gpt 6 astra pro attempt on Math-187

Voevodsky's Smash-Nilpotence Conjecture

Definitions and mathematical statement

For a smooth projective $d$-dimensional variety over $k$, a rational cycle class $\alpha\in\mathrm{CH}^r(X)_{{\mathbb{Q}}}$ is numerically trivial when $\deg(\alpha\cdot\beta)=0$ for every complementary class $\beta\in\mathrm{CH}^{d-r}(X)_{{\mathbb{Q}}}$. It is smash-nilpotent when \[ \alpha^{\boxtimes N}=0\quad\text{in}\quad\mathrm{CH}^{Nr}(X^N)_{{\mathbb{Q}}} \quad\text{for some }N\ge1. \] Here $\boxtimes$ is the external product over $k$, and the exponent may depend on $\alpha$. The conjecture says numerical triviality implies smash-nilpotence for every field and every such cycle. Powers under intersection on $X$, or under composition of correspondences, are different operations. Rational coefficients and the absence of a restriction on the characteristic of $k$ are essential scope choices.

Formulation and concept references: [S1].

Short English statement

Does every rational algebraic cycle invisible to all intersection-number tests become zero after taking sufficiently many external copies of itself?

Research attempt

Current frontier.

The foundational theorem is stronger than a statement about intersection powers but weaker than the conjecture: Voevodsky proves smash-nilpotence for cycles algebraically equivalent to zero. His Proposition 3.1 handles degree-zero cycles on smooth projective curves, and Corollary 3.2 passes to general algebraically trivial cycles [S1]. Numerical triviality is not the hypothesis of that theorem. A 2026 manuscript of Corradini proves a tropical algebraic-triviality analogue and formulates a tropical smash-nilpotence conjecture [E1]. It concerns a different category of cycles and supplies no theorem for all numerically trivial Chow classes over arbitrary fields.

Attack route.

One route would compare numerical and algebraic equivalence after taking an external power. Another would seek tensor control from finite-dimensional motives; composition nilpotence alone is tested below and found inadequate. A third issue, often hidden in both routes, is descent from an algebraic closure. We combine explicit exponent bookkeeping with descent, then isolate the genuinely missing geometric input. All tensor products in the argument are external products over the specified field.

Attempt: descent preserves the exact target.

Descent lemma. If the conjecture is established over every algebraically closed field, it follows over every field, with rational coefficients as in the question.

Proof. Let $\alpha$ be numerically trivial on $X/k$. For a finite extension $L/k$, write $p:X_L\to X$. Every complementary cycle $\beta$ on $X_L$ satisfies \[ [L:k]\deg_L(\alpha_L\cdot\beta) =\deg_k p_*(\alpha_L\cdot\beta) =\deg_k(\alpha\cdot p_*\beta)=0. \] This is the projection formula and the definition of degree over different base fields. A complementary cycle on $X_{\bar k}$ is defined over some finite extension $L/k$, so its pairing with $\alpha_{\bar k}$ is also zero. The assumed algebraically closed case supplies an $N$ with $\alpha_{\bar k}^{\boxtimes N}=0$.

A rational equivalence witnessing this vanishing uses finitely many subvarieties, coefficients, and rational functions. All descend to a finite extension $L/k$, after enlarging it if necessary. Thus $\alpha_L^{\boxtimes N}=0$. For the finite flat map $p_N:(X^N)_L\to X^N$, \[ (p_N)_*p_N^*(\alpha^{\boxtimes N}) =[L:k]\alpha^{\boxtimes N}=0. \] Division by $[L:k]$ in the rational Chow group proves the assertion. Finite extensions need not be separable: finite flat pullback and proper pushforward have the same degree identity. No characteristic-zero hypothesis has entered. Components can be treated in each pure dimension, as in the original statement. $\square$

This proves that passing to an algebraic closure is a legitimate reduction, not an unannounced restriction. It does not prove the algebraically closed case, nor does it give a way to decide whether the rational equivalence over $\bar k$ exists.

Attempt: a computable exponent for a certified class.

Suppose that over a finite extension $L/k$ one has an actual identity \begin{equation}\label{wsref-0-65-71-31-38-32-2d-63-65-72-74-69-66-69-63-61-74-65} \alpha_L=\sum_{i=1}^s(\Gamma_i)_*z_i, \qquad z_i\in\mathrm{CH}^1(C_i)_{{\mathbb{Q}}},\quad\deg z_i=0, \qquad \Gamma_i\in\mathrm{CH}^r(C_i\times X_L)_{{\mathbb{Q}}}, \end{equation} where $C_i/L$ are smooth projective geometrically integral curves of genus $g_i$. This is a certificate to be supplied, not a consequence of numerical triviality. It records the entire cycle identity, not merely agreement of cohomology classes or intersection numbers.

For safety use $N_i=2g_i+1$. These integers are within the vanishing range of Voevodsky's curve theorem [S1], Proposition 3.1, so $z_i^{\boxtimes N_i}=0$. Functoriality of correspondences gives \[ ((\Gamma_i)_*z_i)^{\boxtimes N_i} =(\Gamma_i^{\boxtimes N_i})_* (z_i^{\boxtimes N_i})=0. \] The following bookkeeping argument turns the separate vanishings into a single explicit exponent: \begin{equation}\label{wsref-0-65-71-31-38-32-2d-65-78-70-6f-6e-65-6e-74} \alpha^{\boxtimes N}=0, \qquad N=1+\sum_{i=1}^s(N_i-1) =1+2\sum_{i=1}^s g_i. \end{equation} Indeed, expand the external power of the sum as a sum indexed by ordered lists of $N$ indices. Some index $i$ occurs at least $N_i$ times. After a permutation of factors, that summand contains the vanishing external power of $(\Gamma_i)_*z_i$ as a factor, and hence is zero. All sums and permutations are finite; no infinite tensor completion is used. The descent lemma proves the same exponent over $k$. If the sum is empty, $\alpha_L=0$ and exponent one suffices.

