On the infinite loop spaces of algebraic cobordism and the motivic sphere

We obtain geometric models for the infinite loop spaces of the motivic spectra $\mathrm{MGL}$, $\mathrm{MSL}$, and $\mathbf{1}$ over a field. They are motivically equivalent to $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{lci}(\mathbb{A}^\infty)^+$, $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{or}(\mathbb{A}^\infty)^+$, and $\mathbb{Z}\times \mathrm{Hilb}_\infty^\mathrm{fr}(\mathbb{A}^\infty)^+$, respectively, where $\mathrm{Hilb}_d^\mathrm{lci}(\mathbb{A}^n)$ (resp. $\mathrm{Hilb}_d^\mathrm{or}(\mathbb{A}^n)$, $\mathrm{Hilb}_d^\mathrm{fr}(\mathbb{A}^n)$) is the Hilbert scheme of lci points (resp. oriented points, framed points) of degree $d$ in $\mathbb{A}^n$, and $+$ is Quillen's plus construction. Moreover, we show that the plus construction is redundant in positive characteristic.


Introduction
Let k be a field, let MGL be Voevodsky's algebraic cobordism spectrum over k, and let FSyn be the moduli stack of finite syntomic k-schemes. In [EHK + 20b, Corollary 3.4.2], we showed that there is an equivalence Ω ∞ T MGL L mot (FSyn gp ), where FSyn gp is the group completion of FSyn under disjoint union and L mot is the motivic localization functor. This can be regarded as an algebro-geometric analogue of the identification of Ω ∞ MO with the cobordism space of 0-dimensional compact smooth manifolds. However, unlike in topology, the group completion is necessary in the above equivalence. The goal of this paper is to obtain a group-completion-free description of Ω ∞ T MGL, which is more accessible for some purposes. Let us write FSyn ∞ for the colimit of the sequence where FSyn d ⊂ FSyn is the moduli stack of finite syntomic k-schemes of degree d. Our main result is the following: Theorem 1.1. Let k be a field. Then there is an equivalence Ω ∞ T MGL L mot (Z × FSyn ∞ ) + in H(k), where + denotes Quillen's plus construction (in the ∞-topos of Nisnevich sheaves). Moreover, there is an equivalence (Ω ∞ T MGL)(k) Z × (L A 1 FSyn ∞ )(k) + . If k has positive characteristic, the same equivalences hold without the plus construction.
Here, L A 1 is the "naive" A 1 -localization functor given by the formula (L A 1 F)(X) = colim n∈∆ op F(X × A n ). We recall that FSyn d is motivically equivalent to the Hilbert scheme we obtain a smooth ind-scheme Hilb lci ∞ (A ∞ ) such that the canonical map (forgetting the embedding into A ∞ ) Hilb lci ∞ (A ∞ ) → FSyn ∞ is a motivic equivalence, and in fact an A 1 -equivalence on affine schemes. Thus, we can rewrite Theorem 1.1 as follows: Theorem 1.2. Let k be a field. Then there is an equivalence Ω ∞ T MGL L mot (Z × Hilb lci ∞ (A ∞ )) + in H(k). Moreover, there is an equivalence (Ω ∞ T MGL)(k) Z × (L A 1 Hilb lci ∞ (A ∞ ))(k) + . If k has positive characteristic, the same equivalences hold without the plus construction. Remark 1.3. Unfortunately, we do not know if the plus construction can be removed in characteristic zero. As we will see, this is the case if and only if the cyclic permutation of three points in FSyn 3 (k) becomes homotopic to the identity in (L A 1 FSyn ∞ )(k).
We have similar results for the motivic sphere spectrum 1 and for the special linear algebraic cobordism spectrum MSL. Recall that a framing of a finite syntomic morphism f : Y → X is a trivialization of the cotangent complex L f in K(Y), and an orientation of f is a trivialization of the dualizing sheaf ω f = det(L f ) in Pic(Y). We denote by FSyn fr and FSyn or the moduli stacks of framed and oriented finite syntomic k-schemes, respectively. There are forgetful morphisms  ) and Ω ∞ T MSL L mot (FSyn or,gp ). Let h ∈ FSyn fr (k) be the framed finite syntomic k-scheme consisting of two k-points, one with trivial framing and the other with the opposite framing. We define FSyn or ∞ = colim(· · · → FSyn or 2d +h − − → FSyn or 2d+2 → · · · ).
