The symbol length of K^M_n(ℂ(x_1, …, x_m))/p

The question

The symbol length problem for complex rational function fields [Krashen2024, Problem 2.1.3.12 and §2.1.3.4]: determine, as a function of $m$, $n$ and the prime $p$, the symbol length of $K_n^M(\mathbb{C}(x_1,\dots ,x_m))/p$, that is the least $k$ such that every class is a sum of at most $k$ symbols, or $\infty$ if there is no such $k$. The answer sl m n p : ℕ∞ is characterised by: k is such a bound if and only if sl m n p ≤ k.

For a field $F$ and $\ell \ge 1$, the Milnor K-group $K_n^M(F)/\ell$ is generated by the symbols $\{a_1,\dots ,a_n\}$ with $a_i\in F^{\times }$, and the symbol length of a class is the least number of symbols in a decomposition of the class as a sum of symbols. The symbol length of $K_n^M(F)/\ell$ is the supremum of the symbol lengths of its classes. When $\ell$ is invertible in $F$, the norm residue isomorphism theorem (the Bloch–Kato conjecture, [Voevodsky2011]) gives $K_n^M(F)/\ell \cong H^n(F,{\mu }_{\ell }^{\otimes n})$, so this is also the symbol length of Galois cohomology.

The symbol length problem asks, for a given field $F$ and integers $n$ and $\ell$, for a $k$ such that the symbol length of every class in $K_n^M(F)/\ell$ is at most $k$, for example in terms of a dimension of the field. That a class is a sum of some number of symbols is part of the presentation of $K_n^M(F)$ recalled above, so the content lies entirely in the bound. In degree $2$, over a field containing a primitive $\ell$-th root of unity, Merkurjev and Suslin's theorem [MerkurjevSuslin1983] that the norm residue map is an isomorphism identifies $K_2^M(F)/\ell$ with the $\ell$-torsion of the Brauer group and symbols with symbol algebras of degree $\ell$; that is the form in which the question is usually studied, namely how many symbol algebras are needed to represent a central simple algebra of exponent $\ell$. Becher and Hoffmann [BecherHoffmann2004] named the invariant ${\lambda }_p$ in that degree and asked it for exactly the fields considered below (their Question 4.6). The general form stated here is Problem 2.1.3.12 of Krashen's PCMI notes; Krashen notes there that over complex function fields in at least three variables no bound is known beyond Matzri's in degree $2$ and the case of $\ell$ a power of $2$, which comes from the theory of quadratic forms, and that very little is known in degree $3$ and higher [Krashen2024, §2.1.3.4].

This file states the problem for the rational function fields $\mathbb{C}(x_1,\dots ,x_m)$ and a prime $\ell =p$: determine the symbol length of $K_n^M(\mathbb{C}(x_1,\dots ,x_m))/p$ as a function of $m$, $n$ and $p$, where the symbol length is infinite if there is no bound.

What is known:

• in degree $0$ every symbol is the empty product $1$, so $K_0^M(F)/p=\mathbb{Z}/p$ has symbol length $p-1$ for every field $F$;
• in degree $1$ every class of $K_1^M(F)/p=F^{\times }/(F^{\times }{)}^p$ is a single symbol, so the symbol length is at most $1$;
• $\mathbb{C}(x_1,\dots ,x_m)$ has cohomological dimension $m$ [Serre1997, Ch. II, §4.2], so $K_n^M(\mathbb{C}(x_1,\dots ,x_m))/p=0$ for $n>m$ and the symbol length is $0$;
• $\mathbb{C}(x_1,\dots ,x_m)$ is a $C_m$ field (Tsen–Lang, [Lang1952]), so by Matzri's theorem [Matzri2016, Theorem 8.2] every class in $K_2^M(\mathbb{C}(x_1,\dots ,x_m))/p^t$ is a sum of at most $t(p^{m-1}-1)$ symbols when $m\ge 1$;
• in degree $2$ a tensor product of $m-1$ symbol algebras of degree $p$ over $\mathbb{C}(x_1,\dots ,x_m)$ can be a division algebra, so the symbol length is at least $m-1$ for $m\ge 1$ [BecherHoffmann2004, Proposition 4.5], by an argument going back to Nakayama.