This also proves directly the familiar closure of smash-nilpotent cycles under finite addition, with an explicit bound; compare [S1], Lemma 2.6. For example, a certificate using curves of genera one and two supplies exponent seven. The formula is not asserted to be optimal, and it is not a computable exponent from $X$ and the numerical hypothesis alone: the certificate (eq182-certificate) is additional input.

Attempt: weaken the certificate instead of confusing equivalences.

A more plausible sufficient input for a general numerically trivial $\alpha$ is \begin{equation}\label{wsref-0-65-71-31-38-32-2d-69-6e-74-65-72-6d-65-64-69-61-74-65} \text{for some }m\ge1,\quad \alpha^{\boxtimes m}\text{ is algebraically trivial on }X^m. \end{equation} Voevodsky's theorem applied on $X^m$ then gives an $N$ such that \[ (\alpha^{\boxtimes m})^{\boxtimes N} =\alpha^{\boxtimes mN}=0. \] Conversely, smash-nilpotence gives (eq182-intermediate), since a zero Chow class is algebraically trivial. Thus, using the established theorem, (eq182-intermediate) is an equivalent intermediate target. This reduction is already contemplated in [S1], Section 4; it is stated here to identify exactly what the proposed curve-certificate strategy would have to produce.

Producing the certificate at $m=1$ would ask for more than this reduction. The definitions of numerical and algebraic equivalence do not identify them. Passing to a finite extension, or decomposing a motive into pieces with nilpotent endomorphisms, has not constructed (eq182-intermediate). The next explicit test shows why one tempting substitute for that construction is invalid.

Stress test: composition-square-zero is not tensor-nilpotent.

Let $X={\mathbb{P}}^1_k\times{\mathbb{P}}^1_k$. Denote the two divisor classes by $h_1,h_2$, so $h_1^2=h_2^2=0$ and $\deg(h_1h_2)=1$. The degree-zero correspondence \[ r=h_1\boxtimes h_1 \in\mathrm{CH}^2(X\times X)_{{\mathbb{Q}}} \] is nonzero. The composition formula for external-product correspondences gives \[ r\circ r=\deg(h_1^2)r=0. \] Nevertheless, for every $N\ge1$, \begin{equation}\label{wsref-0-65-71-31-38-32-2d-6e-6f-74-65-6e-73-6f-72} \deg\left(r^{\boxtimes N}\cdot (h_2\boxtimes h_2)^{\boxtimes N}\right)=1. \end{equation} Thus none of the external powers of $r$ vanish. The equality follows by factoring the intersection number over all $2N$ copies of $X$; each factor is $\deg(h_1h_2)=1$. In particular $r$ is not numerically trivial, so it is not a counterexample to the target. It is a counterexample to the auxiliary claim that composition nilpotence alone implies smash-nilpotence. Any use of a Kimura-type nilpotence theorem would require an extra argument exploiting its hypotheses, rather than just changing the multiplication operation in its conclusion.

There is also no useful intersection-power shortcut: a positive-codimension cycle has sufficiently high intersection powers zero on a fixed finite- dimensional variety for dimensional reasons. Its external powers live on varieties of growing dimension, so that automatic vanishing says nothing about (eq182-intermediate).

Stress test: dependence on cohomological standard conjectures.

In any Weil cohomology theory with its Künneth identification, external products satisfy \[ \operatorname{cl}(\alpha^{\boxtimes N}) =\operatorname{cl}(\alpha)^{\otimes N}. \] A nonzero vector over a field has every tensor power nonzero: choose a linear functional not vanishing on it and take its tensor power. Hence smash-nilpotence forces homological triviality. A universal proof of the selected numerical-to-smash implication would therefore imply that numerical and homological equivalence agree in these realizations, the standard-conjecture consequence discussed in [S1], Section 4. This is a warning about the depth of the missing input, not a reason to abandon it or a conditional assumption quietly used in the descent proof.

Outcome.

Outcome: Conditional reduction.

The attempt proves descent without separability restrictions, an explicit external-power bound for supplied curve-correspondence certificates, and the implication from an algebraically trivial external power to the target. It also gives an exact correspondence showing the failure of a composition-nilpotence shortcut. The underlying nilpotence theorem and intermediate reduction are established literature; no independent novelty claim is made for them or for the bookkeeping above.

The missing result is (eq182-intermediate) for an arbitrary numerically trivial rational cycle, or another construction of an external rational-equivalence certificate. Neither the descent nor the tropical analogue supplies it. All-field scope is retained, but the full conjecture has not been resolved.

Sources

• S1 Vladimir Voevodsky, A Nilpotence Theorem for Cycles Algebraically Equivalent to Zero, IMRN1995 no.4,187-198; published IAS snapshot,13pages. https://www.math.ias.edu/vladimir/sites/math.ias.edu.vladimir/files/Nilpotence_theorem_published.pdf. Catalogue formulation source.
• E1 Alexia Corradini, Algebraically trivial tropical cycles are smash-nilpotent, arXiv:2608.21988v1, 22 August 2026, abstract and stated tropical scope. https://arxiv.org/abs/2608.21988v1. Consulted 27 September 2026. Adjacent-category result; no algebraic Chow-group theorem is inferred from it.