Theorem 1.4. Let k be a field. Then there are equivalences If k has positive characteristic, the same equivalences hold without the plus construction.
Our interest in these group-completion-free models is motivated in part by the following conjecture of Mike Hopkins, called the "Wilson space hypothesis": Conjecture 1.6 (M. Hopkins). For every n ∈ Z, the motive M(Ω ∞−n T MGL) ∈ DM(k) is pure Tate. The analog of this conjecture for the complex cobordism spectrum MU is a theorem of Wilson [Wil73], and the analog in C 2 -equivariant homotopy theory was recently proved by Hill and Hopkins [HH18]; these results are subsumed by Conjecture 1.6 for k = C and k = R, respectively. Since the plus construction is invisible to motives, Theorem 1.1 implies that We thus obtain the following geometric reformulation of the case n = 0 of the above conjecture:

A moving lemma
Lemma 2.1. Let k be a field, X a smooth geometrically connected quasi-projective k-scheme of dimension 2, and A ⊂ X a subscheme étale over k. Then there exists a smooth geometrically connected closed subscheme H ⊂ X of codimension 1 containing A.
Proof. If k is infinite, a generic hypersurface section of X of large enough degree containing A is geometrically irreducible and smooth of codimension 1, by [AK79, Theorems 1 and 7]. If k is finite, choose an embedding X ⊂ P n k and a first-order thickeningÃ of A in P n k that is transverse to X. Let P d = Γ (P n k , O(d)) and let is smooth of codimension 1 in X and f |Ã = 0}, (2) f |A = 0; (3) g does not vanish on A ∪ B; (4) Z(f ) ∩ X and Z(g) ∩ X are smooth over k of dimension m − 1.
Proof. Suppose first that k is finite. Proposition 2.3. Let k be a field, X a smooth k-scheme, and U ⊂ X a dense open subscheme. For any k-point x : Spec k → X, there exist framed correspondences α, β ∈ Corr fr k (Spec k, U) with finite étale support and an Proof. Since the question is local around x, we can assume X quasi-projective and geometrically connected [Gro65, corollaire 4.5.14]. By Lemma 2.1, we can find a smooth connected subscheme C ⊂ X of dimension 1 such that x ∈ C and such that C ∩ U is nonempty (hence cofinite in C). Let C be a projective closure of C and L an ample invertible sheaf on C. Shrinking C if necessary, we may assume that C is affine and that Ω C/k and L|C are trivial. Let A = {x}, and let B be the complement of C ∩ U in C with x removed. Then A and B are disjoint finite subschemes of C such that A is étale and C − B is smooth. By Lemma 2.2, we can find an integer d and global sections f and g of L ⊗d with étale vanishing loci, such that Then H is a proper local complete intersection of relative virtual dimension 0 over A 1 k . Moreover, since f and g agree and do not vanish on B, h does not vanish on A 1 × B, so H ⊂ A 1 × C and H is finite over A 1 k (hence syntomic by [EHK + 19, Proposition 2.1.16]). Thus, H defines a finite syntomic correspondence from A 1 k to X. Since Ω C/k and L ⊗d |C are trivial, there exists a framing τ : L H/A 1 k 0 in K(H). Multiplying τ by a unit of k, we can assume that τ restricts to the trivial loop in K({x}). Then (H, τ) is a framed correspondence from A 1 k to X with the desired properties.

Quillen's plus construction and group completion
We shall say that a morphism f : Recall that a modality on T is a factorization system on T (in the sense of Joyal and as explained in [Lur17b, §5.2.8]) whose left class of morphisms is stable under base change [ABFJ20, §2].
Lemma 3.1. In any ∞-topos T, acyclic morphisms form the left class of a modality. In particular, acyclic morphisms are stable under composition, retracts, colimits, cobase change, base change, and finite products.
Proof. Acyclic morphisms are stable under base change since cocartesian squares are (by universality of colimits). By [Lur17b, Proposition 5.5.5.7], it remains to show that the class of acyclic maps is of small generation as a saturated class. For any X ∈ T, the full subcategory of T /X spanned by the acyclic morphisms is accessible by [Lur17b, Proposition 5.4.6.6], being the fiber of the suspension functor. It is thus generated under filtered colimits by a small subcategory. Let C ⊂ Fun(∆ 1 , T) be the union of these small subcategories of T /X as X ranges over a small set of generators of T. Using that acyclic maps are stable under base change, we immediately deduce that C generates the class of acyclic maps under colimits.
