Connectivity and Purity for logarithmic motives

The goal of this paper is to extend the work of Voevodsky and Morel on the homotopy $t$-structure on the category of motivic complexes to the context of motives for logarithmic schemes. To do so, we prove an analogue of Morel's connectivity theorem and show a purity statement for $(\mathbf{P}^1, \infty)$-local complexes of sheaves with log transfers. The homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is proved to be compatible with Voevodsky's $t$-structure i.e. we show that the comparison functor $R^{\overline{\square}}\omega^*\colon \mathbf{DM}^{\textrm{eff}}(k)\to \mathbf{logDM}^{\textrm{eff}}(k)$ is $t$-exact. The heart of the homotopy $t$-structure on $\mathbf{logDM}^{\textrm{eff}}(k)$ is the Grothendieck abelian category of strictly cube-invariant sheaves with log transfers: we use it to build a new version of the category of reciprocity sheaves in the style of Kahn--Saito--Yamazaki and R\"ulling.


Introduction
Voevodsky's category of motivic complexes over a perfect field k is based on a simple idea: most cohomology theories for smooth k-schemes are insensitive to the affine line, i.e. they satisfy A 1 -homotopy invariance. This observation led Voevodsky to introduce as a building block of his theory of motives the category of homotopy invariant sheaves with transfers HI Nis (k), that is, sheaves F for the Nisnevich topology defined on the category of finite correspondences over k such that F (X × A 1 ) ≃ − → F (X) for every smooth k-scheme X. These sheaves enjoy many nice properties: the category HI Nis (k) is a Grothendieck abelian subcategory of the category Shv tr Nis (k) of Nisnevich sheaves with transfers, closed under extensions and equipped with a (closed) symmetric monoidal structure ⊗ HI . Moreover, a celebrated theorem of Voevodsky shows that the cohomology presheaves H n Nis (−, F ) of a homotopy invariant sheaf with transfers F are still A 1 -homotopy invariant. In fact, HI Nis (k) can be identified with the heart of a certain t-structure on the triangulated category DM eff (k), induced by the standard t-structure on the derived to the P 1 -local theory, developed in [Ayo21]. In particular, the statement can be reduced to a purity result for local complexes: Theorem 1.2. (see Theorem 4.4) Let X be a connected fs log smooth k-scheme which is essentially smooth over k (in particular, the underlying scheme X is an essentially smooth k-scheme) such that X is an henselian local scheme. Then the map H i (C(X)) → H i (C(η X , triv)) is injective for every (sNis, )-fibrant complex of presheaves C ∈ Cpx(PSh log (k, Λ)).
Here, we write η X for the generic point of X, and (η X , triv) for η X seen as a log scheme with trivial log structure. The proof is quite long, and for it we use in an essential way the results developed in [BPØ], such as the existence of a number of distinguished triangles in logDA eff (k) and a description of the motivic Thom spaces [BPØ,7.4]: in particular, new ingredients (compared to the argument given by Morel or Ayoub) are required when the log structure on X is not trivial.
We remark that the original formulation of Morel's connectivity theorem was given for the A 1 -localization of presheaves of S 1 -spectra, rather than presheaves of chain complexes. The arguments given in this paper can be easily adapted to that context. Since our main application is about the motivic category introduced in [BPØ], we decided to state the results for Cpx(PSh log (k, Λ)).
Having the analogue of Morel's connectivity theorem at disposal, it is possible to characterize -local complexes of sheaves: Corollary 1.3. (see Corollary 5.5) Let C ∈ D dNis (PSh t (k, Λ)) where t ∈ {log, ltr}. Then the following are equivalent: (a) C is -local (b) the homology sheaves a dNis H i C are strictly -invariant for every i ∈ Z, i.e. their cohomology presheaves are -invariant.
We can then consider the inclusions logDA eff (k, Λ) ֒→ D dNis (PSh log (k, Λ)) logDM eff (k, Λ) ֒→ D dNis (PSh ltr (k, Λ)) that identify logDA eff (k, Λ) and logDM eff (k, Λ) with the subcategories of -local complexes. Using Theorem 1.1 it is easy to show that the truncation functors τ ≤n and τ ≥n preserve the categories of -local complexes, and therefore that the standard t-structures on the categories of (pre)sheaves induce the desired homotopy t-structure on log motives. We denote by CI log dNis (and by CI ltr dNis for the variant with transfers) its heart, which is then identified with the category of strictly -invariant dNis-sheaves. It follows from the fact that the t-structures are compatible with colimits (in the sense of [Lur17]) that CI log dNis and CI ltr dNis are Grothendieck abelian categories. See Theorem 5.7. In particular, the inclusions i : CI log dNis ֒→ Shv log dNis (k, Λ) i tr : CI ltr dNis ֒→ Shv ltr dNis (k, Λ) admit both a left and a right adjoint. Objects of CI log dNis and of CI ltr dNis satisfy the following purity property.
Theorem 1.4. (see Theorem 5.10) Let F ∈ CI log dNis (resp. F ∈ CI ltr dNis ). Then for all X ∈ SmlSm(k) (see the notation below) and U ⊆ X an open dense, the restriction F (X) → F (U) is injective.
In [BPØ], a comparison functor R ω * : DM eff (k, Λ) → logDM eff (k, Λ) has been constructed. Under resolution of singularities, it is known that R ω * is fully faithful, and it identifies DM eff (k, Λ) with the subcategory of (A 1 , triv)-local objects in logDM eff (k, Λ) (see [BPØ,Thm. 8.2.16] and the results quoted there). Even without knowing that R ω * is a full embedding, we can show that it is t-exact with respect to the homotopy t-structures on both sides. In fact, when R ω * is an embedding, it is straightforward to conclude that Voevodsky's homotopy t-structure is induced by the t-structure on logDM eff (k, Λ) via R ω * . See Prop. 5.11.
The good properties of the category of strictly -invariant sheaves CI ltr dNis , deduced from the identification with the heart of the homotopy t-structure, allow us to make a further comparison with the category RSC Nis of reciprocity sheaves of Kahn-Saito-Yamazaki. This is an abelian subcategory of the category of Nisnevich sheaves with transfers Shv tr Nis (k), whose objects satisfy a certain restriction on their sections inspired by the Rosenlicht-Serre theorem on reciprocity for morphisms from curves to commutative algebraic groups [Ser84,III]. See [KSY] and the recollection paragraph below.
In [Sai20b], S. Saito constructed an exact and fully faithful functor (1.4.1) Log : RSC Nis (k) → Shv ltr dNis (k, Z) having as essential image a subcategory of CI ltr dNis . In Section 6 we study its proleft adjoint Rsc : Shv ltr dNis (k, Z) → pro-RSC Nis and in particular its behavior with respect to the lax symmetric monoidal structure (−, −) RSC Nis constructed in [RYS]. See Theorem 6.11 and Corollary 6.12.
The category of reciprocity sheaves RSC Nis is defined in terms of the auxiliary category of modulus pairs, building block of the theory of motives with modulus as developed in [KMSY21a], [KMSY21b] and [KMSY]. In fact, Saito's functor (1.4.1) is itself defined by first "lifting" a reciprocity sheaf to the category of (semipure) sheaves on modulus pairs, and then applying another functor landing in Shv ltr dNis (k, Z). It turns out that such detour may not be necessary. In fact, we can look at the composite functor (1.4.2) ω log CI : CI ltr dNis Shv ltr dNis (k, Z) Shv tr Nis (k, Z) where ω ♯ is the left Kan extension of the restriction functor from smooth log schemes to smooth k-schemes ω : lSm(k) → Sm(k), sending X ∈ lSm(k) to X o , the open subscheme of the underlying scheme X of X where the log structure is trivial. Using a comparison result from [BPØ] (which relies on the resolution of singularities) and our purity Theorem 5.10 we can show that ω log CI in (1.4.2) is faithful and exact (Proposition 7.2). If we assume that furthermore it is full (see Conjecture 7.3), we will denote by LogRec its essential image: if Conjecture 7.3 holds, it is a Grothendieck abelian category, that contains RSC Nis as full subcategory, see Theorem 7.6. Thanks to the purity property for strictly -invariant sheaves, its objects satisfy global injectivity, i.e. for every F ∈ LogRec and U ⊂ X dense open subset of X ∈ Sm(k), the restriction map is injective. See [KSY] for a similar statement for reciprocity sheaves (relying on [Sai20a]). In fact, this would imply that the cohomology presheaves of any reciprocity sheaf F ∈ RSC Nis satisfy global injectivity, see Corollary 7.7.
