Motivic and $\ell$-adic realizations of the category of singularities of the zero locus of a global section of a vector bundle

We study the motivic and $\ell$-adic realizations of the dg category of singularities of the zero locus of a global section of a line bundle over a regular scheme. We will then use the formula obtained in this way together with a theorem due to D.~Orlov and J.~Burke - M.~Walker to give a formula for the $\ell$-adic realization of the dg category of singularities of the special fiber of a scheme over a regular local ring of dimension $n$.


Introduction
The connection between categories of singularities and vanishing cycles is well known, thanks to works of T. Dyckerhoff ([23]), A. Preygel ([45]), A. Efimov ([24]) and many others. Recently, an instance of this fact has been studied in [9].
We start by quickly reviewing the main theorem of [9], that served both as a model and as a motivation for the investigations presented later. The main purpose of the above mentioned paper is to identify a classical object of singularity theory, namely the ℓ-adic sheaf of (inertia invariant) vanishing cycles, with the ℓ-adic cohomology of a non-commutative space (that is defined in loc. cit.), the dg category of singularities of the special fiber. The main result of A. Blanc, M. Robalo, B. Toën and G. Vezzosi's paper ( [9,Theorem 4.39]) reads as follows: Let p : X → S be a proper, flat, regular scheme over an excellent strictly henselian trait S. Let i σ : σ ֒→ S be the embedding of the closed point in S and let p σ : X σ → σ the pullback of p along i σ . Fix a prime number ℓ different from the characteristic of σ. Let Φ p (Q ℓ,X (β)) be the ℓ-adic sheaf of vanishing cycles associated to Q ℓ,X (β) = n∈Z Q ℓ,X (n) [2n]. Denote by I the inertia group (as S is strictly henselian, it coincides with the absolute Galois group of the open point in S) and by (−) hI the (homotopy) fixed points ∞-functor. In [9], the authors define the ℓ-adic realization of dg categories, denoted by R ℓ,∨ S . It is an ∞-functor that associates an ℓ-adic complex to a dg category. For every (derived) scheme Z, let Sing(Z) be the dg category of singularities of Z, that is the dg quotient 1 where Coh b (Z) denotes the dg-category of complexes of quasi coherent O Zmodules with coherent total cohomology and Perf (Z) that of perfect complexes of O Z -modules.

NON COMMUTATIVE SIDE VANISHING CYCLES SIDE
We can ask whether the above diagram makes sense in more general situations. For example, one can start with the datum of a proper, flat and regular scheme over an excellent local regular ring of dimension n. One recovers the case treated in [9] when n = 1.
It is immediate to observe that the left hand side of the diagram makes sense without any change: assume that we are given a proper, flat morphism p : X → S, with X regular and S local, regular of dimension n. Let π = (π 1 , . . . , π n ) be a collection of generators of the closed point of S. Then we can consider the morphism π • p : X → A n S and its fiber X 0 along the origin S → A n S . It makes perfectly sense to consider R ℓ,∨ S (Sing(X 0 )). It comes up that this generalization is related to the following one: one can consider pairs (X, s X ) where X is a regular scheme over S and s X : X → V(L S ) is a morphism towards the total space of a line bundle L S over S. One recovers the situation pictured above when L S = O S is the trivial line bundle. In this situation, we want to compute R ℓ,∨ S (Sing(X 0 )), where X 0 is the fiber of s X : X → V(L S ) along the zero section S → V(L S ).
One can view the former generalization as a particular case of the latter thanks to a theorem of D. Orlov ([42]) and J. Burke-M. Walker ( [11]), which tells us that the dg category of singularities of (X, π • p) is equivalent to the dg category of singularities of (P n−1 X , (π 1 • p) · T 1 + · · · + (π n • p) · T n ∈ O(1)(P n−1 X )). Thus, we only need to find the appropriate generalization of (inertia invariant) vanishing cycles. The first thing we can think of are vanishing cycles over general bases, developed by G. Laumon ([33]) following ideas of P. Deligne and further investigated by many others, including O. Gabber, L.Illusie, F. Orgogozo (see [29], [41]). However, it seems that this is not the right point of view for our purposes. Instead, we will pursue the following analogy: As we will only need to define the analogous of inertia-invariant vanishing cycles, we will not face the problem of filling the empty spot in the mental map above. Nevertheless, we will come back to this matter at the end of the article, presenting a strategy to complete the picture. We will define an appropriate generalization of Φ p (Q ℓ,X (β)) hI (see Definition 4.2.3) and prove a generalization of the formula stated in [9,Theorem 4.39]. Our main theorem will then look as follows: Theorem. 5.2.2 Let X be a regular scheme and let s X be a regular global section of a line bundle L X . Denote X 0 the zero locus of s X . Then R ℓ,∨ X0 (Sing(X 0 )) ≃ Φ mi (X,sX ) (Q ℓ (β))[−1].
Here Φ mi (X,sX ) (Q ℓ (β)) is what we call the monodromy-invariant vanishing cycles ℓ-adic sheaf (see Definition 4.2.6). It coincides with inertia invariant vanishing cycles when we put ourselves in the situation considered in [9].
Using this theorem combined with the above mentioned result of D. Orlov and J. Burke -M. Walker (Theorem 6.1.16), we will deduce the following formula for the situation in which one starts with a regular scheme X and a regular section s X of a vector bundle.
Theorem. 6.2.1 Let f : X → S be a flat morphism, E S be a vector bundle of rank r over S and let s X be a global section of E X = f * E S . Assume that X is a regular scheme and that s X is regular section. The following equivalence holds in Mod R ℓ,∨ X (Sing(X,0)) (Shv Q ℓ (X)) R ℓ,∨ X (Sing(X, s X )) ≃ p * i * Φ mi (P(EX ),Ws X ) (Q ℓ (β))[−1], where W sX is the global section of O P(EX ) (1) associated to s X , i : V (W sX ) → P(E X ) is the closed embedding of the zero locus of W sX in P(E X ) and p : P(E X ) → X is the canonical projection.
In particular, the above theorem applies to the special case E S ≃ O n S and allows us to compute the ℓ-adic realisation of the special fiber of a regular scheme over a local regular noetherian ring of dimension n. More precisely: Corollary. 6.2.2 Assume that S = Spec(A) is a noetherian regular local ring of dimension n and let π 1 , . . . , π n be generators of the maximal ideal. Let p : X → S = Spec(A) be a regular, flat S-scheme of finite type. Let π : S → A n S be the closed embedding associated to π 1 , . . . , π n . Then π • p is a regular global section of O n X . Then the equivalence R ℓ,∨ X (Sing(X, π • p)) ≃ q * i * Φ mi (P n−1 X ,Wπ•p) (Q ℓ (β))[−1] holds in Mod R ℓ,∨ X (Sing(X,0)) (Shv Q ℓ (X)). Here q : P n−1 X = P roj X (O X [T 1 , . . . , T n ]) → X is the canonical projection and i : V (W π•p ) → P n−1 X is the closed embedding determined by the equation W π•p = p * (π 1 ) · T 1 + · · · + p * (π n ) · T n = 0.
We will end this article with some remarks on the following two problems: 1. It seems possible to define a formalism of vanishing cycles in twisted situations, i.e. in the situation in which we have a morphism s X : X → V(L X ) for L X ∈ Pic(X). This is all about completing the empty slot in the mind map above. One should then be able to find Φ mi (X,sX ) (Q ℓ (β)) via a procedure that corresponds to taking homotopy fixed points in the usual situation. A complete account on this formalism will appear in a forthcoming paper in collaboration with D.-C. Cisinski.
2. We will comment the regularity hypothesis that appears both in A. Blanc -M. Robalo -B. Toën -G. Vezzosi's theorem and in the generalization we provide.
Remark. The main body of this article corresponds to [44, Chapter §3], while the "preliminaries" section corresponds Chapter §1 in loc. cit. The only difference lies in section §6, where we have considered a more general setting in the present text.
Remark. Even if the theorems are stated in the category Shv Q ℓ (Z) of ℓ-adic sheaves, they are nevertheless true (and the proves we provide work mutatis mutandis) as BU Z,Q -modules (see section 2.7).

Preliminaries and notation
In this preliminary section we will briefly recall some of the mathematical tools that we will need later. We will also fix the notation that we will use in the other sections.

Some notation and convention
• Even when not explicitly stated, S will always be a regular scheme of finite type over a strictly local noetherian scheme.
• Sm S denotes the category of smooth schemes of finite type over S.
• Sch S denotes the category of separated schemes of finite type over S.
• We will freely use the language of ∞-categories which has been developed in [35], [34]. ∞-category will always mean (∞, 1)-category for us.
• S denotes the ∞-category of spaces.
• We write dg instead of "differential graded".
• We will use cohomological notations. In particular, the differential of a complex increases the degree.
• If we are given a morphism of (derived) schemes f : X → Y and an object E Y ∈ QCoh(Y ), we will write E X instead of f * E Y .

Reminders on dg categories
Remark 2.2.1. For more details on the theory of dg categories, we invite the reader to consult [32], [60] and/or [48].
Let A be a commutative ring. Let dgCat S be the category of small A-linear dg categories together with Alinear dg functors. This category can be endowed with a cofibrantly generated model category structure, where weak equivalences are DK equivalences (see [56]). The underlying ∞-category of this model category coincides with the ∞-localization of dgCat S with respect to the class of DK equivalences. We will denote this ∞-category by dgCat S . Every DK equivalence is a Morita equivalence. We can therefore endow dgCat S with a second cofibrantly generated model category structure by using the theory of Bousfield localizations. In this case weak equivalences are Morita equivalences. Similarly to the previous case, the underlying ∞-category of this model category coincides with the ∞-localization of dgCat S with respect to Morita equivalences. We will label this ∞-category by dgCat idm S . LetĈ c denote the dg category of perfect C op -dg modules. Then dgCat idm S is equivalent to the full subcategory of dgCat S spanned by dg categories C for which the Yoneda embedding C ֒→Ĉ c is a DK equivalence.
To summarize, we have the following pair of composable ∞-localizations is a left adjoint to the inclusion dgCat idm S ֒→ dgCat S under the identification mentioned above. At the level of objects, it is defined by the assignment T →T c .
It is possible to enhance dgCat S and dgCat idm S with symmetric monoidal structures. Furthermore, if we restrict to the full subcategory dgCat lf S ⊆ dgCat S of locally flat (small) dg categories, we get two composable symmetric monoidal ∞-functors More details on the Morita theory of dg categories can be found in [59]. One of the most recurrent operations that occur in this work is that of forming quotients of dg categories: given a dg category C together with a full sub dg category C ′ , both of them in dgCat idm S , we can consider the pushout C ∐ C ′ 0 in dgCat idm S . Here 0 denotes the final object in dgCat idm S , i.e. the dg category with only one object whose endomorphisms are given by the zero homcomplex. We denote this pushout by C/C ′ and refer to it as the dg quotient of C ′ ֒→ C. Equivalently, The dg category C/C ′ can also be obtained as the image in dgCat idm S of the pushout C ∐ C ′ 0 formed in dgCat S . Its homotopy category coincides with (the idempotent completion of) the Verdier quotient of H 0 (C) by the full subcategory H 0 (C ′ ) (see [22]).
Compact objects in dgCat idm S are dg categories of finite type over A, as defined in [64]. In particular, Moreover, dgCat idm S is equivalent to the ∞-category of small, idempotent complete, A-linear stable ∞-categories ( [19]).
When S is a non affine scheme, the symmetric monoidal ∞-category of dgcateogories over S is defined as the limit Here there are some dg categories that we will use: let X be a derived scheme (stack).
• QCoh(X) will denote the dg category of quasi-coherent O X -modules. It can be defined as follows. If X = Spec(B), then QCoh(X) = Mod B , the dg category of dg modules over the dg algebra associated to B via the Dold-Kan equivalence. In the general case, QCoh(X) = lim ← −Spec(B)→X Mod B (the functoriality being that induced by base change).
• Perf (X) will denote the full sub dg category of QCoh(X) spanned by perfect complexes. If X = Spec(B), then Perf (B) is the smallest subcategory of Mod B which contains B and that is stable under the formation of finite colimits and retracts. More generally, an object E ∈ QCoh(X) is perfect if, for any g : Spec(B) → X, the pullback g * E ∈ Perf (B). Perfect complexes coincide with dualizable objects in QCoh(X) and, under some mild additional assumptions (that will always be verified in our examples), with compact objects (see [12]).
Moreover, as we assume to work in the noetherian setting, we can consider: • Coh b (X) will denote the full sub dg category of QCoh(X) spanned by those cohomologically bounded complexes E (i.e. H i (E) = 0 only for a finite number of indexes) such that H * (E) is a coherent H 0 (O X )-module.
• Coh − (X) will denote the full dg category of QCoh(X) spanned by those cohomologically bounded above complexes E (i.e. H i (E) = 0 for i >> 0) such that H * (E) is a coherent H 0 (O X )-modules. These are also known as pseudo-coherent complexes.
• Let p : X → Y be a proper morphism locally almost of finite type. By [28,Cpt.4 Lemma 5.1.4], we have an induced ∞-functor p * : Coh b (X) → Coh b (Y ). We denote Coh b (X) Perf (Y ) the full subcategory of Coh b (X) spanned by those objects E such that p * E ∈ Perf (Y ).

Derived algebraic geometry
Derived algebraic geometry is a broad generalization of algebraic geometry whose building blocks are simplicial commutative rings, rather then commutative rings. It is usually better behaved in the situations that are typically defined bad in the classical context, e.g. non-transversal intersections. The main idea is to develop algebraic geometry in an homotopical context: instead of saying that two elements are equal we rather say that they are homotopic, the homotopy being part of the data. In this article derived schemes appear exclusively as (homotopy) fiber products of ordinary schemes. It is necessary to allow certain schemes to be derived, as if we restrict ourselves to work with discrete (i.e. classical) schemes, some important characters do not appear (e.g. the algebra which acts on certain dg categories of (relative) singularities). The ideas and motivations that led to derived algebraic geometry go back to J.-P. Serre (Serre's intersection formula, [52]), P. Deligne (algebraic geometry in a symmetric monoidal category, [21]), L. Illusie, A. Grothendieck, M. André, D. Quillen (the cotangent complex, [3], [30], [46], [53]) . . . however, the theory nowadays relies on solid roots thanks to the work of J. Lurie ([36]) and B. Toën -G. Vezzosi ( [65], [66]). For a brief introduction to derived algebraic geometry we refer to [63]. We will denote the ∞-category of derived schemes by dSch.
By definition, each derived scheme X has an underlying scheme t 0 (X) (its truncation). Indeed, the assignment X → t 0 (X) is part of an adjunction ι : Sch ⇄ dSch : t 0 . (2.3.1) A derived scheme is affine if its underlying scheme is so. There is an equivalence of ∞-categories dSch aff ≃ sRing op (2.3.2) where dSch aff is the full subcategory of dSch spanned by affine derived schemes and sRing is the ∞-category of simplicial rings. For any simplicial commutative ring A, we denote Spec(A) the associated derived scheme. The ∞-category dSch has all finite limits. In particular, it has fiber products. For example, if we consider a diagram

Spec(B) → Spec(A) ← Spec(C)
of affine derived schemes, the fiber product is equivalent to Spec(B ⊗ L A C), the spectrum of the derived tensor product.
Consider two (underived) S-schemes X, Y (S being underived itself). Then the fiber product computed in dSch (denoted X × h S Y ) might differ from the one computed in Sch (denoted X × S Y ). However, they are related by the formula

ℓ-adic sheaves
We shall briefly introduce the ∞-category of ℓ-adic sheaves, following [27]. Fix a prime number ℓ, which is invertible in each residue field of our base scheme S, where S is a regular scheme of finite type over a strictly local noetherian scheme 2 . Let Shv(X, Z/ℓ d Z) be the full subcategory of the ∞-category Fun(Sch op et , Mod Z/ℓ d Z ) spanned by (hypercomplete) étale sheaves. Here Sch et denotes the category of étale sheaves and Mod Z/ℓ d Z the ∞-category of Z/ℓ d Zmodules. The ∞-category Shv(X, Z/ℓ d Z) is compactly-generated, its compact objects being constructible sheaves (see [9,Proposition 3.38]). In what follows, we will denote Shv c (X, Z/ℓ d Z) the full subcategory of Shv(X, Z/ℓ d Z) spanned by compact objects.
The ring homomorphisms Z/ℓ d Z → Z/ℓ d−1 Z induce a sequence of ∞-functors and it follows from [27,Proposition 2.2.8.4] that the image of a constructible sheaf is again constructible, yielding We can then consider the limit of the diagram of ∞-categories This ∞-category can be identified with the full subcategory of Shv(X, Z) generated by ℓ-complete constructible sheaves, i.e. by those objects F ∈ Shv(X, Z) such that We will refer to this ∞-category with the ∞-category of constructible ℓ-adic sheaves.
The pushforward for a morphism f : X → Y of S-schemes of finite type induces an ∞-functors at the level of constructible ℓ-adic sheaves It admits a left adjoint that, at the level of objects, takes a constructible ℓ-adic sheaf to the ℓ-completion of its pullback. We next consider the ind-completion of such categories: the ∞-category of ℓ-adic sheaves. It is then a formal fact that we have a couple of adjoint functors, also called the pushforward and the pullback, defined at the level of ℓ-adic sheaves. Finally, we consider the localization of Shv ℓ (X) with respect to the class of morphisms {F → F[ℓ −1 ]}, obtaining the ∞-category of Q ℓ -adic sheaves Shv Q ℓ (X).

Stable homotopy categories
In this section we will briefly recall the constructions and main properties of SH S and of SH nc S , the stable ∞-category of schemes and the stable ∞-category of non-commutative spaces (a.k.a. dg categories).

The stable homotopy category of schemes
The stable homotopy category of schemes was first introduced by F. Morel and V. Voevodsky in their celebrated paper [40]. The main idea is to develop an homotopy theory for schemes, where the role of the unit interval -which is not available in the world of schemes -is played by the affine line. It was first developed using the language of model categories. We will rather use that of ∞-categories, following [48] and [49]. The two procedures are compatible, as shown in [48] and [49].
Let S = Spec(A) denote an affine scheme. One can produce the unstable homotopy category of schemes as follows: one considers the ∞-category Fun(Sm op S , S) of presheaves (of spaces) on Sm S . Its full subcategory spanned by Nisnevich sheaves, Sh N is (Sm S ) is an example of an ∞-topos (in the sense of [35]). Then one has to consider its hypercompletion Sh N is (Sm S ) hyp , which coincides with the localization of Fun(Sm op S , S) spanned by objects that are local with respect to Nisnevich hypercovers. If we further localize with respect to the projections {A 1 X → X}, we obtain the unstable homotopy ∞-category of schemes, which we will denote by H S . It is also important that the canonical ∞-functor Sm S → H S can be promoted to a symmetric monoidal ∞-functor with respect to the Cartesian structures. One then considers the pointed version of H S , H S * : it comes equipped with a canonical symmetric monoidal structure H ∧ S * and there is a symmetric monoidal ∞-functor H × S → H ∧ S * . The final step consists in stabilization: in classical stable homotopy theory one forces stabilization by inverting S 1 . However, in this context, there exist two circles, the topological circle S 1 := ∆ 1 /∂∆ 1 and the algebraic circle G m,S . One then stabilizes H S * by inverting S 1 ∧ G m,S ≃ (P 1 S , ∞) := cof ib(S ∞ − → P 1 S ) 3 -this can be done using the machinery developed in [49, §2.1]. As a result, we obtain the presentable, symmetric monoidal, stable ∞-category SH ⊗ S := H ∧ S * [(P 1 S , ∞) −1 ], called the stable homotopy ∞-category of schemes. It is moreover characterized by the following universal property (see [49,Corollary 2.39]): there is a symmetric monoidal ∞-functor Σ ∞ + : Sm × S → SH ⊗ S and for any presentable, symmetric monoidal pointed ∞-category D ⊗ , the map induced by Σ ∞ + is fully faithful and its image coincides with those symmetric monoidal ∞-functors F : Sm × S → D ⊗ that satisfy • Nisnevich descent, is an invertible object in D. Remark 2.5.2. It can be shown, using results of Ayoub ([4], [5]), ) and the machinery developed by Gaitsgory-Rozenblyum ( [28]) and , [38], [39]), that the assignment S → SH S defines a sheaf It is possible to mimic the procedure described in the previous paragraph to construct a presentable, symmetric monoidal, stable ∞-category SH nc S defined starting with non-commutative spaces rather than schemes, a.k.a. dg categories. The role of smooth schemes is played by dg categories of finite type in this context. Moreover, there exists an analog of Nisnevich square of non-commutative spaces (see [48] and [49] One then considers the category of presheaves Fun(Nc op S , S) and its localization with respect to the class of morphisms {j(U ) ∐ j(X) j(V ) → j(W )} determined by Nisnevich squares of non-commutative spaces, where j : Nc S → Fun(Nc op S , S) is the Yoneda embedding. We further localize with respect to the morphisms {X ⊗ Perf (S) → X ⊗ Perf (A 1 S )} 5 and obtain a presentable symmetric monoidal ∞-category H nc,⊗ S . Pursuing the analogy with the commutative case, we should now force the existence of a zero object in H nc S and then stabilize with respect to the topological and algebraic circles. It comes out that the situation is simpler in the non-commutative case: let denote the ∞-functor that we obtain if we force the topological circle to be invertible. Then • SH nc S is pointed due to the convention of Remark 2.5.5.
• The non-commutative motive is invertible in SH nc,⊗ S ([49, Proposition 3.24]): indeed, it is equivalent to Perf (S) ≃ 1 nc and therefore it is not necessary to force G m,S to be invertible.

The bridge between motives and non-commutative motives
We have now at our disposal the following picture: The existence of the dotted map, which we name perfect realization, is granted by the universal property of SH ⊗ S . Indeed • ψ • Perf (•) sends (ordinary) Nisnevich squares to pushout squares in SH nc S : this is a consequence of the compatibility of non-commutative Nisnevich squares with the classical ones and of the definition of SH nc S , • A 1 -invariance is forced by construction, • ψ(P 1 S , ∞) is an invertible object in SH nc,⊗ S . 5 Since the Yoneda embedding is symmetric monoidal (when we consider the Day convolution product on the ∞-category of presheaves), then it suffices to localize with respect to the morphism j(Perf (S)) → j(Perf (A 1 S )).
The fact that R Perf commutes with colimits is also guaranteed by the universal property of SH ⊗ S . As both SH S and SH nc S are presentable ∞-categories, the adjoint functor theorem [ In particular, the ∞-functor M factors trough the full subcategory of SH S spanned by modules over BU S : (2.6.3) In [9], the authors introduced a dual version of this ∞-functor. Consider the endomorphism of SH nc S induced by the internal hom: M ∨,⊗ (2.6.5) where the vertical map on the left is given by the inclusion of dgCat ft,⊗ S in its Ind-completion dgCat idm,⊗ S and the oblique ψ ⊗ is induced by the universal property of the Ind-completion. Let T ∈ dgCat idm S . Then M ∨ (T ) is the sheaf of spectra X ∈ Sm S → HK(Perf (X) ⊗ S T ): . Moreover, M ∨ has the following nice properties • it is lax monoidal (it is a composition of lax monoidal ∞-functors), • it commutes with filtered colimits (see [9,Remark 3.4]), • it sends exact sequences of dg categories to fiber-cofiber sequences in Mod BUS (SH S ) ⊗ (see [9,Corollary 3.3]).
We will refer to M ∨ as the motivic realization of dg categories.

ℓ-adic realization of dg categories
We will need a way to associate an ℓ-adic sheaf to a dg category. We follow the construction given in [9, §3.6, §3.7], which relies on results of J. Ayoub and D.-C. Cisinski -F. Déglise. Let HQ be the Eilenberg-MacLane spectrum of rational homotopy theory. Then we get which identifies the ∞-category on the right hand side with non-torsion objects of SH S . Similarly, if one puts BU S,Q := HQ ⊗ BU S , then one gets where ν is the free generator in degree (1) [2].
By using the theory of h-motives developed by the authors in [14] one can define an ℓ-adic realization ∞-functor strongly compatible with the 6-functors formalism and Tate twists, at least for noetherian schemes of finite Krull dimension. Then, using the equivalence we obtain an ℓ-adic realization ∞-functor . We will use the notation and refer to this ∞-functor as the ℓ-adic realization of dg categories.
3 Motivic realization of twisted LG models Notation 3.0.1. Let S be a noetherian (not necessarely affine) regular scheme. We will label Sch S the category of separated S-schemes of finite type.

The category of twisted LG models
Consider the category Sch S and let L S be a line bundle on S. Then we can consider the category of Landau-Ginzburg models over (S, L S ), defined as follows: • Objects consists of pairs (X, s X ), where p : X → S is a (flat) S-scheme and s X is a section of L X := p * L S .
• Composition and identity morphisms are clear.
We will denote this category by LG (S,LS) . Remark 3.1.1. If L S is the trivial line bundle, then LG (S,LS) coincides with the category of usual Landau-Ginzburg models over S (as defined for example in [9]). Indeed, in this case V(L S ) = Spec OS (O S [t]). Therefore, for any X ∈ Sch S , a section of L X consists of a morphism O X [t] → O X , i.e. of a global section of O X .
where s x ⊞ s Y = p * X s X + p * Y s Y . This is clearly (weakly) associative and the unit is given by (S, 0), where 0 stands for the zero section of L S .
Notice that this tensor product is induced by the abelian group structure on V(L S ) = Spec S (Sym OS (L ∨ S )). Remark 3.1.3. If L S is the trivial line bundle, then ⊞ coincides with the tensor product on the category of LG modules over S.
We will now exhibit twisted LG models as a fibered category.
LG be the category defined as follows: • Objects are triplets (f : Y → X, L X , s Y ) where f is a flat morphism between S-schemes, L X is a line bundle on X and s Y is a global section of f * L X .
• Given two objects (f i : , a morphism from the first to the second is the datum of a commutative diagram and of an isomorphism α : g * X L X2 → L X1 such that s Y1 corresponds to s Y2 under the isomorphism By abuse of notation, we will say in the future that g * Y (s Y2 ) = s Y1 if this condition is satisfied. We will denote such a morphism by (g, α).
• Composition and identities are defined in an obvious way.
We will refer to this category as the category of twisted Landau-Ginzburg models of rank 1 over (S, L S ) (twisted LG models for short).
Notice that there is a functor and let (f 2 : Y 2 → X 2 , L X2 , s Y2 ) be an object of LG over (X 2 , L X2 ). Consider the morphism (g, α) : (f 1 : where g Y is the projection Y 1 → Y 2 (which is flat as it is the pullback of a flat morphism), f 1 is the projection Y 1 → Y 2 and s Y1 is g * Y (s Y2 ). It is clear that it is a morphism of LG over (g X , α). We need to show that it is cartesian. Consider ((r, p), β) (3.1.5) Then the universal property of Y 1 gives us an unique morphism r : Z → Y 1 such that the compositions with f 1 and g Y are p • h and q respectively. We just need to show that r * (s Y1 ) = s Z . But this is clear since Remark 3.1.6. Let (X, L X ) be an object of X∈SchS BG m,S (X). Then the fiber of (X, L X ) along π is LG (X,LX ) . Definition 3.1.7. We say that a collection of maps {(g i , α i ) : (U i , L Ui ) → (X, L X )} i∈I in X∈SchS BG m,S (X) is a Zariski covering if {g i : U i → X} i∈I is so. They clearly define a pre-topology on X∈SchS BG m,S (X) 6 . We will refer to the corresponding topology by Zarisky topology on X∈SchS BG m,S (X).
LG is a stack over X∈SchS BG m,S (X) endowed with the Zariski topology.
Proof. This is a simple consequence of the fact that a morphism of schemes is uniquely determined by its restriction to a Zariski covering and that line bundles are Zariski sheaves.
We conclude this section with the following observation:

a Zariski covering in
X∈SchS BG m,S (X). Then the canonical functor is a symmetric monoidal equivalence. 6 Notice that pullbacks exists in X∈Sch S BG m,S (X): the pullback of (Y 1 , L Y 1 ) with the projections defined in an obvious way Proof. Consider the functors It is easy to see that they respect the tensor structure. Therefore, we get the desired symmetric monoidal functor (3.1.6) in the (big) category of symmetric monoidal categories and symmetric monoidal functors. Then, in order to prove that is a symmetric monoidal equivalence, it suffices to show that the underlying functor LG ⊞ (Ui,LU i ) )).
(3.1.8) As the functor which forgets the symmetric monoidal structure of a category is a right adjoint, it preserves limits. Thus, The assertion now follows from the previous lemma.

The dg category of singularities of a twisted LG model
Let (X, s X ) be a twisted LG model over (S, L S ). The section s X defines a closed sub-scheme of X. Since we are not assuming that the section is regular, some derived structure might appear. More precisely: let O X → L X be the morphism of O X -modules associated to s X . Then, taking duals, it defines a morphism L ∨ X → O X . If we apply the relative spectrum functor, we get a morphisms Spec OX (O X ) = X → Spec OX (Sym OX (L ∨ X )) = V(L X ), i.e. a section of the vector bundle associated to L X . By abuse of notation, we will label this morphism by s X . We can also consider the zero section X → V(L X ), which is the morphism associated to 0 ∈ L X (X).
is an lci morphism (Zariski locally, it is just the zero section a scheme Y in the affine line A 1 Y ). It is well known that this class of morphisms is stable under derived pullbacks. In particular, i : X 0 → X is a derived lci morphism.
Remark 3.2.3. There is a truncation morphism t : π 0 (X 0 ) → X, where π 0 (X 0 ) is the classical scheme associated to X 0 . Whenever s X is a regular section, the truncation morphism is an equivalence in the ∞-category of derived schemes. Remark 3.2.4. Of major importance for our purposes is that i is an lci morphism. Indeed, by [61], if f : Y → Z is an lci morphism of derived schemes and E ∈ Perf (Y ), then f * E ∈ Perf (Z). In particular, we get that Perf (X 0 ) is a full subcategory of Coh b (X 0 ) Perf (X) , the full subcategory of Coh b (X 0 ) spanned by those objects E ∈ Coh b (X 0 ) such that f * E ∈ Perf (X).
The dg category of absolute singularities of a derived scheme is a noncommutative invariant (in the sense of Kontsevich) which captures the singularities of the scheme.   . . . which has nontrivial cohomology in every even negative degree.
Since i is an lci morphism of derived schemes, by [61] and by [28] the pushforward induces a morphism i * : Sing(X 0 ) → Sing(X). Sing(X, s X ) := f iber i * : Sing(X 0 ) → Sing(X) . Remark 3.2.9. It is a consequence of the functoriality properties of the pullback and that of taking quotients and fibers in dgCat idm S that the assignments (X, s X ) → Sing(X, s X ) can be organized in an ∞-functor We will need to endow the ∞-functor (3.2.5) with the structure of a lax monoidal ∞-functor. In order to do so, we will introduce a strict model for Coh b (X 0 ) Perf (X) , following the strategy exploited in [9].
This is a morphism of Sym OX (L ∨ X )-modules (induced by the multiplication of Sym OX (L ∨ X )). Base changing along the morphism Sym OX (L ∨ X ) → O X induced by s X we obtain that the cdga associated to X 0 is the spectrum of L ∨ Example 3.2.11. Consider the zero section of L X . Then the cdga associated to the structure sheaf of X × h V(LX ) X is the Koszul algebra in degrees [−1, 0] (recall that we use cohomological conventions) Define Coh s (B, s) as the dg category of K(B, s)-dg modules whose underlying B-dg module is perfect. Since X is an affine scheme, every perfect B-dg module is equivalent to a strictly perfect one. Therefore, we can take Coh s (B, s) as the dg category of K(B, s)-dg moodules whose underlying B-module is strictly perfect. More explicitly, such an object corresponds to a triplet (E, d, h) where (E, d) is a strictly perfect B-dg module and is a morphism such that Notice that Coh s (B, s) is a locally flat A-linear dg category. This follows immediately from the fact that the underlying complexes of B-modules are strictly perfect and from the fact that B is a flat A-algebra.
This strict dg category is analogous to the one that has been introduced in [9]. It gives us a strict model for the ∞-category Coh b (X 0 ) Perf (X) . Indeed, the following result holds: Construction 3.2.14. We can endow the assignment (B, s) → Coh s (B, s) with the structure of a pseudo-functor: let f : (B, s) → (C, t) be a morphism of affine twisted LG models, i.e. a morphism of A-algebras f : B → C such that the induced morphism id ⊗ f : L B → L C sends s to t. Then we can define the dg functor This yields a pseudo-functor Construction 3.2.15. We can endow the pseudo-functor (B, s) → Coh s (B, s) with a weakly associative and weakly unital lax monoidal structure. For any pair of twisted affine LG models (B, s), (C, t), we need to construct a morphism For a pair (B, s), let Z(s) denote Spec(K(B, s)). Moreover, let Then consider the following diagram where φ is the morphism corresponding to  Consider the two projections , which immediately implies that it is strictly bounded. In order to see that each component where A is the unit in dgCat lf A , i.e. the dg category with only one object whose endomorphism algebra is A, is defined by the assignment It is clear that (3.2.16) and (3.2.17) satisfy the associativity and unity axioms, i.e. they enrich the pseudo-functor (3.2.11) with a lax monoidal structure Denote this class of morphisms in Pairs-dgCat lf A by W DK . Also notice that the symmetric monoidal structure on dgCat lf A induces a symmetric monoidal structure on Pairs-dgCat lf It is clearly associative and unital, where the unit is (A, {id}). We will denote this symmetric monoidal structure by Pairs-dgCat lf,⊗ A . Note that this symmetric monoidal structure is compatible to the class of morphisms in W DK , as we are working with locally-flat dg categories.
Then as in [9, Construction 2.34, Construction 2.37], we can construct a symmetric monoidal ∞-functor and compose it with the functor We get, after a suitable monoidal left Kan extension, the desired lax monoidal ∞-functor Lemma 3.2.16. Let S = Spec(A) be a regular noetherian ring. Let L be a line bundle over S. The following equivalence holds in CAlg(dgCat idm,⊗ Proof. The first equivalence is an immediate consequence of the regularity hypothesis on S. As we have remarked in the Example (3.2.11), the cdga assocaiated via the Dold-Kan correspondence to S 0 is In particular, we get the equivalence The endomorphism dg algebra RHom can be computed explicitly by means of the following L ∨ 0 − → A-resolution of A: which is quasi isomorphic to A. However, when we ask for (L ∨ 0 − → A)-linearity, the copies of L ⊗n in odd degree disappear, as the local generators ε in degree . Therefore we find that RHom This shows that there exists an equivalence Notice that both these objects carry a canonical algebra structure. In order to conclude that the two commutative algebra structures coincide, we consider the dg functor ) is a strict cdga, seen as a commutative algebra object in QCoh(A). Similarly to [9,Lemma 2.39] The same arguments given in loc.cit. hold mutatis mutandis in our situation and therefore we obtain a symmetric monoidal functor which preserves quasi-isomorphisms. If we localize both the l.h.s. and the r.h.s. we thus obtain a symmetric monoidal ∞-functor from which one recovers the equivalence of symmetric monoidal dg categories Then, for a noetherian regular scheme S with a line bundle L S , we obtain a lax monoidal ∞-functor (3.2.37) is endowed with an action of Coh b (S 0 ) (recall that we are assuming that S is regular). Similarly to [9, Remark 2.38] we can describe this action. Consider the diagram  implies that following [45] and [9].
where ν is the generator in degree 2. More explicitly, which, at the level of objects, is defined by Proof. Let M ∈ Perf (S 0 ). By definitions this means that, for every Zariski open f : Spec(A) → S 0 , the pullback f * M ∈ Perf (A). Consider our covering U. Then, for every i ∈ I, M |Ui is a perfect B i -module. By the same argument given above, we get equivalences Proof. This follows by the fact that the categories involved satisfy Zariski descent, by the previous lemmas and by [9,Proposition 2.43].
There is an equivalence of dg categories Proof. By definition, is the subcategory spanned by locally u-torsion modules in Coh b (X 0 ) Perf (X) . We will show that they coincide with the subcategory of perfect complexes, using [9,Proposition 2.45]. Suppose that M is in Perf (X 0 ). Then, for every affine open covering j : perfect. This proves the lemma.
Corollary 3.2.24. Let (X, s X ) be a twisted LG model over (S, L S ). Then the exact sequence in dgCat idm is equivalent to  Proof. This is an obvious consequence of the previous lemmas and of the fact that the ∞-functor