For X ∈ T, we denote by X → X + the final object in the ∞-category of acyclic maps out of X, which exists by Lemma 3.1. The functor X → X + is called the plus construction. Note that acyclic morphisms are connected and become equivalences after a single suspension; it follows that the canonical map X → X + induces an isomorphism on π 0 , and that it is an equivalence whenever X admits a structure of E 1 -group (i.e., when X is a loop space [Lur17a, Theorem 5.2.6.15]).
In the ∞-topos Spc of spaces, the above construction coincides with Quillen's plus construction which was used to define algebraic K-theory [Qui72,§12]. More precisely, X → X + is the initial map that kills the maximal perfect subgroups of the fundamental groups of X, and hence it is an equivalence if and only if the fundamental groups of X are hypoabelian (i.e., have no nontrivial perfect subgroups). We refer to [Rap19] for a discussion of acyclic morphisms in Spc and for a proof of this fact.
Let M be a commutative monoid in T and let m : * → M be a global section. We denote by Mod M (T) the ∞-category of M-modules, i.e., objects of T with an action of M (which is again an ∞-topos). The full subcategory of M-modules on which m acts invertibly is reflective, and we denote by E → E[m −1 ] the associated localization functor. It is easy to check that this functor preserves finite products. In particular, There is then a canonical map of M-modules tel m (E) → E[m −1 ], but unlike in the case of 1-topoi, it is not invertible in general. For example, if T = Spc and F = n 0 BΣ n is the free commutative monoid on a single element a, then F[a −1 ] is the group completion of F, but tel a (F) Z × BΣ ∞ does not admit a monoid structure since its fundamental groups are not abelian. Proof. Since acyclic maps are closed under colimits and finite products, it suffices to prove the result for E = M. The classifying ∞-topos for pointed commutative monoids is a presheaf ∞-topos (namely, presheaves on the pushout of (Spc fin * ) op ← (Spc fin ) op → (CMon(Spc) fin ) op in the ∞-category of ∞-categories with finite limits), so we are immediately reduced to the case T = Spc. Let F be the free commutative monoid on an element a. The element m induces a morphism of commutative monoids F → M sending a to m, and we have tel m (M) = M ⊗ F tel a (F) and M[m −1 ] M ⊗ F F gp , where ⊗ F is the tensor product of spaces with F-action. Since acyclic maps are closed under colimits and finite products, M ⊗ F (−) preserves acyclic maps. We can therefore replace (M, m) by (F, a), and in particular we can assume that π 0 (M)[m −1 ] is a group. In this case we must show that the canonical map tel m (M) + → M gp is an equivalence, which is the classical McDuff-Segal group completion theorem [MS76]. We recall a proof due to Nikolaus [Nik17]. Note that the plus construction preserves finite products and hence commutative monoids. Consider the commutative square The left vertical is an equivalence since the plus construction is a left localization of Spc. The lower horizontal map is an equivalence by Proposition 5.1, since the cyclic permutation of order 5 becomes trivial in the hypoabelianization of Σ 5 . The top horizontal map is a stable equivalence by the localization theory of E ∞ -ring spectra. Hence the right vertical map is a stable equivalence. Since E ∞ -groups are simple, the right vertical map is in fact an equivalence, so the top horizontal map is an equivalence as well. In what follows we will use the plus construction in Spc and in the ∞-topos of Nisnevich sheaves on Sm k .

Proofs of the main results
Recall that FSyn ∞ = colim d→∞ FSyn d . In the notation of Section 3, there is a canonical map which is an equivalence on connected schemes (in particular, it is a Zariski-local equivalence).
Proof. We first observe that L mot (FSyn gp ) is the group completion of L mot (FSyn) in Nisnevich sheaves. To prove this we may replace k by a perfect subfield, and the claim follows from [EHK + 19, Theorem 3.4.11] since the objectwise group completion of L mot (FSyn) is an A 1 -invariant presheaf with framed transfers. By Corollary 3.4, it remains to show that the monoids π nis 0 L mot (FSyn ∞ ) and π 0 (L A 1 FSyn ∞ )(k) are groups, which follows from Proposition 4.1.
Except for the statement about positive characteristic, which we shall prove in Section 5, Theorem 1.1 follows from Corollary 4.2 and [EHK + 20b, Corollary 3.4.2(i)].
If S is a scheme and a ∈ O(S) × , we denote by a ∈ FSyn fr (S) the finite syntomic S-scheme S framed by the image of a under the canonical map O(S) × → ΩK(S). We write n for the alternating sum 1 + −1 + 1 + · · · with n terms, and we write h for 2 = 1 + −1 . Recall that we have FSyn fr ∞ = colim d→∞ FSyn fr 2d and FSyn or ∞ = colim d→∞ FSyn or 2d , where the transition maps are given by adding h. In the notation of Section 3, we have equivalences   We denote by GW the presheaf of Grothendieck-Witt rings [Kne77], and by GW = K MW 0 the presheaf of unramified Grothendieck-Witt rings [Mor12, §3.2]. Note that if K is a field then GW(K) = GW(K) [Mor12, Lemma 3.10], whereas in general the relationship between GW(R) and GW(R) is more subtle.

Lemma 4.5. Let R be a regular local ring over a field. Then the group GW(R) is generated by a for a ∈ R × .
Proof. If R has characteristic 2, we know that GW(R) = GW(R) [BH21, Theorem 10.12], and GW(R) has the claimed property by [HM73,Corollary I.3.4]. Suppose therefore that R has characteristic 2. By Popescu's theorem [Stacks, Tag 07GC], we can assume R essentially smooth over a perfect field. The result now follows from Lemma 4.6(iii) below.
Lemma 4.6. Let k be a perfect field of characteristic 2. Let I ⊂ GW denote the kernel of the rank map GW → Z, and write I n ⊂ GW for the nth power Zariski subsheaf of ideals.
(i) Each sheaf I n is strictly A 1 -invariant (in particular a Nisnevich sheaf ).
(iii) For n 1, the sheaf I n is Zariski locally generated by n-fold Pfister forms a 1 , . . . , a n with a i ∈ O × . The sheaf GW is Zariski locally generated by a for a ∈ O × .
Proof. Denote by I n ⊂ GW the sheaf defined in [Mor12, Example 3.34]. Thus each I n is a strictly A 1 -invariant subsheaf of ideals, I n I m ⊂ I n+m , and I n (K) = I n (K) for finitely generated field extensions K/k. Moreover I 1 is the kernel of the rank map, so that I 1 = I. For X ∈ Sm k , write N n (X) ⊂ ν n (X) for the subgroup generated by global logarithmic differentials, i.e., expressions of the form [a 1 , . . . , a n ] = da 1 /a 1 ∧ · · · ∧ da n /a n with a i ∈ O(X) × . It is clear that N n is a subpresheaf of ν n . Let X ∈ Sm k be connected with generic point η, and consider the following diagram N n (X) (I n /I n+1 )(X) ν n (η) (I n /I n+1 )(η), f f where I n /I n+1 is the quotient in the Nisnevich topology. It follows from [GL00, Theorem 8.3] that ν n is strictly A 1 -invariant, and hence the left-hand vertical map is injective. Since strictly A 1 -invariant sheaves form an abelian subcategory of Nisnevich sheaves, I n /I n+1 is strictly A 1 -invariant and the right-hand vertical map is an injection. By [Kat82], there is an isomorphism f as displayed, which is determined by f ([a 1 , . . . , a n ]) = a 1 , . . . , a n . This formula implies that f restricts to a morphismf , and that the mapsf (for various X) assemble into a morphism of presheaves. By [Mor15, Theorem 1.2], L zar N n = ν n and hence we have constructed a map L zarf : ν n → I n /I n+1 . Since L zarf induces an isomorphism on fields, it is an isomorphism. Let R be an essentially smooth local k-algebra. For n 1, let J n ⊂ I n (R) denote the subgroup generated by n-fold Pfister forms. Consider the exact sequence The map J n ⊂ I n (R) → ν n (R) is surjective, since it sends a 1 , . . . , a n to [a 1 , . . . , a n ]. It follows that ν n (R) I n (R)/I n+1 (R). It also follows that I n (R) = J n +I n+1 (R) and hence by induction that I n (R) = J n +I m (R) for any m > n. By [HM73, Theorem III.5.10 and preceeding paragraph] and the unramifiedness of I m , we have I m (R) = 0 for m > dim R. Thus J n = I n (R) = I n (R), and so I n = I n . If G ⊂ GW(R) denotes the subgroup generated by a for a ∈ R × , then G → Z is surjective and we similarly get GW(R) = G + I(R) = G. All claims follow.
