Homotopy invariance of Euler class groups over finite fields and the top Chow-group comparison in characteristic 2 (Das's Questions 7.10 and 1.4)

The theory of Euler class groups of commutative rings, introduced by Nori and developed by Bhatwadekar and Sridharan, assigns to a smooth affine algebra $R$ of Krull dimension $d$ an obstruction group $E^d(R)$ that detects when a rank-$d$ projective $R$-module of trivial determinant splits off a free summand. Work of Asok and Fasel, revisited and sharpened by Das, identifies $E^d(R)$ with the Barge–Morel Chow–Witt group $\widetilde{CH}^d(X)$ and the weak Euler class group $E_0^d(R)$ with the top Chow group $CH^d(X)$ for $X=\mathrm{Spec}(R)$, but only over base fields that are infinite (perfect) and, for $d\ge 3$, of characteristic different from 2; and the homotopy invariance $E^d(R)\simeq E^d(R[T])$ is proved only over infinite fields. Das's preprint explicitly leaves open two questions that share one objective: free this theory from its residual base-field hypotheses. First (Question 1.4): is the canonical morphism $E_0^d(R) \to CH^d(X)$ still an isomorphism for $d\ge 3$ when the infinite perfect base field has characteristic 2? Second (Question 7.10): is the canonical morphism $E^d(R) \to E^d(R[T])$ (always injective) an isomorphism when $R$ is smooth of finite type of dimension $d\ge 3$ over a finite field? No resolution of either question is known in the literature to date.

Background

A finitely generated algebra $R$ over a field $k$ is called smooth affine (of Krull dimension $d$) when it is of finite type over $k$ and $X=\mathrm{Spec}(R)$ is a regular variety; the prototypical objects are coordinate rings of affine open subsets of smooth projective varieties. A classical theme of commutative algebra is which projective modules (direct summands of finite free modules) over such rings exist and when they acquire a free summand. To organize this question, Nori introduced and Bhatwadekar and Sridharan developed the Euler class group $E^d(R)$: its generators are pairs $(I,\omega_I)$ where $I\subset R$ is an ideal of height $d$ whose conormal module $I/I^2$ is generated by $d$ elements, and $\omega_I$ is a local orientation, i.e. an equivalence class, for the natural action of $SL_d(R/I)$, of surjections $(R/I)^d \twoheadrightarrow I/I^2$; one imposes the relations that $(I,\omega_I)=0$ whenever the orientation lifts to an actual surjection $R^d \twoheadrightarrow I$ (a global orientation), together with additivity relations for intersections of such ideals. The weak Euler class group $E_0^d(R)$ is the variant in which orientations are only considered up to the action of $GL_d(R/I)$. The group $E^d(R)$ is an obstruction group in the following sense: to a projective $R$-module $P$ of rank $d$ with a trivialization $\chi:R \stackrel{\sim}{\to} \wedge^d P$ of its determinant (top exterior power) one associates an Euler class $e_d(P,\chi)\in E^d(R)$, and under mild hypotheses on $R$, $e_d(P,\chi)=0$ exactly when $P$ has a free summand. On the geometric side, $CH^d(X)$ denotes the Chow group of zero cycles on $X$ (codimension-$d$ cycles modulo rational equivalence), and $\widetilde{CH}^d(X)$ the Chow–Witt group of Barge and Morel, a quadratic refinement of the Chow group built from Milnor–Witt K-theory for which the standard comparison machinery requires $2$ to be invertible in $k$. A guiding analogy, and the source of the field's name for it, is homotopy invariance: an invariant $F$ of rings is homotopy invariant if $F(R)\simeq F(R[T])$ for every regular ring $R$. For $F=K_0$ (the Grothendieck group of finitely generated projective modules) this is the content of the Bass–Quillen conjecture, equivalently the statement that every regular ring has property E (every finitely generated projective $R[T]$-module is extended from $R$); property E is known for regular rings of dimension at most 2 (Murthy) and, by Lindel and Popescu, for regular rings containing a field and for rings whose local rings are unramified or excellent Henselian (a Henselian local ring is one over which Newton-style root lifting holds: simple roots of polynomials modulo the maximal ideal lift to actual roots; 'excellent' is a standard finiteness condition on completions), but remains open in ramified mixed characteristic (regular rings in which some prime number is neither zero nor invertible, so that neither a field nor all primes are available). For Euler class groups, homotopy invariance of $E^d$ fails functoriality (the assignment is not a functor on all ring maps) yet one may still ask whether the canonical map is an isomorphism. The comparison program culminated in theorems of Asok and Fasel, obtained through $\mathbb{A}^1$-homotopy theory (Morel–Voevodsky's homotopy category of schemes; for instance the Segre class map identifies $E^d(R)$ with homotopy classes of maps from $X$ to the split smooth quadric $Q_{2d}$ of dimension $2d$): for $k$ infinite perfect with $\mathrm{char}(k)\neq 2$, there is a functorial isomorphism $E^d(R)\simeq \widetilde{CH}^d(X)$ and the canonical morphism $E_0^d(R)\to CH^d(X)$ is an isomorphism (with the case $d=3$ originally requiring $\mathrm{char}(k)=0$, using their classification of vector bundles on smooth affine threefolds). In the 2019 preprint, Das recast the foundations, extended these comparisons (removing, for $d\ge 3$, the characteristic-zero hypothesis but still needing $\mathrm{char}(k)\neq 2$, since by his Remark 1.3 the isomorphism $E_0^d(R)\to CH^d(X)$ is obtained from the Chow–Witt comparison where the assumption enters), and proved homotopy invariance $E^d(R)\simeq E^d(R[T])$ for regular $R$ of dimension $d\ge 3$ essentially of finite type over an infinite field (Theorem 7.9), noting that the canonical morphism is always injective; his proof for $d\ge 4$ builds on the excision and obstruction-theory results of Mandal and Yang. Two residual restrictions on the base field survive: characteristic 2 in the top Chow comparison, and finiteness of the base field in homotopy invariance. Removing them is precisely the content of the two questions stated below. The paper also recalls Bass–Quillen (property E) as Conjecture 3.3 and uses property E as a hypothesis in several of its theorems; that famous conjecture is context here — its general validity, open in ramified mixed characteristic by the results quoted above, is a separate problem and is outside the scope of this task.

