ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck pending

How statement and proof provenance work

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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Faithful flatness retaining a nilpotent class

Example

Assume the Axiom of Choice. Let $X = Spec (C [x, y] / (x^{2}))$ and let $p$ be the closed $C$ -point at the origin. Write $T = C^{N^{2}}$ for the ring of coefficient arrays $(a_{i, j})_{i, j \geq 0}$ , with pointwise addition and finite Cauchy multiplication $(a b)_{i, j} = \sum_{r = 0}^{i} \sum_{s = 0}^{j} a_{r, s} b_{i - r, j - s}$ ; this formal ring is denoted $C ⟦ x, y ⟧$ , with $x$ and $y$ the arrays supported at $(1, 0)$ and $(0, 1)$ , respectively, and its unit is supported at $(0, 0)$ with coefficient $1$ . No convergence condition is imposed. Its algebraic local ring is $A = (C [x, y] / (x^{2}))_{(x, y)}$ , and its analytic local ring is $B = C {x, y} / (x^{2})$ . The canonical map $A \to B$ is faithfully flat with common maximal-ideal completion $T / (x^{2} T)$ , and the class of $x$ is a nonzero element whose square is zero in both rings.

Example references

Serre, local algebraic-to-holomorphic comparison and flat-pair appendix

Anschütz, analytification of locally finite-type schemes

MIT OpenCourseWare, Lecture 22: GAGA

Facts & Assumptions

Given: AC and the local algebraic and analytic rings of the scheme and point specified in the example.

[A1]

AC says that every family of nonempty sets has a choice function (The Axiom of Choice).

[F1]

The local algebraic-to-analytic map for a finite-type complex scheme at a closed $C$ -point is faithfully flat and identifies the maximal-ideal completions by extending the same finite defining ideal (Faithful flatness of finite-type algebraic-to-analytic local rings).

[F2]

The formal ring $T = C ⟦ x, y ⟧$ is the coefficient-array ring defined in Common completions of algebraic and holomorphic local rings and their finite-type quotients: its elements are arrays $(a_{i, j})_{(i, j) \in N^{2}}$ and multiplication is finite Cauchy convolution. The monomial $x$ has its only nonzero coefficient, $1$ , at $(1, 0)$ . For every $f \in T$ , the coefficient of $x^{1} y^{0}$ in $x^{2} f$ is zero, since no decomposition $(1, 0) = (2, 0) + (i, j)$ with $(i, j) \in N^{2}$ exists. This is a formal coefficient calculation, with no convergence hypothesis.

Verification

1.1givenA1F1

Apply [F1] with $n = 2$ and $I = (x^{2}) \subseteq (x, y)$ to the finite-type complex scheme $X$ at its closed point $p$ with residue field $C$ . The map $A \to B$ is faithfully flat, and both completions identify with $Q : = T / (x^{2} T)$ . Both completion maps send the class of $x$ to $x + x^{2} T \in Q$ .

2.1F1F2step 1.1

In the formal ring $T$ , the coefficient of $x^{1} y^{0}$ in $x$ is $1$ , whereas every element of $x^{2} T$ has coefficient $0$ there by [F2]. Thus $x \notin x^{2} T$ , so $x + x^{2} T$ is nonzero in $Q$ . If the class of $x$ were zero in either $A$ or $B$ , its image under that ring's completion map would be zero in $Q$ , contradicting step 1.1 and this nonvanishing.

3.1givenstep 1.1step 2.1∎

By the quotient definitions in the example, $x^{2} = 0$ in $A$ and $B$ . Step 2.1 proves that $x$ is nonzero in both, so it is a nonzero nilpotent class preserved by the faithfully flat comparison.

Faithful flatness retaining a nilpotent class

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources