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LetC ⊂Sch/k be an admissible category. Fix an objectX ofC and an integern≥1. By definition of an admissible category,C is closed under finite products and quotients under finite groups. Thennth fold productX×nis an object ofC, hence, the quotient

X×n/Σn is also in C. Denote this quotient by Symn(X). Then, we have a functor Symn:C → C. It is immediate to observe that Symn(Spec(k)) is isomorphic to the point Spec(k) for n≥1. By convention, Sym0 will be the constant endofunctor of C which sends an object X of C to the point Spec(k).

Let us fix n ∈_N. Since C is a small category and ∆opS is cocomplete, Theorem 3.7.2 of [4] asserts the existence of the left Kan extension of the composite

C Sym−→nC −→h S

along the Yoneda embedding h.

Definition 2.3.31. We denote by Symn_g the above left Kan extension, and call it the

nth-foldgeometric symmetric powerof Nisnevich sheaves.

Explicitly, Symn_g is described as follows. For a sheafX inS, we denote by (h↓X) the comma category whose objects are arrows of the form hU → X for U ∈ ob (C). Let F_X : (h ↓ X) → S be the functor which sends a morphism hU → X to the representable sheafhSymn_U. Then, Symn

g(X) is nothing but the colimit of the functor

F_X.

Definition 2.3.32. The endofunctor Symn_g of Definition 2.3.31 induces an endofunctor of ∆opS. We call it the nth-fold geometric symmetric power of simplicial Nisnevich sheaves. By abuse of notation, we denote this endofunctor by the same symbol Symn_g if no confusion arises.

Example 2.3.33. Fix a natural number n. For each k-scheme X in C, the nth fold geometric symmetric power Symn_g(hX) of the representable functor hX coincides with the representable functorhSymn_X. The section Symn

g(hX)(Spec(k)) is nothing but the set of effective zero cycles of degreen onX.

Remark 2.3.34. Since Symn_g : S → S preserves the point Spec(k), it induces an endofunctor ofS∗, and hence an endofunctor of ∆opS∗.

Warning 2.3.35. As many statements hold similarly for pointed and unpointed (sim- plicial) sheaves, we shall use the same symbol Symn_g to denote the nth fold geometric symmetric power both pointed and unpointed (simplicial) sheaves if no confusion arises.

Lemma 2.3.36. Left adjoint functors preserves left Kan extensions, in the following sense. Let L:E → E0 be a left adjoint functor. If LanGF is the left Kan extension of

a functor F:C →E along a functorG:C →D, then the composite L◦LanGF is the

Proof. See [33, Lemma 1.3.3].

Lemma 2.3.37. For every natural n, the endofunctor Symn_g of ∆opS is isomorphic to the compositionγn◦λn. Similarly,Symng as an endofunctor of∆opS∗ is isomorphic

to the composition γn,+◦λn,+.

Proof. Since the functors Symn_g, γn and λn are termwise, it is enough to show that Symn_g, as a endofunctor ofS, is isomorphic to the composition of ˜Λn with (colimΣn)

∗_.

Indeed, as the functor (colimΣn)

∗ _{is left adjoint, Lemma 2.3.36 implies that the com-}

posite S Λ˜n // SΣn (colimΣn) ∗ / /_{S (2.26)}

is the left Kan extension of the composite

C Λn //

CΣn h //SΣn (colimΣn)

∗

/

/_S

along the embeddingh:C →S. Now, in view of the commutativity of diagram (2.25), the preceding composite is isomorphic to the composite

C Λn //

CΣn colimΣn //C h //S _,

but it is isomorphic to the compositeC Sym

n

−→ C →h S, which implies that the composite (2.26) is isomorphic to Symn_g, as required.

We denote by ¯C+ the full subcategory of coproducts of pointed objects of the form

(hX)+ inS∗ for objectsX inC. For every object X inC, the pointed sheaf (hX)+ is

isomorphic toh(X+). Indeed, (hX)+is by definition equal to the coproducthXqhSpec(k)

and this coproduct is isomorphic to the representable functorh_XqSpec(k) which is equal

toh(X+).

Similarly, we denote by C¯Σn

+ the full subcategory of coproducts of pointed objects

(hX)+ inS∗Σn for objectsX inCΣn.

Theorem 2.3.38 (Voevodsky). Let f:X →Y be a morphism in ∆opC¯+. If f is an

A1-weak equivalence in ∆opS∗, then Symng(f) is an A1-weak equivalence.

Proof. By Lemma 2.3.37, Symn_g is the composition γn,+◦λn,+. The idea of the proof

is to show that γn,+ and λn,+ preserve A1-weak equivalences between objects which termwise are coproducts of representable sheaves. The functor λn,+ sends morphisms

ofWNis,+∪ PNis,+ between objects in ∆opC¯+toA1-weak equivalences between objects in ∆opC¯Σn

+ . Sinceλn,+ preserves filtered colimits, Lemma 2.20 of [41] implies thatλn,+

preservesA1-weak equivalence as claimed. Similarly, in view of the class given in (2.24), we use again Lemma 2.20 of loc.cit. to prove that γn,+ sends A1-weak equivalences between objects in ∆opC¯Σn

We define the functor Φ : ∆opC¯+→H∗(CNis,A1) as the composite ∆opC¯+,→∆opS∗→H∗(CNis,A1),

where the first arrow is the inclusion functor and the second arrow is the localization functor with respect to theA1-weak equivalences.

Lemma 2.3.39. Let C be an admissible category. The functor

Φ : ∆opC¯+ →H∗(CNis,A1)

is a strict localization, that is, for every morphism f in H∗(CNis,A1), there is a mor-

phismg of ∆opC¯+ such that the image Φ(g) is isomorphic to f.

Proof. By Theorem 2.5 of [30, page 71], the category H∗(CNis,A1) is the localization of the categoryH∗(CNis) with respect to the image ofA1-weak equivalences trough the canonical functor. Then, it is enough to prove that the canonical functor from ∆opC¯+

toH∗(CNis) is a strict localization. Indeed, let f:X →Y be a morphism of pointed

simplicial sheaves on the siteCNis representing a morphism inH∗(CNis). The functorial

resolutionQproj gives a commutative square

Qproj_(X) Qproj(f) // Qproj_(Y) X _f //_Y

where the vertical arrows are object-wise weak equivalences. Since the object-wise weak equivalences are local weak equivalences, the vertical arrows of the above diagram are weak equivalences. This implies thatfis isomorphic toQproj(f) inH∗(CNis). Moreover,

by Corollary 2.3.21, the morphismQproj(f) is in ∆opC¯+.

Corollary 2.3.40. For each integern≥1, there exists the left derived functorLSymn_g

fromH∗(CNis,A1)to itself such that we have a commutative diagram up to isomorphism ∆opC¯+ Φ Symng _// ∆opS∗ H∗(CNis,A1) _LSymn g / /_H∗(CNis,A1₎ (2.27)

Proof. By Theorem 2.3.38, the functor Symn_g preserves A1-weak equivalences between objects in ∆op_C¯_{+. Hence, the composite}

∆opC¯+ Symn

g

−→ ∆opS∗ −→H∗(CNis,A1)

sends A1-weak equivalences to isomorphisms. Then, by Lemma 2.3.39 there exists a functor LSymn_g such the diagram (2.27) commutes and for every simplicial sheaf X, the object LSymn_g(X) is isomorphic to Symn_g(Qproj_{(X)) inH}

∗(CNis,A1).