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I have just uploaded my paper [1] to the arXiv. It arose first as an attempt to better understand the spherical Witt vector functor constructed in Burklund–Schlank–Yuan [2]. Their proof uses deformation theory and an existence result of Lurie. My proof uses instead the notion of transmutation.

After figuring this out, I went on a bit of a wild goose chase, aiming to understand how to solve Grothendieck’s problem (see for example [7]) and capture all of the information about a finite type nilpotent space $X$ from its $\bZ$-cochains $\bZ^X$. The idea I came up with for this was a notion of binomial derived rings, which are derived $\lambda$-rings (derived in the sense of Bhatt, Brantner, Mathew, and Raksit) equipped with compatible trivializations of their $\psi^p$ Adams operations.

At the time, I was unware of the work of Kriz [4] in characteristic $p$ or the recent work of Horel [3] over $\bZ$. Last week, another perspective came out in a paper [5] of Kubrak, Shuklin, and Zakharov [5].

All three papers [1,3,5] on binomial rings give slightly different $\infty$-categorical models. Horel uses cosimplicial binomial rings with weak equivalences inverted, Kurbak–Shuklin–Zakharov directly derive the binomial ring monad, and I work with derived $\lambda$-rings with additional trivializations of the Frobenii. My applications are to spherical Witt vectors and spherical lifts of these binomial rings. Horel studies an integral version of the Grothendieck–Teichmüller group. Kubrak–Shuklin–Zakharov used derived binomial rings to give a definition of integral cochain models of the Kato–Nakayama spaces arising in log geometry.

I want to explain here one problem, not stated in [1], which I’d be interested to understand. Let $\CAlg_{\bF_p}^\perf$ denote the category of perfect commutative $\bF_p$-algebras and let $\DAlg_{\bF_p}^\perf$ denote the $\infty$-category of perfect derived commutative $\bF_p$-algebras, i.e., those derived commutative rings $R$ such that the Frobenius $\varphi\colon R\rightarrow R$ is an equivalence. Consider the right Kan extension $F$ of the inclusion $\CAlg_{\bF_p}^\perf\rightarrow\DAlg_{\bF_p}^\perf$ along itself.

Question. Is $F$ equivalent to the identity functor of $\DAlg_{\bF_p}^\perf$?

It is possible, using Toën’s affinization theorem (see [7] for an overview), to show that the analogous result for non-perfect rings is yes: the right Kan extension of the inclusion of discrete commutative $\bF_p$-algebras into coconnective derived commutative $\bF_p$-algebras along itself is the identity functor of $\DAlg_{\bF_p}^{\mathrm{ccn}}$. However, this does not help answer our question as the inverse limit perfection $R^\perf$ of a discrete commutative $\bF_p$-algebra is typically derived because of a non-vanishing $\lim^1$.

The question has a positive answer if and only if for each perfect derived commutative $\bF_p$-algebra $R$, the natural map $R\rightarrow\lim_{R\rightarrow S}S$ is an equivalence, where the limit ranges over all maps from $R$ to discrete perfect commutative $\bF_p$-algebras.


[1] Antieau, Spherical Witt vectors and integral models for spaces, arXiv:2308.07288.

[2] Burklund, Schlank, Yuan, The chromatic Nullstellensatz, arXiv:2207.09929.

[3] Horel, Binomial rings and homotopy theory, arXiv:2211.02349.

[4] Kriz, p-adic homotopy theory, Topology Appl. 52 (1993), no. 3, 279–308.

[5] Kubrak, Shuklin, Zakharov, Derived binomial rings I: integral Betti cohomology of log schemes, arXiv:2308.01110.

[6] Mandell, Cochains and homotopy type, Pub. Math. IHES 103 (2006), 213-246.

[7] Toën, Le problème de la schématisation de Grothendieck revisité, Épijournal Géom. Algébrique 4 (2020), Art. 14.