Construction 3.2.25. Consider the following lax monoidal ∞-functor
LG op,⊞ is defined on objects by the assignment (X, s) → Sing(X, s).

The motivic realization of Sing(X, s X )
Recall from Section 2.6 that there is a motivic realization lax monoidal ∞- with the following properties • M ∨ S sends exact sequences of dg categories to fiber-cofiber sequences in Mod BUS (SH ⊗ S ). Our main scope in this section is to study the motivic realization of the dg category Sing(X, s X ) associated to (X, s X ) ∈ LG (S,LS) , under the assumption that X is regular. The first important fact is the following one: ≃ BU X (see section 2.5.1 for notation). Proof. Consider the construction given in Section 2.6 with S replaced by X: Notice that since the ∞-functor above is right lax monoidal and Perf (X) is a commutative algebra in dgCat idm,op,⊗ , there is a canonical morphism of commutative algebras It suffices to show that they represent the same ∞-functor. Let Y be an object in SH ⊗ X . Then . The fact that the equivalence holds in CAlg(SH ⊗ X ) follows immediately from the conservativity of the forgetful functor along M ∨ X coincides with multiplication by the class [E] ∈ HK 0 (X) in the commutative algebra BU X , which we denote m E . In a formula : The next step will be understanding the motivic realization of the category of coherent bounded complexes Coh b (Z) of an S-scheme Z, at least when it is possible to regard it as a closed subscheme of a regular S-scheme X.
Proof. Consider the exact sequence of dg categories where Coh b (X) Z denotes the subcategory of objects in Coh b (X) whose support is in Z.
Recall that, if X is a scheme, an O X -linear dg category is a sheaf of dg categories on the Zariski site of X. Then the above sequence of dg categories is exact by definition and if we apply M ∨ X to it we obtain a fiber cofiber sequence in SH X (see [69,Remark 2 The regularity hypothesis imposed on X implies that Coh b (X) ≃ Perf (X) and Coh b (U ) ≃ Perf (U ). If we apply M ∨ X and use the previous proposition, we obtain which is a fiber-cofiber sequence in SH X (since M ∨ X sends exact triangles of dg categories to fiber-cofiber sequences). Moreover, M ∨ X is compatible with pushforwards and M ∨ X (j * ) ∼ j * . As the spectrum BU of non-connective homotopyinvariant K-theory is compatible with pullbacks (see [15]), the previous fibercofiber sequence is nothing but where the map on the right is induced by the unit of the adjunction (j * , j * ). In particular, we get a canonical equivalence and therefore we are left to show that we have an equivalence where V is an affine open subscheme of X.
Notice that there is a canonical morphism The collection of objects Σ ∞ + Y ⊗ BU X , where Y ∈ Sm X , forms a family of compact generators of Mod BUX (SH X ). As M ∨ X commutes with filtered colimits, it suffices to show that the morphism .  is an equivalence of spectra. Notice that By [45], , and therefore the spectrum above coincides with HK(Coh b (Y × X Z)), which by the homotopy-invariance of G-theory and by the theorem of the Heart ([7, Corollary 6.4.1]) coincides with the G-theory of Y × X Z. In the same way, we obtain that The claim now follows from Quillen's dévissage.
An anonymous referee has pointed out that in order to deal with O X -linear dg categories one can use the theory of 1-affiness of D. Gaitsgory ([26]). One can use this to give an alternative proof of the proposition above.
As a final observation, we remark that the assignemnt Proof. By the proof of the previous proposition, . Then the theorem of the Heart and the computation above allows us to conclude. Now consider a twisted LG model (X, s X ) over (S, L S ) and assume that X is a regular scheme. As above, let X 0 i − → X be the derived zero section of s X in X and let j : X U = X − X 0 → X be the corresponding open embedding. In this case, we have that Sing(X, s X ) ≃ Sing(X 0 ). Consider the diagram in dgCat idm X and its image in SH X By the previous results, the second diagram can be rewritten and completed as follows where i = i • t : π 0 (X 0 ) → X is the closed embedding of the underlying scheme of X 0 in X.
Remark 3.3.6. The morphism BU X → i * i * BU X can be factored as  we see that i * i * is equivalent to the dg functor LG (S,LS) and assume that X is regular. Then there is a fiber-cofiber sequence in Mod BUX (SH X )  In particular, if we apply the ∞-functor i * , we get the following fiber-cofiber sequence:  Proof. The second statement is an immediate consequence of the first and of the equivalence i * j ! ≃ 0. The first fiber-cofiber sequence can be obtained by applying the octahedreon axiom to the triangle which appears in diagram (3.3.19) and by the fact that   where both squares are (homotopy) cartesian. Notice that in this case π 0 (S 0 ) = S and therefore i = i • t = id. Then, the fiber-cofiber sequence of the previous corollary gives us an equivalence is commutative up to coherent homotopy, to get the first desired equivalence it suffices to show that M ∨ S,Q (t * ) is an equivalence. This is true as i • t = id and t • i is homotopic to the identity (see [9,Remark 3.31]). Proof. Let V := V(L S ). Consider the equivalence that we get from the localization sequence of (i 0 , j 0 ) where the map c is the one induced by the counit i 0 * i ! 0 BU V,Q → BU V,Q . Notice that i 0 : S → V is a closed embedding between regular schemes. In particular, absolute purity holds. It follows from [15,Remark 13.5.5] that the composition corresponds to 1 − m L ∨ S , as the conormal sheaf of i 0 : S → V is L ∨ S . If we apply i ! we obtain