Remark 4.7. One may prove that for any local ring R, GW(R) is generated by elements of the form a for a ∈ R × . It follows from Lemma 4.6(iii) that for any regular local k-algebra R with fraction field K, the ring GW(R) ⊂ GW(K) = GW(K) coincides with the image of GW(R) → GW(K). Proof. Let f : L mot (Z × FSyn fr ∞ ) → L mot (FSyn fr,gp ) be the canonical map. As in the proof of Corollary 4.2, L mot (FSyn fr,gp ) is the group completion of L mot (FSyn fr ) in Nisnevich sheaves. For the first equivalence, it suffices by Corollary 3.4 to show that f induces an isomorphism on π nis 0 . The π nis 0 of the right-hand side is isomorphic to the unramified Grothendieck-Witt sheaf GW, and this isomorphism sends a to a [EHK + 20a, Corollary 3.3.11]. If X is an essentially smooth henselian local scheme, then GW(X) is generated by a for a ∈ O(X) × by Lemma 4.5, and since a is invertible in the left-hand side by Lemma 4.3, we deduce that f is an effective epimorphism. Note that a surjective map of discrete monoids whose codomain is a group is injective if and only if its kernel is trivial. It therefore remains to show that the fiber of f over 0 is connected. By [Mor05, Lemma 6.1.3], it suffices to check this on finitely generated separable field extensions of k. Thus, it suffices to show that π 0 L mot (Z × FSyn fr ∞ )(k) is a group for any field k; indeed, by Corollary 3.4 this implies that f is acyclic on k-points, and in particular connected. Since we have a surjection , it suffices to show that the left-hand side is a group, which also implies the second equivalence (by Corollary 3.4 again).
Let T ∈ FSyn fr d (k) be a framed finite syntomic k-scheme of degree d. Choosing an embedding of T in A m and lifting the framing of T to a trivialization of its conormal sheaf, we obtain a k-point of the smooth k-scheme Hilb fr d (A m ). We now apply Proposition 2.3 with X = Hilb fr d (A m ) and U ⊂ X the finite étale locus: we obtain framed finite étale k-schemes A and B with a framed cobordism T A ∼ B. Now B is a sum of framed finite étale schemes of the form (Spec L, a ) with L/k a finite separable field extension and a ∈ L × . We are therefore reduced to proving that (Spec L, a ) is invertible in π 0 L A 1 (Z × FSyn fr ∞ )(k). By Lemma 4.3, it is enough to prove that, for some a ∈ L × , both (Spec L, a ) and (Spec L, −a ) are invertible. If f (x) is a monic polynomial such that L k[x]/(f (x)), then (Spec L, ±f (x) ) ϕ(±f ). By [EHK + 19, Proposition B.1.4], ϕ(±f ) is framed cobordant to ±1 [L : k] , which is clearly invertible in π 0 (Z×FSyn fr ∞ )(k). This completes the proof for FSyn fr . The proof for FSyn or is exactly the same, using that the unit map

Removing the plus construction in positive characteristic
The following proposition is an elaboration of [Rob15, Proposition 2.19].
Proposition 5.1. Let C be an E k -monoidal ∞-category with 2 k ∞ and let x ∈ C. Let C[x −1 ] be the E k -monoidal ∞-category obtained from C by inverting x, and let tel x (C) be the C-module colimit of the sequence Consider the following assertions: (1) The cyclic permutation of x ⊗3 becomes homotopic to the identity in tel x (C).
(2) For some n 2, the cyclic permutation of x ⊗n becomes homotopic to the identity in tel x (C).
Proof. The implication (1) ⇒ (2) is trivial. Note that assertion (3) is equivalent to the assertion that x acts invertibly on the telescope tel x (C). Consider the commutative diagram where each square commutes via the cyclic permutation of x ⊗n . Let Seq be the 1-skeleton of the nerve of the poset N and let σ : Seq → ∞-Cat be the cone given by the first row mapping to the colimit of the second row. Then the action of x on tel x (C) is the induced map colim(σ |Seq) → σ (∞). Under assumption (2), we obtain an equivalent cone if we replace each cyclic permutation in the above diagram by the identity, which is trivially a colimiting cone. This proves (2) ⇒ (3).