Acknowledgements. The authors would like to thank Joseph Ayoub for a careful reading of a preliminary version of this manuscript, and for suggesting an improvement that allowed us to weaken the original assumptions on Theorem 4.4. We are also grateful to Kay Rülling for useful comments, to Doosung Park for many conversations on the subject of this paper, and to Shuji Saito for his interest in our work. Finally, we thank the referee for their detailed report.
Notations and recollections on log geometry. In the whole paper we fix a perfect base field k and a commutative unital ring of coefficients Λ. Let S be a Noetherian fine and saturated (fs for short) log scheme. We denote by lSm(S) the category of fs log smooth log schemes over S. We are typically interested in the case where S = Spec(k), considered as a log scheme with trivial log structure.
For X ∈ lSm(S), we write X ∈ Sch(S) for the underlying S-scheme, where S is the scheme underlying S. We also write ∂X for the (closed) subset of X where the log structure of X is not trivial. Let SmlSm(S) be the full subcategory of lSm(S) having for objects X ∈ lSm(S) such that X is smooth over S. By e.g. [BPØ,A.5.10], if X ∈ SmlSm(k), then ∂X is a strict normal crossing divisor on X and the log scheme X is isomorphic to (X, ∂X), i.e. to the compactifying log structure associated to the open embedding (X \ ∂X) → X. If X, Y ∈ lSm(S), we will write X × S Y for the fiber product of X and Y over S computed in the category of fine and saturated log schemes: it exists by [Ogu18, Cor. III.2.1.6] and it is again an object of lSm(S) using [Ogu18, Cor. IV.3.1.11]. Unless S has trivial log structure, the underlying scheme X × S Y does not agree with X × S Y . See [Ogu18, §III.2.1] for more details.
We denote by PSh log (S, Λ) the category of presheaves of Λ modules on lSm(S). It has naturally the structure of closed monoidal category. If τ is a Grothendieck topology on lSm(S) (see below), we write Shv log τ (S, Λ) for full subcategory of PSh log (S, Λ) consisting of τ -sheaves. We typically write a τ for the τ -sheafification functor.
Let SmlSm(S) be the category of fs log smooth S-schemes X which are essentially smooth over S, i.e. X is a limit lim ← −i∈I X i over a filtered set I, where X i ∈ SmlSm(S) and all transition maps are strictétale (i.e. they are strict maps of log schemes such that the underlying maps f ij : X i → X j areétale) For (X, ∂X) ∈ SmlSm(S) and x ∈ X, let ι : Spec(O X,x ) → X, be the canonical morphism. Then the local log scheme (Spec(O X,x , ι * (∂X)) is in SmlSm(S).
We frequently allow F ∈ PSh log (S, Λ) to take values on objects of SmlSm(S) by setting F (X) := lim − →i∈I F (X i ) for X as above.
Notations and recollections on reciprocity sheaves. We briefly recall some terminology and notations from the theory of modulus sheaves with transfers, see [KMSY21a], [KMSY21b], [KSY], and [Sai20a] for details.
A modulus pair X = (X, X ∞ ) consists of a separated k-scheme of finite type X and an effective (or empty) Cartier divisor X ∞ such that X := X \ |X ∞ | is smooth; it is called proper if X is proper over k. Given two modulus pairs X = (X, X ∞ ) and Y = (Y , Y ∞ ), with opens X := X \ |X ∞ | and Y := Y \ |Y ∞ |, an admissible left proper prime correspondence from X to Y is given by an integral closed subscheme Z ⊂ X × Y which is finite and surjective over a connected component of X, such that the normalization of its closure Z N → X × Y is proper over X and satisfies Let PSh tr (k) be Voevodsky's category of presheaves with transfers. Recall from [Sai20a, Def. 1.34] that F ∈ PSh tr (k) has reciprocity if for any X ∈ Sm(k) and a ∈ F (X) = Hom PSh tr (Z tr (X), F ), there exists X = (X, X ∞ ) ∈ MSm(X) such that the mapã : Z tr (X) → F corresponding to the section a factors through h 0 (X ). Here MSm(X) is the category of objects X ∈ MCor such that X − |X ∞ | = X, and h 0 (X ) is the presheaf defined as where = (P 1 , ∞) (we will use the same notation for the log scheme in lSm(k)), and the tensor product refers to the monoidal structure in MCor, see [KMSY21a]. It is easy to see that RSC is an abelian category, closed under sub-objects and quotients in PSh tr (k). On the other hand, it is a theorem [Sai20a, Thm. 0.1] that RSC Nis = RSC ∩ NST is also abelian, where NST = Shv tr Nis (k) is the category of Nisnevich sheaves with transfers.

Preliminaries on logarithmic motives
In this Section we review the construction and the basic properties of the categories logDM eff (k, Λ) and logDA eff (S, Λ) of motives, with and without transfers, as introduced in [BPØ]. The standard reference for properties of log schemes is [Ogu18]. The definitions in this section work for a quite general base log scheme S, but in the rest of the paper we will mostly deal with the case S = Spec(k).
2.1. Topologies on logarithmic schemes. Recall from [BPØ, 3.1.4] that a cartesian square of fs log schemes for the reduced scheme structures. We say that Q is a dividing distinguished square (or elementary dividing square) if Y ′ = X ′ = ∅ and f is a surjective proper logétale monomorphism. According to [BPØ,A.11.9], surjective proper logétale monomorphisms are precisely the log modifications, in the sense of F. Kato [Kat]. We similarly say that Q is a (strict) Zariski distinguished square if f and g are (strict) open immersions (note that "strict" here is redundant, since open immersions in the category of log schemes are automatically strict).
Definition 2.1. The strict Nisnevich cd-structure (resp. the dividing cd-structure) is the cd structure on lSm(S) associated to the collection of strict Nisnevich distinguished squares (resp. of elementary dividing squares), and the dividing Nisnevich cd structure is the union of the strict Nisnevich and of the dividing cd-structures.
The associated Grothendieck topologies on lSm(S) are called the strict Nisnevich and the dividing Nisnevich topology respectively. Mutatis mutandis, we define the (strict) Zariski and the dividing Zariski topologies on lSm(S) in a similar fashion.
Let S be a Noetherian fs log scheme such that S has finite Krull dimension. According to [BPØ,Prop. 3.3.30], the strict Nisnevich and the dividing Nisnevich cd structures on lSm(S) are complete, regular and quasi-bounded with respect to the dividing density structure ([BPØ, Def. 3.3.22]). In particular, any X ∈ lSm(S) has finite cohomological dimension. When S = Spec(k), we can bound the dNis cohomological dimension by the Krull dimension of the underlying scheme, according to the following Proposition.
Proposition 2.2. (see [BPØ,Cor. 5.1.4]) Let F ∈ Shv log dNis (k, Λ) and let X ∈ lSm(k). Let d = dim(X). Then H i dNis (X, F X ) = 0 for i ≥ d + 1. Remark 2.3. Since the dividing Nisnevich cd-structure is clearly squareable in the sense of [BPØ,Def. 3.4.2], one can apply [BPØ,Theorem 3.4.6] to get a bound on the dNis cohomological dimension for any X ∈ lSm(S) in terms of the dimension of a log scheme computed using the dividing density structure: this is, for a general log scheme X, larger than the Krull dimension of the underlying scheme X (see [BPØ,Ex. 3.3.25]). In view of [BPØ,Rmk. 3.3.27], for S = Spec(k) and X ∈ lSm(k) such dimension agrees with the Krull dimension.
The dividing Nisnevich cohomology groups are, a priori, difficult to compute. The situation looks better for X ∈ SmlSm(k) thanks to the following result.
Theorem 2.4. [BPØ, Theorem 5.1.8] Let C be a bounded below complex of strict Nisnevich sheaves on SmlSm(k). Then for every X ∈ SmlSm(k) and i ∈ Z there is an isomorphism where X Sm div is the category of smooth log modifications Y → X of X. A formula similar to (2.4.1) holds for X ∈ lSm(S) as in the following Theorem.
Theorem 2.5. [BPØ, Theorem 5.1.2] Let S be a Noetherian fs log scheme, and let C be a bounded below complex of strict Nisnevich sheaves on lSm(S). Then for every X ∈ lSm(S) and i ∈ Z there is an isomorphism where the colimit runs over the set X div of log modifications of X (not necessarily smooth).
The following result comes in handy to produce long exact sequences: Lemma 2.6. Let X, Y ∈ SmlSm, let D X ⊆ X and D Y ⊆ Y be Cartier divisors such that D X + |∂X| and D Y + |∂Y | have simple normal crossings. Suppose that is a Zar-(resp. Nis-) distinguished square in Sm. Let ∂X + and ∂Y + be the log structures induced by the divisors D X +|∂X| and D X +|∂Y |, and let X + := (X, ∂X + ) and Y + := (Y , ∂Y + ). Then, for every complex C ∈ PSh ltr (k, Λ) which is sZar-(resp. sNis-) fibrant the following square is a homotopy pullback.
Proof. Let τ be either Zar or Nis. Since the log structures on X − D X (resp Y − D Y ) induced by X and X + (resp. Y and Y + ) are the same, the following squares are sτ -distinguished: Moreover, the canonical maps X + → X and Y + → Y , whose underlying maps of schemes are the identities of X and Y , make the following diagram commutative: Since C is sτ -fibrant, the big rectangle and the square on the right are homotopy pullbacks. Hence, the square on the left is a homotopy pullback.
2.2. log correspondences. Following [BPØ], we denote by lCor(k) the category of finite log correspondences over k. It is a variant of the Suslin-Voevodsky category of finite correspondences Cor(k) introduced in [Voe00], see [MVW06]. It has the same objects as lSm(k), and morphisms are given by the free abelian subgroup such that the closure V ⊂ X × Y is finite and surjective over (a component of) X and such that there exists a morphism of log schemes V N → Y , where V N is the fs log scheme whose underlying scheme is the normalization of V and whose log structure is given by the inverse image log structure along the composition V N → X × Y → X. See [BPØ, 2.1] for more details, and for the proof that this definition gives indeed a category.
Additive presheaves (of Λ-modules) on the category lCor(k) will be called presheaves (of Λ-modules) with log transfers. Write PSh ltr (k, Λ) for the resulting category. We have a natural adjunction where by convention γ ♯ is left adjoint to γ * , which is left adjoint to γ * . Here γ : lSm(k) → lCor(k) is the graph functor. For a topology τ on lSm(k), a presheaf with log transfers F is a τ -sheaf if γ * F is a τ -sheaf. We denote by Shv ltr τ (k, Λ) ⊂ PSh ltr (k, Λ) the subcategory of τ -sheaves. By [BPØ,Prop. 4 2.3. Effective log motives. We fix again a Noetherian fs log scheme S and a field k, and let C be either lSm(S) or lCor(k). We start by recalling some standard facts. The category Cpx(PSh(C, Λ)) of unbounded complexes of presheaves is equipped with the usual global (projective) model structure (W, Cof, Fib), where the weak equivalences are the quasi-isomorphisms and the fibrations are the degreewise surjective maps (see, for example, the remark after [HPS97, Thm. 9.3.1] or [Ayo07, Proposition 4.4.16]).
Let τ be a topology on C (and we require that τ is compatible with transfers when C = lCor(k)). Recall that a morphism of complexes of presheaves F → G in The left Bousfield localization of the global model structure on Cpx(PSh(C, Λ)) with respect to the class of τ -local equivalences exists and the resulting model structure (W τ , Cof, Fib τ ) is called the τ -local model structure (see, for example, [Ayo07,Prop. 4.4.31]). The maps in W τ are precisely the τ -local equivalences. It is well known that the homotopy category of Cpx(PSh(C, Λ)) with respect to the local model structure, denoted D τ (PSh(C, Λ)), is equivalent to the unbounded derived category D(Shv τ (C, Λ)) of the Grothendieck abelian category of τ -sheaves Shv τ (C, Λ).
For any X ∈ C, we write RΓ τ (X, −) : D τ (PSh(C, Λ)) → D(Λ) for the right derived functor of the global section functor Γ(X, −). The τ -(hyper) cohomology of X with values in a complex of presheaves C is then computed as H * τ (X, a τ (C)) = H * (RΓ τ (X, a τ C)). Finally, let S := (P 1 S , ∞ S ) ∈ C, with S = Spec(k) if C = lCor(k). Definition 2.7. The (τ, S )-local model structure on Cpx(PSh(C, Λ)) is the (left) Bousfield localization of the τ -local model structure with respect to the class of maps for all X ∈ C and n ∈ Z.
General properties of the Bousfield localization (see e.g. [Ayo07, Définition 4.2.64, Proposition 4.2.66]) imply that a complex of presheaves C is (τ, S )-fibrant if and only if it is τ -fibrant (i.e. fibrant for the τ -local model structure) and the morphisms C(X) → C(X × S S ) induced by the projection, are quasi-isomorphisms for every X ∈ C.
The interested reader can verify that Definition 2.9 is equivalent to [BPØ, Def.

5.2.1]
We collect now some well-known facts about the (τ, S )-local model structure, for τ ∈ {sNis, dNis} that we are going to use later. Recall that Cpx(PSh log (S, Λ)) is a closed monoidal model category with respect to the global model structure by [Ayo07,Lemme 4.4.62]. We write Hom(−, −) for the internal Hom functor.
2.11. Let X ∈ lSm(S) and let λ : X → S be the structural morphism. We have an induced functor λ * : PSh log (S, Λ) → PSh log (X, Λ) given by precomposition with λ. The functor λ * and its left Kan extension λ ! induce two adjoint functors on the categories of complexes: Since λ * is exact, it preserves by definition global fibrations and global weak equivalences, hence λ ! preserves global cofibrations and (2.11.1) is a Quillen adjunction. In fact, by e.g. [Ayo07,Thm. 4.4.51], the same holds for the τ -local model structure where τ is a topology on lSm(S); in particular, λ * preserves τ -fibrant objects.
2.12. We end this section with a computation of the localization functor where Cpx(PSh log (S, Λ)) (τ, S ) denotes the subcategory of (τ, S )-local objects. By general properties of the Bousfield localization, L comes equipped with a natural transformation λ : id → L, and the pair (L, λ) is unique up to a unique natural isomorphism.
An explicit description of the localization functor has been worked out by Ayoub in [Ayo21, Section 2] for the P 1 -localization. We spell out the construction for presheaves without transfers and for τ ∈ {sNis, dNis}.
We obtain an endofunctor Φ equipped with a natural transformation ϕ : id → Φ, and we define the endofunctor Φ ∞ by taking the colimit of the following sequence: By construction, the functor Φ ∞ comes equipped with a natural transformation Proof. We follow the same pattern of the proof in [Ayo21], and we divide the proof in two steps. First, we need to show that for any complex of presheaves C, the morphism C → Φ ∞ (C) is a (τ, S )-local equivalence. After that, we have to prove that Φ ∞ (C) is fibrant for the (τ, S )-local model structure.
We begin by observing that for all F ∈ Cpx(PSh log (S, Λ)), the tensor product Indeed, the subcategory of (τ, S )-locally acyclic complexes is a triangulated subcategory of D τ (PSh log (S, Λ)) which is stable by direct sums, and by construction it contains all the objects of the form Λ( red S ) ⊗ Λ Λ(X) for any X ∈ lSm(S). Next, note that since the homotopy fiber of ϕ C is given by which is then (τ, S )-locally acyclic in virtue of what we just observed, ϕ C is a (τ, S )-local equivalence for all complexes C. Since filtered colimits preserve (τ, S )local equivalences, we conclude that the map C → Φ ∞ (C) is a (τ, S )-local equivalence.
We move to the second part of the proof. By construction, the map Φ •n (C) → Φ •n+1 (C) factors through Φ •n (C) τ , which are by construction τ -fibrant. Hence Φ ∞ (C) is a filtered colimit of τ -fibrant objects.
Finally, we need to show that Φ ∞ (C) is S -local, which is equivalent to show that Hom S ( red S , Φ ∞ (C)) is acyclic. The argument in the proof of part (B) of [Ayo21, Thm. 2.7] goes through without changes. We leave the verification to the reader.

Lemma 2.15. Let S be a Noetherian scheme of finite Krull dimension and let
Proof. We argue as in [Ayo07, Proposition 4.5.62]. For τ = sNis, it follows from [Sta20, Tag 0737], using that S is Noetherian of finite Krull dimension. For τ = dNis, we have that for every X ∈ lSm(S), and every filtered system where (1) and (3) follow from Thm. 2.5, and (2) follows from the fact that each Y is also Noetherian of finite Krull dimension. This implies that filtered colimits preserve dNis-fibrant objects.

The connectivity theorem following Ayoub and Morel
In this section we show a -analogue of the A 1 -connectivity theorem of Morel [Mor05, Thm. 6.1.8], adapting the argument of Ayoub in [Ayo21, Section 4]. As in [Ayo21], we exploit the notion of preconnected complex (see Definition 3.3 below), and we reduce the proof of the connectivity Theorem 3.2 to a purity statement, namely Theorem 4.4, whose proof will be given in section 4. The reader should note that while the results in this section are direct analogues of the results in [Ayo21], new ingredients are necessary to prove the purity Theorem, and this is where our arguments diverge from [Ayo21].
Throughout this section, we fix a ground field k and we work with the categories of presheaves and τ -sheaves on lSm(k) for τ ∈ {sZar, sNis, dNis}. Recall from [BPØ, Lemma 4.7.2] that Shv log dNis (k, Λ) is equivalent to the category Shv dNis (SmlSm(k), Λ), of sheaves defined on the full subcategory SmlSm(k) ⊂ lSm(k). If X = (X, ∂X) ∈ SmlSm(k) and x ∈ X is any point, we consider Spec(O X,x ) ∈ SmlSm(k) with the logarithmic structure induced by the pullback of ∂X.
Definition 3.1. Let n ∈ Z and let C be a complex of presheaves on a site (C, τ ). We say that C is locally n-connected (for the topology τ ) if the homology sheaves a τ H j (C) are zero for j ≤ n.
The main result of this section is the following: Theorem 3.2. Assume that k is a perfect field and let τ ∈ {sNis, dNis}. Let C ∈ Cpx(PSh log (k, Λ)) be locally n-connected for the τ -topology. Then for any (τ, )fibrant replacement C → L, the complex L is locally n-connected.
The proof will be given at the end of this section, assuming Theorem 4.4. We need some preliminary definitions, cfr. with [Ayo21, Déf. 4.5].
(i) A complex C of presheaves is called generically n-connected if for all X ∈ SmlSm(k) with X connected and generic point η X , the homology groups H j (C(η X )) are zero for j ≤ n (ii) A complex C of presheaves is called n-preconnected if for all X ∈ SmlSm(k), the homology groups H j (C(X)) are zero for j ≤ n − dim(X).
(2) If C ∈ Cpx(PSh log (k, Λ)) is locally n-connected for a topology τ where the cohomological dimension equals the Krull dimension of the underlying scheme, then . We will prove some technical result that will be needed later. Here we let τ be either sZar, sNis or dNis.
The result for τ = dNis then can be deduced from the case sNis. Indeed, using Lemma 3.6 below, we get in particular the required vanishing holds for dNis as well.
Lemma 3.6. Let τ ∈ {sZar, sNis}. Let F be a presheaf of Λ-modules on the small site X τ such that for every τ -cover X ′ → X and Proof. This is [Ayo21, Lemma 4.9]; we reproduce part of the proof in our setting for completeness and to take care of some subtleties. Observe that the forgetful functor f : SmlSm(k) → Sm(k) that sends X to the underlying scheme X defines an isomorphism of the small sites f X : X sNis ≃ − → X Nis (and similarly for sZar and Zar): the inverse functor sends anétale scheme g : U → X to the morphism of log schemes U → X, where U is the log scheme having U as underlying scheme and log structure given by the inverse image log structure along g (note that this would be false for the dNis-topology). A presheaf F on X sNis (resp. on X sZar ) gives then canonically a presheaf F on X Nis (resp. X Zar ), by setting , and by abuse of notation we drop the underline and write simply F for both presheaves on X sNis or on X Nis (and the same for the Zariski case).
The rest of the proof of the Lemma goes through as in [Ayo21, Lemme 4.9]. See loc.cit. for more details.
Proof. Follows from the fact that H −j (L(X)) = H j τ (X, C) and Proposition 3.5. We have the following set of elementary properties of n-preconnected complexes.
Lemma 3.8. (see [Ayo21,Lemme 4.11]) Let C be an n-preconnected complex of presheaves on lSm(k): Proof. The argument of [Ayo21, Thm. 4.12] goes through. We have an explicit description of C given by Theorem 2.14. Let Proof of Theorem 3.2. We give a proof for τ = dNis, since the case τ = sNis is identical. Let C ∈ Cpx(PSh log (k, Λ)) be a complex of presheaves, locally nconnected for the dNis topology. Since the Krull dimension of any X ∈ lSm(k) agrees with the dNis-cohomological dimension by Proposition 2.2, the fact that C is locally n-connected is equivalent to ask that, for any X ∈ SmlSm(k), we have If G is a dNis-local fibrant replacement of C, this implies that H is n-preconnected (see Remark 3.4(2)), and by Proposition 3.9, any (dNis, )-fibrant replacement L of C is then n-preconnected as well. In particular, it is generically n-connected.
We are left to show that every (dNis, )-fibrant complex L which is generically n connected is also locally n-connected. Consider the canonical map a dNis H i (L)(X) → H i (L)(η X , triv) for any X ∈ SmlSm(k) with X connected and generic point η X . Here we write (η X , triv) to indicate the essentially smooth log scheme given by the scheme η X with trivial log structure. By Corollary 4.6 below (this is where the assumption that k is perfect is used), this map is injective. This implies that a dNis H i (L)(X) = 0 for any X ∈ SmlSm(k) and i < n, i.e. the homology sheaves a dNis H i (L) are zero for i < n, proving the claim.

Purity of logarithmic motives
Throughout this section, we fix a base field k, and a (sNis, )-fibrant complex of presheaves C ∈ Cpx(PSh log (k, Λ)). (1) Z maps isomorphically to e(Z), i.e. there is a Nisnevich distinguished square of schemes In particular, e(Z) is closed in P 1 Y and it is disjoint from ∞ Y . We now divide the proof in two parts. Case (i): Let us suppose that X has trivial log structure. In this case we have two sNis-distinguished squares We define the following objects of D(Λ): C Z (X) = hofib(C(X) → C(U)), Since C is (sNis, )-fibrant, it is in particular sNis-fibrant and therefore the three left vertical arrows of the following diagram (4.1.1) , s Y and t respectively, are quasi-isomorphisms. Let now α ∈ H i (C(X)) such that α |U = 0, hence there exists β ∈ H i (C Z (X)) such that α = δ(β). By the quasi-isomorphism above, there exists a unique β P 1 In particular, there exists α 0 ∈ H i (C 0 (P 1 Y )) such that δ 0 (α 0 ) = α P 1 Y . We will conclude by showing that rδ 0 is the zero map.