3.39)
which under the equivalence BU S,Q ≃ i ! BU V,Q corresponds to 1 − m L ∨ S . Construction 3.3.11. Let (X, s X ) ∈ LG (S,LS) . Consider the morphism Since j : X U → X is the complementary open subscheme to the zero locus of the section s X , it follows that j * L ∨ X ≃ O X U . In particular, Then we obtain a morphism LG (S,LS) and assume that X is regular and that Proof. This follows immediately from the octahedron property applied to the triangle in the following diagram and from the compatibility of 1 − m L ∨ X with pullbacks.

The formalism of vanishing cycles
We shall begin with a quick review of the formalism of vanishing cycles. We hope that this will be useful to understand the analogy which led to the definition of monodromy-invariant vanishing cycles.
Notation 4.1.1. Throughout this section, A will be an excellent henselian trait and S will denote the associated affine scheme. Label k its residue field and K its fraction field. Let σ be Spec(k) and η be Spec(K). Fix algebraic closures k alg and K alg of k and K respectively. We will consider: • The maximal separable extension k sep of k inside k alg . We will use the notationσ = Spec(k sep ).
• The maximal unramified extension K unr of K in K alg . We will use the notation η unr = Spec(K unr ).
• The maximal tamely ramified extension K t of K inside K alg . We will use the notation η t = Spec(K t ).
• The maximal separable extension K sep of K inside K alg . We will use the notationη = Spec(K sep ).
Moreover, we will fix an uniformizer π.
Remark 4.1.2. It is well known that there is an equivalence between the category of separable extensions of k and that of unramified extensions of K. In particular, Gal(k sep /k) ≃ Gal(K unr /K). This, together with the fundamental theorem of Galois theory, implies that there is an exact sequence of groups The Galois group on the left is called the inertia group and it is usually denoted by I. The chain of extensions K unr ⊆ K t ⊆ K sep gives us the following decomposition of I: The Galois group I w is called wild inertia group, while I t is called tame inertia group. See [51] for more details. In particular, using the equivalence of Shv Q ℓ (X η unr ) with the ∞-category of ℓ-adic sheaves on the generic geometric fiber endowed with a continuous action of I Shv Q ℓ (Xη) I , we have that Shv Q ℓ (X) is the recollement of Shv Q ℓ (Xσ) and Shv Q ℓ (Xη) I .
Remark 4.1.5. Notice that we have the following diagram where each arrow is an equivalence:  We will label the ∞-category of ℓ-adic sheaves onỸ et ×σ etη unr et by Shv Q ℓ (Y ) I , as it is the ∞-category of ℓ-adic sheaves on Y endowed with a continuous action of I. The ∞-category Shv Q ℓ (Y ) It of ℓ-adic sheaves on Y endowed with a continuous action of I t identifies with the full subcategory of Shv Q ℓ (Y ) I such that the induced action of I w is trivial. Definition 4.1.7. Let p :X →S be anS-scheme. Let E ∈ Shv Q ℓ (X η unr ). The ℓ-adic sheaf of tame nearby cycles of E is defined as (4.1.8) Analogously, the ℓ-adic sheaf of nearby cycles of E is defined as Ψ(E) :=ī * j * E |Xη ∈ Shv Q ℓ (Xσ) I . (4.1.9) Remark 4.1.8. With the same notation as above, the following equivalence holds in Shv Q ℓ (Xσ) It : Definition 4.1.9. Let p :X →S be anS-scheme. Let F ∈ Shv Q ℓ (X). The unit of the adjuction (j * t , j t * ) induces a morphism in Shv Q ℓ (Xσ) It where we regard the object on the left endowed with the trivial I t -action. The cofiber of this morphism is by definition the ℓ-adic sheaf of tame vanishing cycles of F, which we will denote by Φ t (F). In a similar way, we define the ℓ-adic sheaf of vanishing cycles Φ(F) as the cofiber of the morphism (called the specialization morphism in literature)ī * F → Ψ(FXη ) (4.1.12) induced by the unit of the adjunction (j * ,j * ).

Monodromy-invariant vanishing cycles
Context 4.2.1. Assume that S is strictly henselian, i.e. that k is a separably closed field and that p : X → S is a proper morphism.
For our purposes, we are interested in the image of Φ(Q ℓ,X ) via the ∞-functor It is then important to remark that it is possible to determine p σ * Φ(Q ℓ,X ) hI without ever mentioning this ∞-functor. Notice that Φ(Q ℓ,X ) hI is the cofiber of the image of the specialization morphism via (−) hI : is a fiber-cofiber sequence in Shv Q ℓ (X σ ).