Problem Statement

Let $k$ be a field and $R$ a smooth affine $k$-algebra of Krull dimension $d\ge 3$, with $X=\mathrm{Spec}(R)$. Resolve, in the stated generality, the two questions left open in Das's preprint arXiv:1901.10686, which together ask whether the Euler class group theory — its top Chow-group comparison and its homotopy invariance — can be freed from the residual restrictions on the base field. (1) (Question 1.4 of the source.) Theorem 1.2(2) of the source states that for $k$ infinite and perfect, the canonical morphism $E_0^d(R) \longrightarrow CH^d(X)$ from the weak Euler class group to the Chow group of zero cycles is an isomorphism if $d=2$, or if $d\ge 3$ and $\mathrm{char}(k)\neq 2$. Can the characteristic assumption be removed for $d\ge 3$: is $E_0^d(R) \to CH^d(X)$ an isomorphism for every smooth affine $k$-algebra $R$ of dimension $d\ge 3$ over an infinite perfect field $k$ of characteristic $2$? (2) (Question 7.10 of the source.) Let now $k$ be a finite field and let $R$ be smooth of finite type over $k$ with $\dim(R)=d\ge 3$. Is the canonical morphism $\varphi: E^d(R) \longrightarrow E^d(R[T])$ an isomorphism of Euler class groups? (The morphism $\varphi$ is injective in general, so the question is whether it is surjective.) Both questions are posed in the full stated generality; no additional restrictions on $d$ (beyond $d\ge 3$), on the field, or on the ring are part of the question.

References

1. Mrinal Kanti Das, Remarks on Euler class groups and two conjectures, preprint (2019), arXiv:1901.10686, DOI:10.48550/arXiv.1901.10686, https://arxiv.org/abs/1901.10686 (source of Questions 1.4 and 7.10 and of Conjecture 3.3)
2. Aravind Asok, Jean Fasel, Euler class groups and motivic stable cohomotopy, with an appendix by Mrinal Kanti Das, Journal of the European Mathematical Society 24 (2022), no. 8, 2775–2822, DOI:10.4171/JEMS/1156, https://doi.org/10.4171/JEMS/1156 (arXiv:1601.05723, https://arxiv.org/abs/1601.05723)
3. Aravind Asok, Jean Fasel, A cohomological classification of vector bundles on smooth affine threefolds, Duke Mathematical Journal 163 (2014), no. 14, 2561–2601, DOI:10.1215/00127094-2819299, https://doi.org/10.1215/00127094-2819299
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