Since C is -local, the projection π : Y → Y induces a quasi-isomorphism π * : C(Y ) ≃ − → C( Y ). Since clearly π factors through the natural map Y → P 1 Y , we have a commutative diagram and this immediately shows that rδ 0 factors through an acyclic complex, as required. Case (ii): Let us now suppose that dim(X) = 1 and ∂X is nontrivial, supported on a finite set of k-rational points. If x ∈ |∂X|, then we can suppose X = (X − |∂X|, triv) and conclude as before (this in fact does not use the assumption on the dimension of X). So let's assume that x ∈ |∂X|: since dim(X) = 1, by replacing X with an open neighborhood of x we can suppose |∂X| = x = Z.
After replacing X with an open neighborhood of x we have a sNis distinguished square .
Since x is a k-rational point, we conclude that k = k(x) and e(x) is a k-rational point of P 1 k . We drop the subscript k for simplicity. Write as before: Since C is (sNis, )-fibrant, hence sNis fibrant, the left vertical arrow of the following diagram is a quasi-isomorphism. Now, since C is -local, the complex C(P 1 , e(x)) is quasiisomorphic to C(Spec(k)), and by choosing any k-rational point of P 1 −e(x) splitting the projection (P 1 − e(x)) → Spec(k), we see that the map is injective for every i ∈ Z. This, together with the commutativity of (4.1.2), allows us to conclude.
Corollary 4.2. Let τ be either sZar, sNis or dNis (i) Let X ∈ Sm(k). Then the following map is injective: where η X is the generic point of X and (X, triv) denotes the scheme X seen as log scheme with trivial log structure. (ii) Let X ∈ SmlSm(k) such that dim(X) = 1 and |∂X| is supported on a finite number of k-rational points. Then the following map is injective: where η X is the generic point of X.
Proof. We begin by observing that maps in (i) and (ii) exist since H i C(η X , triv) = a τ H i C(η X , triv). We first prove (i). Let α ∈ a τ H i (C(X, triv)) be a section such that α |η X = 0. Let V → X be a τ -cover such that there exists β ∈ H i (C(V, triv)) mapping to the image of α in a τ H i C(V, triv). Let η V be the disjoint union of the generic points of V . The following diagram is clearly commutative hence β maps to zero in H i (C(η V , triv)). By Lemma 4.1(i), for all x ∈ V there exists an open neighborhood V x such that β → 0 in H i (C(V x , triv)). Since we can cover V by the V x , and since for every topology τ as in the statement open sieves are covering, we conclude that β maps to zero in a τ H i C(V, triv), hence α = 0, since (V, triv) → (U, triv) is a τ -cover. This proves (i). The proof of (ii) is similar, replacing (V, triv) with (V, ∂X |V ) and using Lemma 4.1(ii).
In order to prove Theorem 4.4, we need the following technical result, which is well known to the experts. Recall that an henselian k-algebra is said to be of geometric type if there exists X ∈ Sm(k) and x ∈ X such that R ∼ = O h X,x , the henselization of the local ring O X,x at x. Lemma 4.3. Let k be a perfect field, R a henselian k-algebra of geometric type. Let p ⊆ R such that R/p is essentially smooth over k. Then the map R p → k(p) has a section.
Proof. Let κ be the residue field of R. By the properties of henselian k-algebras of geometric type (see for example [Sai20a, Lemma 6.1]), there exists a regular sequence t 1 . . . t n ∈ R such that R ∼ = κ{t 1 . . . t n }, the henselization of the local ring of A n κ at (0), and p = (t r+1 , . . . t n ), hence R/p ∼ = κ{t 1 . . . t r }.
Theorem 4.4. Let X ∈ SmlSm(k) such that X is an henselian local scheme. Then the map is injective.
If dim(X) = 1 and n = 0. Then (4.4.1) is injective by Corollary 4.2 (i). Assume then that dim(X) = 1 but n > 0. Then ∂X is supported on the closed point x (note that ∂X is automatically irreducible, since X is 1-dimensional and local). By Lemma 4.3, the map Spec(k(x)) → X has a retraction, hence X ∈ SmlSm(k(x)) and |∂X| is supported on a k(x)-rational point.
Suppose now that dim(X) > 1 and n = 0. Then again (4.4.1) is injective by Corollary 4.2 (i). We now pass to the case dim(X) > 1 and n ≥ 1. For every 1 ≤ r ≤ n, let η Dr ∈ X be the generic point of D r and ι Dr : D r → X the inclusion. For Y ∈ SmlSm(k), we write c(Y ) for the number of irreducible components of the strict normal crossing divisor ∂Y .
We make the following Claim: Claim 4.5. Assume the induction hypothesis above, i.e. suppose that Theorem 4.4 holds for every Y ∈ SmlSm(k) local henselian such that dim(Y ) ≤ n − 1 and c(Y ) ≥ 0 and with dim(Y ) = dim(X) and c(Y ) ≤ n − 1. Then, for every U ⊆ X dense open such that U ∩D n ⊆ D n is dense, the restriction map We postpone the proof of Claim 4.5 and complete the proof of the Theorem. Since filtered colimits are exact in the category of Λ-modules, we get from Claim 4.5 an injective map: Let O X,η Dn be the local ring of X at η Dn : it is a discrete valuation ring with generic point η X and infinite residue field k(η Dn ). Since O X,ηDn is the localization of a henselian k-algebra at a prime ideal generated by a regular sequence, we can apply Lemma 4.3 to get a map Spec(O X,η Dn ) → Spec(k(η Dn )) that splits η Dn → Spec(O X,η Dn ), hence (Spec(O X,η Dn ), ι * Dn ∂X) ∈ SmlSm(k(η Dn )) and |ι * Dn ∂X| is a k(η Dn )-rational point. Let λ : Spec(k(η Dn )) → Spec(k). We argue as above: since C is (sNis, )-fibrant in Cpx(PSh log (k, Λ)), λ * C is (sNis, )-fibrant in Cpx(PSh log (k(η Dn ), Λ)) (see again Remark 2.11), hence by Corollary 4.2 (ii) we have an injective map: ֒→ H i (λ * C(η X , triv)) = H i (C(η X , triv)).
Combining (4.5.1) with (4.5.2), we get the desired injectivity. This reduces the proof of Theorem 4.4 to the proof of Claim 4.5.
Proof of Claim 4.5. Let X − := (X, ∂X − ) ∈ SmlSm(k), where ∂X − is the strict normal crossing divisor D 1 + . . . + D n−1 . Since c(X − ) = n − 1, by hypothesis (this is the induction assumption on the number of components of ∂X), the map H i C(X − ) → H i C(η X , triv) is injective.
Let U be an open dense subset of X such that U ∩D n is dense in D n and U ∩D i = ∅ if i = n, and set U := (U, ∂X |U ). Write U − := (U, ∂X − |U ) = (U, triv). Hence we have a commutative diagram: (4.5.3) (2) (3) where (1), (2) and (3) are injective since they all factor the injective map H i C(X − ) → H i C(η X , triv). Since X is Henselian local of dimension r ≥ n with closed point x, there exists an isomorphism ε : X ∼ = Spec(k(x){t 1 , . . . , t r }). Without loss of generality, we can assume that t r is a local parameter for D n , so that ε induces an isomorphism D n ∼ = Spec(k(x){t 1 , . . . , t r−1 }). Hence the map henselization at 0 induces a pro-Nisnevich square 2 of (usual) schemes: (4.5.4) p By Lemma 2.6, the square (4.5.5) is a filtered colimit of homotopy pullbacks, hence it is itself a homotopy pullback. (4.5.6) Hence up to refining U we can suppose that U itself fits in a pro-Zariski square like (4.5.6), so again using Lemma 2.6 and the fact that a filtered colimit of homotopy pullbacks is itself a homotopy pullback, we get the following homotopy pullback square: (4.5.7) We conclude that for C sNis-fibrant the squares (4.5.5) and (4.5.7) induce the following equivalences: where the last isomorphisms come from the definition of the motivic Thom space [BPØ,Def. 7.4.3], the fact that X is local and U ⊆ X is an open immersion, hence N Dn/X − ∼ = D n × A 1 and N Dn∩U/U − ∼ = (D n ∩ U) × A 1 . Here, Hom • (K, C) ∈ D(Λ) for K ∈ Cpx(PSh log (k, Λ)) is the mapping complex. In particular, we get for every i ∈ Z the following commutative diagram: where the top horizontal sequence is exact and the bottom horizontal sequence is exact in the middle. We will now show that for every i, the natural map is injective, where Z = X − U : assuming this, by diagram chase in (4.5.8) we finally conclude that the map H i (C(X)) ֒→ H i (C(U)) is injective for every U as above. We can use [BPØ, Proposition 7.4.5] (note that the condition that C is (sNis, )fibrant is enough) to compute the motivic Thom spaces: we get a commutative diagram where the rows are split exact sequences (4.5.9) We have that H i C(− × P 1 ) = H i (Hom((P 1 , triv), C))(−) and Hom((P 1 , triv), C) is (sNis, )-fibrant since C is (see Lemma 2.10). By induction on dimension we conclude that the middle vertical map of (4.5.9) is injective, and since the rows in (4.5.9) are split-exact sequences, the right vertical map is a retract of the middle one, hence it is injective. This concludes the proof.