In degree $2$, Becher and Hoffmann ask whether that lower bound is the answer, that is whether the symbol length is exactly $m-1$ [BecherHoffmann2004, Question 4.6]. For $m=n=2$, the smallest case not settled above, this asks whether every class of $K_2^M(\mathbb{C}(x_1,x_2))/p$ is a single symbol; they call it "a striking open question" and note that it holds for $p\le 3$ by a theorem of Artin [BecherHoffmann2004, Introduction]. Since $m-1$ does not involve $p$, their question is an instance of the expectation recorded in [Krashen2024, §2.1.3.4] that a bound can be taken independent of $\ell$. See also the survey [Krashen2016].

The definitions MilnorK, MilnorK.symbol, MilnorK.grade and MilnorK.symbolLengthBounds are in FormalConjecturesForMathlib/FieldTheory/MilnorKTheory.lean.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Other.SymbolLengthMilnorKTheory.

theorem symbol_length_problem :
    ∀ m n p : ℕ, p.Prime → ∀ k : ℕ,
      k ∈ symbolLengthBounds (MvRatFunc (Fin m) ℂ) p n ↔ symbol_length_problem.answer m n p ≤ k

What counts as progress

• A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• symbol_length_problem.variants.bounded — The symbol length problem in the form of [Krashen2024, Problem 2.1.3.12]: is there, for every m, n and prime p, a bound k such that every class in K^M_n(ℂ(x_1…
• symbol_length_problem.variants.uniform — The expectation recorded in [Krashen2024, §2.1.3.4] that the bound can be taken independent of the prime: is there, for every m and every n ≥ 1, a k such that…
• symbol_length_problem.variants.degree_two_isLeast — Becher and Hoffmann's question [BecherHoffmann2004, Question 4.6]: is the symbol length of K^M_2(ℂ(x_1, …, x_m))/p exactly m - 1?

References

• [Krashen2024] D. Krashen, PCMI Summer School (not quite) notes, notes for the minicourse Field arithmetic and the complexity of algebraic objects, PCMI Graduate Summer School on Motivic Homotopy Theory, 2024, draft of 24 July 2024, pdf.
• [BecherHoffmann2004] K. J. Becher and D. W. Hoffmann, Symbol lengths in Milnor $K$-theory, Homology Homotopy Appl. 6 (2004), no. 1, 17–31, doi:10.4310/HHA.2004.v6.n1.a3.
• [MerkurjevSuslin1983] A. S. Merkurjev and A. A. Suslin, *K-cohomology of Severi–Brauer varieties and the norm residue homomorphism*, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), 1011–1046; English translation Math. USSR-Izv. 21 (1983), no. 2, 307–340, doi:10.1070/IM1983v021n02ABEH001793.
• [Krashen2016] D. Krashen, *Period and index, symbol lengths, and generic splittings in Galois cohomology*, Bull. London Math. Soc. 48 (2016), 985–1000, doi:10.1112/blms/bdw060.
• [Matzri2016] E. Matzri, Symbol length in the Brauer group of a field, Trans. Amer. Math. Soc. 368 (2016), 413–427, doi:10.1090/tran/6326, arXiv:1402.0332.
• [Lang1952] S. Lang, On quasi algebraic closure, Ann. of Math. (2) 55 (1952), 373–390, doi:10.2307/1969785.
• [Serre1997] J.-P. Serre, Galois cohomology, Springer Monographs in Mathematics, Springer, 1997, doi:10.1007/978-3-642-59141-9.
• [Voevodsky2011] V. Voevodsky, On motivic cohomology with Z/l-coefficients, Ann. of Math. (2) 174 (2011), 401–438, doi:10.4007/annals.2011.174.1.11.

Source and licence

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