If C is E 3 -monoidal and x ∈ C is invertible, then the cyclic permutation of x ⊗3 is homotopic to the identity. Indeed, π 0 Aut(x ⊗3 ) is an abelian group and the cyclic permutation of order 3 becomes trivial in the abelianization of Σ 3 . This proves (3) ⇒ (1).
Example 5.2. Let R be a derived commutative ring. The cyclic permutation of R 3 is induced by a matrix in SL 3 (Z) and hence is A 1 -homotopic to the identity. Applying Proposition 5.1 to the E ∞ -space (L A 1 Vect)(R), we deduce the well-known fact that the canonical map is an A 1 -equivalence on derived commutative rings, where Z is the constant sheaf with value Z and Vect ∞ = colim n Vect n . This explains why the plus construction is not needed in the equivalences (over a regular base) Ω ∞ T KGL L mot (Z × Vect ∞ ) L mot (Z × Gr ∞ (A ∞ )). Lemma 5.3. Let p be a prime number. Over F p , there exists a sequence of C p -equivariant framed cobordisms between p with nontrivial C p -action and −1 p with trivial C p -action. In particular, for any F p -scheme S and any α ∈ FSyn fr (S), the action of C p on pα in (L A 1 FSyn fr )(S) is trivial.
with the framing τ induced by −f (t, x). The group C p acts on X over F p [t] by x → x + t. As this action fixes −f (t, x), we obtain an action of C p on (X, τ). The fiber X 1 consists of p points cyclically permuted and trivially framed: indeed, the framing on the point Spec On the other hand, the fiber X 0 is Spec F p [x]/(x p ) with trivial action and framing induced by −x p . By [EHK + 19, Proposition B.1.4], X 0 is framed cobordant to −1 p .
Remark 5.4. Let S be an affine F p -scheme, M an étale sheaf of E ∞ -spaces on Sm S , and m ∈ M(S). Then the action of C p on m p in (L A 1 M)(S) is trivial, because Bé t C p is A 1 -contractible on affine F p -schemes by Lemma 5.5 below. This does not apply directly to FSyn fr , which is not an étale sheaf. However, by [EHK + 19, §4.2.36], there is an E ∞ -map FEt → FSyn fr where FEt is the moduli stack of finite étale schemes, which is an étale sheaf. This gives an alternative proof of the last statement of Lemma 5.3.
Lemma 5.5. The presheaf Bé t C p is A 1 -contractible on affine F p -schemes.
Proof. By the Artin-Schreier sequence [Stacks, Tag 0A3J], we have a fiber sequence of presheaves where G a is the additive group scheme. Since L A 1 Bé t G a * , it suffices to prove that this sequence remains a fiber sequence after applying L A 1 . This follows from a general criterion for the geometric realization of a cartesian square to remain cartesian [Lur17a, Lemma 5.5.6.17], which applies because π 0 (Bé t G a (X)) = H 1 et (X, O) = * for X an affine scheme [Stacks, Tags 03P2 and 01XB]. The following proposition completes the proofs of all the theorems in the introduction.
Proposition 5.6. Let k be a field of positive characteristic. Then the canonical maps L mot (FSyn ∞ ) → L mot (FSyn ∞ ) + (L A 1 FSyn ∞ )(k) → (L A 1 FSyn ∞ )(k) + are equivalences. The same holds for FSyn fr ∞ and FSyn or ∞ . Proof. By Lemma 5.3 and Proposition 5.1, the left-hand sides are commutative monoids, and they are grouplike by Corollary 4.2 and Proposition 4.8. Hence they coincide with their plus constructions.
Remark 5.7. Let C be a smooth curve over a field k of characteristic zero. If X is a finite flat C-scheme with an action of the cyclic group C n , then the locus of points c ∈ C such that C n acts trivially on the fiber X c is clopen. Indeed, that locus is the equalizer of a pair of sections of a finite flat C-subgroup scheme of Aut(X/C), which is necessarily étale. In particular, unlike in positive characteristic, there cannot exist a C n -equivariant cobordism between a finite syntomic k-scheme with nontrivial C n -action and one with trivial C n -action. The group homomorphism C n → π 1 (L A 1 FSyn ∞ )(k) could nevertheless be trivial for other reasons.