Corollary 4.6. Let X ∈ SmlSm(k) and let τ be either sNis or dNis. Then the following map is injective: where η X is the generic point of X.
Proof. The case where τ = dNis follows from the case of sNis. Indeed, since filtered colimits are exact in the category of Λ-modules, and since for all Y ∈ X div , the map Y → X is birational, so that η Y = η X , we get Thus, from now on let τ = sNis. For all x ∈ X, let X h x be the henselization of X at x with log structure induced by the log structure of X, and let η(X h x ) be its fraction field, which is a field extension of η X . We have a diagram The map ( * 1) is injective by the sheaf condition, the map ( * 2) is injective by Theorem 4.4 and the fact that injective morphisms are stable under arbitrary products in Λ-modules. Hence the map ( * 3) is injective, which concludes the proof.

The homotopy t-structure on logarithmic motives
The goal of this section is to generalize to the logarithmic setting the results of Morel on the existence of the homotopy t-structure on the category of motives. Having the connectivity theorem 3.2 at disposal, the proofs are fairly straightforward.
Recall that the triangulated categories  [BPØ,Prop. 4.7.5] (with transfers), which hold for the dNis-topology but not for the strict Nisnevich topology. We write τ ≥n and τ ≤n for the (homologically graded) truncation functors on D dNis (PSh(lSm(k), Λ)) and τ tr ≥n and τ tr ≤n for the (homologically graded) truncation functors on D dNis (PSh ltr (lSm(k), Λ)). In view of (5.0.1) and (5.0.2), we will work with the category of sheaves on SmlSm(k) without further notice, and simply write Shv log dNis (k, Λ) (resp. Shv ltr dNis (k, Λ)) for the abelian category of sheaves (resp. of sheaves with log transfers). The proof of the following theorem is formally identical to [Ayo21,Thm. 4.15].
Theorem 5.1. Let C ∈ D dNis (PSh(SmlSm(k), Λ)), and suppose that C is -local (see Definition 2.8). Then for all n ∈ Z, the truncated complexes τ ≥n C and τ ≤n C are -local.
Proof. Up to shifting, we can clearly assume that n = 0, and by the standard properties of the t-structure, it is enough to show the statement for τ ≥0 C. Since C is -local, the natural map τ ≥0 C → C factors through L(τ ≥0 C) as where L(τ ≥0 C) is any (dNis, )-fibrant replacement. We have by Theorem 3.2 that L(τ ≥0 C) is locally −1-connected, so the map ℓ factors as By the universal property of τ ≥0 we get that ℓ 0 e o = id τ ≥0 C . Hence, τ ≥0 C is a direct summand of L(τ ≥0 C), so it is -local as required.
Corollary 5.2. Let C ∈ D dNis (PSh ltr (SmlSm(k), Λ)), and suppose that C islocal. Then for all n ∈ Z, the truncated complexes τ tr ≥n C and τ tr ≤n C are -local. Proof. As in the proof of Theorem 5.1, it is enough to prove the statement for τ ≥0 C. Recall that the graph functor γ : SmlSm(k) → SmlCor(k), which sends a map X → Y to the finite correspondence X γ(f ) − − → Y induced by its graph, is faithful: the category SmlCor is, by definition, the full subcategory of lCor(k) consisting of all objects in SmlSm(k) (it is denoted lCor SmlSm /k in [BPØ]). Presheaves with log transfers on SmlSm(k) are, by definition, presheaves (of Λ-modules) on SmlCor(k).
It is immediate so see that γ * is t-exact, conservative and preserves flasque sheaves, hence and for all X ∈ SmlSm(k) and F ∈ D dNis (PSh ltr (SmlSm(k), Λ)), we have RΓ(X, γ * F ) = RΓ(X, F ) In particular F is -local if and only if γ * F is. To prove the Corollary, it is then enough to show that γ * (τ tr ≥0 C) is -local. But since γ * is t-exact, we have γ * (τ tr ≥0 C) = τ ≥0 γ * C, which is -local by Theorem 5.1.
Analogously to [Sai20b], we denote by CI log dNis (resp. CI ltr dNis ) the full subcategory of Shv log dNis (k, Λ) (resp. Shv ltr dNis (k, Λ)) of strictly -invariant sheaves. Remark 5.4. Note that the above definition is slightly non-standard: in the context of reciprocity sheaves we typically write CI Nis for the category of -invariant Nisnevich sheaves, without "strictness" condition, i.e. without asking the property that the cohomology presheaves are -invariant. If F ∈ CI Nis is moreover semipure in the sense of [Sai20a, Def. 1.28], the fact that the cohomology presheaves are -invariant (at least when restricted to the subcategory MCor ls defined in loc.cit.) is indeed a difficult result due to S. Saito, [Sai20a,Thm. 9.3]. In the A 1 -invariant context, the analogous statement is due to Voevodsky [MVW06,§24].
The heart of this t-structre is naturally equivalent to CI log dNis (resp. CI ltr dNis ), which is then a Grothendieck abelian category.
Proof. The first assertion follows directly from Theorem 5.1 (resp. Corollary 5.2), the second from Corollary 5.5 and Proposition 5.6. The fact that the heart of a t-structure is abelian is well-known [BBD82].
Next, note that the homotopy t-structure is clearly accessible in the sense of [Lur17, Definition 1.4.4.12].
Moreover, filtered colimits commute with cohomology, hence if {F α } is a filtered system of (dNis, ) fibrant objects, then lim − → F α is (dNis, ) fibrant since it is dNisfibrant (as observed in the proof of Theorem 2.14) and So if H i F α = 0 for i ≥ 0 and all α, then Hence the t-structure is compatible with colimits in the sense of [Lur17, 1.3.5.20].
Proof. We prove the assertion for CI log dNis , since the statement for CI ltr dNis is identical. First, note that if the right adjoint h 0 exists, then for F ∈ Shv dNis and X ∈ SmlSm(k), we have ih 0 (F )(X) = Hom Shv dNis (a dNis Λ(X), ih 0 F ) = Hom CI log dNis (h 0 (a dNis Λ(X)), h 0 F ) = Hom Shv dNis (ih 0 (a dNis Λ(X)), F ). as required. Hence we only have to prove that h 0 exists.
By the Special Adjoint Functor Theorem (see [Mac71,p. 130]), a functor between two Grothendieck abelian categories has a right adjoint if and only if it preserves all (small) colimits, so we need to show that this holds for i : CI log dNis → Shv log dNis (k, Λ), i.e. that CI log is closed under small colimits in Shv log dNis (k, Λ). As it was observed in the proof of Theorem 5.7, CI log is stable under filtered colimits. Since (small) colimits are filtered colimits of finite colimits, it is enough to show that CI log is stable under finite limits. Since it is an abelian subcategory, it is enough to show that it is stable under cokernels.
