotent irrelevant ideal gives empty Proj

Example

Let $k$ be a field and let $S = k [ε] / (ε^{2})$ be graded by $\deg ε = 1$ and $\deg k = 0$ , so that $S_{0} = k$ and $S_{+} = (ε)$ with $ε^{2} = 0$ . Then $S_{+}$ is a nilpotent ideal, $Proj S = \emptyset$ , and yet $Spec S$ is nonempty: the ring $k [ε] / (ε^{2})$ has the single prime ideal $(ε)$ . So emptiness of Proj is not emptiness of the spectrum, and it is detected by the nilpotency of the irrelevant ideal.

Facts & Assumptions

Given: A field $k$ , the graded ring $S = k [ε] / (ε^{2})$ with $\deg ε = 1$ .

[F1]

$Proj S$ is the set of homogeneous prime ideals $p$ with $S_{+} \subseteq̸ p$ ; a prime contains every nilpotent element; and $S = ⨁_{d \geq 0} S_{d}$ with $S_{1} = k ε$ and $S_{d} = 0$ for $d \geq 2$ . (Points of Proj of a graded ring)

[F2]

$Proj S = \emptyset$ if and only if every homogeneous element of $S_{+}$ is nilpotent; if $S_{+}$ is finitely generated this is equivalent to $S_{+}$ being nilpotent. (Empty Proj and irrelevant torsion)

[F3]

In the ring $k [ε] / (ε^{2})$ every prime ideal contains $ε$ , so $(ε)$ is the unique prime, and it is maximal; the localisation $S_{ε}$ is the zero ring. [algebra]

Verification

technique · direct: identify $S_+$, apply the emptiness criterion, and exhibit the point of $\operatorname{Spec}S$

1.1algebra

The irrelevant ideal is nilpotent. Here $S_{+} = k ε$ consists of the multiples of $ε$ , and $S_{+}^{2} = k ε^{2} = 0$ ; hence every element of $S_{+}$ is nilpotent and $S_{+} = (ε)$ is finitely generated.

1.2F1F3algebra

The standard chart is empty. For the homogeneous element $ε$ of degree $1$ we have $S_{(ε)} = (S [ε^{- 1}])_{0}$ , and $S [ε^{- 1}] = 0$ because $ε$ is nilpotent; hence $D_{+} (ε) = Spec S_{(ε)} = Spec 0 = \emptyset$ , and since $ε$ generates $S_{+}$ this is the only standard open.

2.1F1F2step 1.1

Proj is empty. By step 1.1 every homogeneous element of $S_{+}$ is nilpotent, so [F2] gives $Proj S = \emptyset$ ; equivalently, any homogeneous prime $p \subseteq S$ contains the nilpotent $ε$ , hence contains $S_{+} = (ε)$ and is excluded from $Proj S$ .

3.1F1F3algebra

The spectrum is nonempty. In $k [ε] / (ε^{2})$ the element $ε$ is nilpotent but nonzero, so the ideal $(ε)$ is proper; every prime contains the nilpotent $ε$ , so $(ε)$ is the unique prime ideal and $Spec S = {(ε)} \neq \emptyset$ , in contrast with step 2.1.

4.1

Conclusion. Steps 1.1 and 2.1 show that the nilpotent irrelevant ideal produces empty Proj, while step 3.1 shows that the underlying ring still has a point; the two conclusions are consistent because Proj discards exactly the primes containing all of $S_{+}$ . [F1, F2, step 2.1, step 3.1] \qed

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Constructions of Schemes, Sections 27.8-27.21 (standard reference, not scraped)
Ravi Vakil, The Rising Sea, 29 August 2022, Sections 4.5, 7.4, 9.3, 10.6, 17.4, 17.6, 18.2 (standard reference, not scraped)