Let F, G ∈ CI log dNis and let F → G be a map in Shv dNis . Then we have that where F dNis and G dNis denote the dNis-fibrant replacements. Since F and G are strictly -local, F dNis and G dNis are (dNis, )-fibrant, hence Cofib(F dNis → G dNis ) is also (dNis, )-fibrant.
Note that every X ∈ SmlSm(k), we have an isomorphism Hom(Λ(X), G dNis ) ∼ = Hom(Λ(X × ), G dNis ), since by adjunction we have From this it easily follows that Hom(F, G dNis ) is -local, hence we conclude by Theorem 5.7 Theorem 5.10. Let F ∈ CI log dNis (resp. F ∈ CI ltr dNis ). Then for all X ∈ SmlSm(k) and U ⊆ X an open dense, the restriction F (X) → F (U) is injective.
Proof. As before, we give a proof for the version without transfers. Let F [0] → G be a dNis-fibrant replacement. Since F [0] is -local, G is (dNis, )-fibrant. Since F = a dNis H 0 G, the result follows from Theorem 4.6. where ω * denotes as usual the restriction functor, ω ♯ its left Kan extension and ω * the right Kan extension. Since λ is left adjoint to ω, we have λ * = ω ♯ . By construction, ω * and ω ♯ are t-exact for the global t-structures.
The adjunction (ω ♯ , ω * ) is a Quillen adjunction with respect to the dNis-local model structure on the left hand side and the Nis-local model structure on the right hand side, see [BPØ,4.3.4], and with respect to the (dNis, )-local model structure on the left hand side and the (Nis, A 1 )-local model structure on the right side, see [BPØ,4.3.5] and induces therefore the following derived adjunctions: (5.10.2) Lω ♯ : D dNis (Cpx(PSh ltr (k, Λ))) D Nis (Cpx(PSh tr (k, Λ))) : Rω * .
Similar adjunctions hold for the categories without transfers.
Proposition 5.12. The functor R ω * is t-exact with respect to Voevodsky's homotopy t-structure on DM eff and to the homotopy t-structure on logDM eff of Theorem 5.7.
To conclude, we need to show that R ω * • τ DM ≤n ∼ = τ logDM ≤n • R ω * . But since Rω * is t-exact and (5.12.2) commutes, we have The same argument applies to the truncation τ ≥n , so that we can conclude.
Remark 5.13. Assume that k satisfies resolution of singularities. Then the functor R ω * is fully faithful, and its essential image is identified with the subcategory of A 1local objects in logDM eff by [BPØ,Thm. 8.2.16]. It follows from Proposition 5.12 that under R ω * , the homotopy t-structure on DM eff is induced by the homotopy t-structure on logDM eff .
Corollary 5.14. The functor L ω ♯ is right t-exact.
Proof. This follows immediately from the fact that its right adjoint is t-exact (in particular, left t-exact).

Application to reciprocity sheaves
In this section, we discuss some applications to the theory of reciprocity sheaves. As above, for X ∈ SmlSm(k), let |∂X| be the strict normal crossing divisor supporting the log structure of X. We will call the modulus pair (X, |∂X| red ) the associated reduced modulus pair. We remark that the assignment X → (X, |∂X| red ) does not give rise to a functor from SmlSm(k) to MCor, since a priori there is no control on the multiplicities of the divisor ∂X in the pullback along a morphism in SmlSm(k).
However, thanks to [Sai20b] there exists a functor Log : RSC Nis (k) → Shv ltr dNis (k, Z) where for X = (X, ∂X) ∈ SmlSm(k) we have Here, ω CI : RSC Nis → CI Nis is the functor defined in [ It follows that for every set A, the map is a quasi-isomorphism. Thus A Log(G a ) → A I • is a sZar-local equivalence, hence a sNis-local equivalence, so A I • is an injective resolution of A Log(G a ).
We conclude that In particular, A Log(G a ) is strictly invariant. On the other hand, by [KSY16, Remark 6.1.2], if A is infinite A G a does not belong to RSC Nis .
Using the above-defined functors, we can compute the sections of Log(F ) on X ∈ SmlSm(k) for F ∈ RSC Nis as follows. Write X = (X, ∂X) and X o = X − |∂X|. Choose a normal compactification j : X ֒→ Y with the property that X o → X → Y is open dense, and such that the complement Y − X o = D + ∂X Y for some effective Cartier divisors D and ∂X Y on Y satisfying Y − |D| = j(X) and ∂X Y ∩ X = ∂X as reduced Cartier divisors. Such a compactification is called a Cartier compactification of X, and it always exists (cfr. [KMSY21a, Def.1.7.3]). Then we have Since CI is closed under colimits, and h 0 and i are left adjoints, we conclude i h 0 colim MPST F i = colim MPST i h 0 F i , so that the colimit is in RSC, as required.
Remark 6.5. For X, Y ∈ SmlSm(k), we have by e.g. [Ogu18, III.2] where the divisors D X and D Y support M X and M Y respectively, and the functor (−) fs does not change the support. We conclude that the associated reduced modulus pair of X × Y is X ⊗ Y.
Lemma 6.6. Log has a pro-left adjoint Rsc, given by the formula for every G ∈ Shv ltr dNis (k, Z). Proof. It follows directly from Saito's theorem [Sai20b, Thm. 6.3] that Log preserves finite limits, so the existence of a pro-left adjoint is formal (see e.g. [AGV72a,I.8.11.4]). In the rest of the proof we characterize the pro-adjoint explicitly: such description will be used later in the computation. Let F ∈ RSC Nis and G ∈ Shv ltr dNis (k, Z). For any X ∈ SmlSm(k), let (X, D) be the associated reduced modulus pair and choose X a Cartier compactification of X. Set D ′ := X ′ \ X.
Remark 6.10. If C is a category equipped with a monoidal structure ⊗ (in particular, associative), then the category pro-C is equipped with the level-wise monoidal struc- [FI07,11]. Since the construction (6.7.1) gives a monoidal structure on RSC Nis only in a weak sense (in particular, associativity is not known to hold), we need to verify explicitly that the level-wise assignment 6.8 is indeed well defined. Note that the argument is ad hoc, and only proves the existence of a bi-functor at the level of pro-categories.
The functoriality statement of the previous Proposition implies in particular that if (F i ) i∈I and (G j ) j∈J are diagrams in pro-RSC Nis , then there is a natural map (6.10.1) In general, there is no reason to expect that (6.10.1) is an isomorphism (see also [FI07,Ex. 11.2] for a similar problem). Using the explicit description of the pro-left adjoint to Log, we get then the following result.
Theorem 6.11. For F, G ∈ CI ltr dNis , there exists a natural map Rsc(F ⊗ ltr G) → (Rsc(F ), Rsc(G)) pro RSC . Proof. The tensor product in Shv ltr dNis (k) is given by Day convolution from the monoidal structure on SmlSm(k). So, if F = colim X↓F a dNis Z tr (X) and G = colim Y ↓G a dNis Z tr (Y ). Then Using the explicit description of the functor Rsc given in the proof of Lemma 6.6, we get

Consider now the natural maps
they give a natural map ) RSC By definition the last term is equal to

RSC
Hence we obtained a natural map

RSC
Finally, as observed in (6.10.1) there is a natural map RSC and the last term is equal to (Rsc(F ), Rsc(G)) pro RSC . Corollary 6.12. Let F, G ∈ RSC, then there exists a natural map  Finally, since Log((F, G) RSC ) ∈ CI ltr dNis , the previous map factors through the localization h 0 (Log(F ) ⊗ ltr Log(G)) = Log(F ) ⊗ CI ltr dNis Log(G), giving the desired map.

Log reciprocity sheaves
In this final section, we state a conjecture that will allow us to construct full subcategories LogRec and LogRec tr respectively of Shv Nis (k, Λ) and Shv tr Nis (k, Λ) such that RSC Nis ⊆ LogRec tr .
Our definition generalizes the construction of [KSY] and it is very similar in spirit. Proof. Exactness follows from the exactness of i tr and ω tr ♯ (resp. i tr and ω tr ♯ ). To show faithfulness, it is enough to show that for all F ∈ CI log dNis (resp. CI ltr dNis ), the unit map F → ω CI log ω log ♯ F (resp.F → ω CI ltr ω ltr ♯ F ) is injective. By [BM21, Theorem 5.10] we have that for all X ∈ SmlSm(k), Because F is -local, the map u (resp. u tr ) factors through ω CI log ω log ♯ F (resp. ω CI ltr ω ltr ♯ F ), which concludes the proof.
We now state a conjecture, that we hope to prove soon: Conjecture 7.3. The functors ω log CI and ω ltr CI are full. Definition 7.4. Assume Conjecture 7.3 Let LogRec (resp. LogRec tr ) denote the essential image of ω log CI (resp. ω ltr CI ), i.e. categories of sheaves F ∈ Shv Nis (resp. Shv tr Nis ) such that there exists G ∈ CI log dNis (resp. CI ltr dNis ) such that F = ω CI G. By definition, ω log CI (resp. ω ltr CI ) induces an equivalence between CI log dNis (resp. CI ltr dNis ) and LogRec (resp. LogRec tr ) with quasi-inverse the restriction of ω CI . Remark 7.5. Assume Conjecture 7.3. Let F ∈ LogRec and let G ∈ CI ltr dNis such that F = ω ♯ G. We deduce some immediate properties: (1) For all X ∈ Sm and U ⊆ X dense open, Theorem 5.10 implies that F (X) ֒→ F (U) is injective.
Theorem 7.6. Assume Conjecture 7.3. The category RSC Nis is a full subcategory of LogRec. In particular, (7.6.1) Log = ω CI log i RSC Proof. Since RSC Nis is a full subcategory of Shv Nis (k, Λ), it is enough to show that for every F ∈ RSC Nis there exists G ∈ CI log dNis such that F = ω ♯ G. By [Sai20b, Section 4] we have that (7.6.2) ω ♯ Log(F )(X) = ω CI F (X, ∅) = F (X) Hence RSC Nis is a full subcategory of LogRec. Finally, since ω ♯ is an equivalence, (7.6.1) follows directly from (7.6.2).
Then the cohomology of F satisfies for every n ≥ 0 and Y henselian local essentially smooth k-scheme with generic point η Y .
Proof. It follows immediately from Theorem 7.6 and Remark 7.5.
Let i RSC (resp. i log RSC ) denote the inclusion of RSC Nis in Shv tr Nis (k) (resp. in LogRec). Recall by [KSY] that i RSC has a pro-left adjoint ρ such that for X ∈ Sm(k) and X a Cartier compactification with D = X − X, then ρ(Z tr (X)) = "lim"ω ! h 0 (X, nD).
Proposition 7.8. Assume Conjecture 7.3. The functor i log RSC has a pro-left adjoint ρ log , which factors ρ. In particular, Rsc = ρ log ω log CI Proof. Since i RSC = i LogRec i log RSC and i LogRec is fully faithful, for F ∈ Shv tr Nis G ∈ RSC Nis we have that Hom pro-RSC (ρi LogRec F, G) = Hom Shv tr Nis (i LogRec F, i LogRec i log RSC G) = Hom Shv tr Nis (i LogRec F, i LogRec i log RSC G) = Hom LogRec (F, i log RSC G). Finally, for F ∈ CI ltr dNis and G ∈ RSC Nis , we have that Hom pro-RSC (Rsc(F ), G) = Hom CI ltr dNis (F, Log(G)) = Hom CI ltr dNis (F, ω CI log i log RSC G) = Hom Shv tr Nis (i LogRec ω log CI F, i RSC G) = Hom pro-RSC (ρ log ω CI F, G) Remark 7.9. Assume Conjecture 7.3. Since CI ltr dNis is a symmetric monoidal Grothendieck abelian category, then LogRec is symmetric monoidal with tensor product given by F ⊗ LogRec G := ω ♯ (h 0 (ω CI log F ⊗ ltr ω CI log G)). By 6.12, for all F, G ∈ RSC Nis we have a map F ⊗ LogRec G → (F, G) RSC If ch(k) = 0, this map is not an isomorphism (see below). We do not know whether we expect it to be an isomotphism when ch(k) = 0: this would prove that ( , ) RSC Nis defines a monoidal structure on RSC Nis . 7.10. Let F, G ∈ RSC Nis and let F ′ ⊆ ω CI F such that ω ! F ′ = F (in the language of [RS], F ′ corresponds to a semi-continuous conductor of F different from the motivic conductor). By construction, there exists a canonical map (7.10.1) ω ! (F ′ ⊗ Nis CI ω CI G) → ω ! (ω CI F ⊗ Nis CI ω CI G) = (F, G) RSC This map is surjective: let Q be the cokernel of the inclusion F ′ → ω CI F such that ω ! Q = 0. Hence, since ⊗ CI ω CI G is right exact, there is a right exact sequence F ′ ⊗ Nis CI ω CI G → ω CI F ⊗ Nis CI ω CI G → Q ⊗ Nis CI ω CI G → 0 and since Q⊗ Nis CI ω CI G is a quotient of Q⊗ MNST ω CI G and ω ! is exact and monoidal in MNST, we conclude ω ! (Q ⊗ CI ω CI G) = 0, which shows the surjectivity of (7.10.1).
The kernel of (7.10.1) incapsulates the obstruction to the associativity of ( , ) RSC , and it seems to be very difficult to compute in general. We know that it is not trivial if ch(k) = 0: see [RYS,Theorem 4.17] and [RYS,Theorem 5.19] for an explicit computation.
On the other hand, we do not have any counterexamples if ch(k) = 0, hence we do not know whether to expect that the map above is an isomorphism. In this direction, we have the following result: Proposition 7.11. Assume Conjecture 7.3. Let F, G ∈ RSC Nis . Then for all F ′ ⊆ ω CI F (in MNST) such that ω ! F ′ = F , the canonical map F ⊗ LogRec G → (F, G) RSC factors through ω ! (F ′ ⊗ Nis CI ω CI G). Proof. Let (−) log be the functor of [Sai20b] and recall that Log(F ) = (ω CI F ) log . Since Log(F ) = (F ′ ) log by construction, we can look at the diagram that makes the diagram above commutative. By adjunction, it is enough to construct a map (F ′ ) log → Hom Shv ltr dNis (Log(G), (ω CI F ⊗ Nis,sp CI ω CI G) log ).
Remark 7.12. For F ∈ Shv Nis , we denote by h 0 A 1 (F ) the biggest A 1 -local subsheaf as defined in [RS,4.34]: for U ∈ Sm, h 0 A 1 (F )(U) := Hom(h A 1 0 (U), F ). On the other hand, for U ֒→ X a Cartier compactification such that X is proper and smooth over k and X − U is a simple normal crossing divisor, then for X = (X, ∂X) ∈ SmlSm(k) such that ∂X is supported on X − U, by [BPØ,Proposition 8.2.4] we have that h A 1 0 (X) = ω ♯ h 0 (X) Hence if F ∈ LogRec, then h 0 A 1 (F )(U) = Hom(ω ♯ h 0 (X), F ) = ω CI log F (X). Here we underline that this does not depend on X, as long as X is proper.
We conclude with this observation: for X as above and X ∈ MCor the associated reduced modulus pair, by [RS,Corollary 4.36] if F ∈ RSC Nis , we have that Hom(ω ! h ,sp 0 (X ), F ) = h 0 A 1 F = Hom(ω log CI h 0 (X), F ) This implies that ω ! h ,sp 0 (X ) ∼ = ω log CI h 0 (X) In particular, by [Sai20b, Corollary 2.6 (3)], we have that Log(ω ! h ,sp 0 (X )) = h ,sp 0 (X ) log hence, by the fact that ω log CI is an equivalence on LogRec, we have that h ,sp 0 (X ) log ∼ = h 0 (X) ∼ = ω * h A 1 0 (U) Again, we stress that these isomorphisms do not depend on X nor X , as long as X is proper.