Definition 4.2.3. Consider the canonical morphism in Shv
induced by the base change natural transformation p * j 0 * → j * p * η . We will refer to it as the monodromy-invariant specialization morphism. We will refer to the cofiber of this morphism in Shv Q ℓ (X σ ) as monodromy-invariant vanishing cycles, which we will denote Φ hm p (Q ℓ ). In order to justify the choice of the name of this morphism, we shall prove the following: We shall view both arrows as maps p σ * Q ℓ,Xσ ⊗ Q ℓ,σ Q hI ℓ,σ → p σ * i * j * Q ℓ,Xη . Since the tensor product defines a cocartesian symmetric monoidal structure on CAlg(Shv Q ℓ (σ)) (see [34]), it suffices to show that the maps of commutative algebras obtained from p σ * (sp) hI and p σ * sp hm p by precomposition with p σ * Q ℓ,Xσ → p σ * Q ℓ,Xσ ⊗ Q ℓ,σ Q hI ℓ,σ and Q hI ℓ,σ → p σ * Q ℓ,Xσ ⊗ Q ℓ,σ Q hI ℓ,σ respectively are equivalent. It is essentially the proof of the main theorem in [9] that the precompositions of p σ * (sp) hI with these two maps are induced by the canonical morphisms 1 → j * j * and 1 → p η * p * η . We are left to show that the same holds true if we consider sp hm p instead of (sp) hI . Recall that sp hm p is induced by the base change morphism p * j 0 * → j * p * η , i.e. the morphism which corresponds under adjunction to j 0 * → j 0 * p η * p * η , induced by the unit of the adjunction (p * η , p η * ). From this we obtain the following diagram

(4.2.4)
In particular the composition i * Xη is homotopic to the map p σ * Q ℓ,Xσ → p σ * i * j * Q ℓ,Xη induced by 1 → j * j * by the following commutative triangle Notice that what we said above is actually true for any ℓ-adic sheaf in the image of p * : Shv Q ℓ (S) → Shv Q ℓ (X).
The advantage of this reformulation is that it allows to define inertia invariant vanishing cycles without ever mention the inertia group. We will adopt it for the situation we are interested in.
Assume that S is a noetherian regular scheme.
Definition 4.2.6. Let (X, s X ) be a twisted LG model over (S, L S ). Consider the diagram obtained from the zero section of the vector bundle associated to L X : Define the monodromy-invariant specialization morphism associated to (X, s X ) as the map in Shv Q ℓ (π 0 (X 0 )) We will refer to the cofiber of sp mi (X,sX ) as monodromy invariant vanishing cycles of (X, s X ), that we will denote Φ mi (X,sX ) (Q ℓ ). Proposition 4.2.7. Let (X, s X ) be as above and assume X regular. There is an equivalence The ℓ-adic sheaf i * 0 j 0 * Q ℓ,U is the cofiber of the morphism i *  The absolute purity isomorphism is given by the class cl(X) ∈ H 2 X (V X , Q ℓ (1)), whose image in H 2 (X, Q ℓ (1)) is c 1 (N ∨ X|VX ), the first Chern class of the conormal bundle (see [25]). The map Q ℓ,X → Q ℓ,X (1) [2] corresponding to this class is the image of the right vertical arrow in the diagram above via the ∞-functor −(1) [2]. It suffices to notice that N ∨ X|VX ≃ L X to conclude.

2-periodic ℓ-adic sheaves
We shall now approach the comparison between monodromy-invariant vanishing cycles and the ℓ-adic realization of the dg category of singularity Sing(X 0 ). In order to do so, we need to work with the category of Q ℓ,S (β)-modules in Shv Q ℓ (S). Recall that there is an adjucntion of ∞-functors given by − ⊗ Q ℓ,S Q ℓ,S (β) and by the forgetful functor. Notice that Mod Q ℓ,• (Shv Q ℓ (•)) defines a fibered category over the category of schemes, which satisfies Grothendieck's six functors formalism.
Definition 5.1.1. With the same notation as above let be the arrow in Mod Q ℓ,X (β) (Shv Q ℓ (X)) induced by the base change morphism s * X j 0 * → j * s * U . Denote by Φ mi (X,sX ) (Q ℓ (β)) its cofiber.
Proof. This is analogous to [9,Proposition 4.28]. As a first step, notice that both * -pullbacks and * -pushforwards commute with Tate and usual shifts. Since − ⊗ Q ℓ Q ℓ (β) commutes with * -pullbacks, we have Using the equivalence Q ℓ,VX (β) ≃ i∈Z Q ℓ,VX (i)[2i], we see that We shall show that the canonical map is an equivalence. This follows immediately from the fact that the * -pushforward commutes with filtered colimits and from the equivalence i∈Z Q ℓ, . This shows that i * s * X j 0 * Q ℓ,UX ⊗ Q ℓ,X Q ℓ,X (β) ≃ i * s * X j 0 * Q ℓ,UX (β).
The same argument applies to show that To show that the two maps are homotopic, it suffices to show that they are so before applying i * . Notice that is compatible with the * -pullback and with the !-pushforward, which coincides with the *pushforward for s U as it is a closed morphism. For what we have said above, is compatible with * -pullbacks. The last assertion follows as − ⊗ Q ℓ Q ℓ (β) is exact.
LG (S,LS) and assume that X is a regular scheme. The following equivalence holds in Mod Q ℓ,X (β) (Shv Q ℓ (X)) Proposition 3.8], c 1 (L) is nilpotent and therefore the Chern character of L can be written as 1 + c 1 (L) + 1 2 c 1 (L) 2 + · · · + 1 m! c 1 (L) m for some m ≥ 1. Then The last equivalence follows from the fact that, by [ and, as we have remarked above, We are finally ready to state our main theorem: Let (X, s X ) be a twisted LG model over (S, L S ). Assume that X is regular and that 0, s X : X → V(L X ) are Tor-independent. The following equivalence holds in Mod i * s * X j0 * Q ℓ,U X (β) (Shv Q ℓ (X 0 )): where the i * s * X j 0 * Q ℓ,UX (β)-module structure on the l.h.s. is the one induced by the equivalence of commutative algebras i * R ℓ,∨ X (Sing(X, 0)) ≃ i * s * X j 0 * Q ℓ,UX (β).
Proof. We start by noticing that i = s 0 : X 0 → X. Indeed, i 0 • s 0 = s X • i and both i 0 and s X are sections of the canonical morphism V X = V(L X ) → X. In particular (5.2.8) Since X is supposed to be regular, Sing(X, s X ) ≃ Sing(X 0 ). By Propositions 3.3.8 and 5.2.1 we have a fiber-cofiber sequence in Mod i * s * X j0 * Q ℓ,U X (β) (Shv Q ℓ (X 0 )). On the other hand, by Proposition 5.1.2, we have another fiber-cofiber sequence Both f and g in these two fiber-cofiber sequences are defined by the universal property of the object in the middle. Therefore, if we consider the diagram  where the middle vertical arrows induce the morphism f and g respectively, it suffices to showf ∼g. By definition,f is induced by the counit Q ℓ,X (β) → j * j * Q ℓ,X (β), whileg is induced by the base change morphism b.c. : s * X j 0 * Q ℓ,UX (β) → j * s * U Q ℓ,UX (β): It also suffices to show that the two arrows are homotopic before applying i * . Consider the diagram (5.2.13) If we analyse the commutative triangle on the left, we see that the objlique arrow corresponds to the unit of the adjunction ((s X • j) * , (s X • j) * ). Indeed, by definition, the vertical arrow is and the horizontal arrow is the one induced by the unit of (j * 0 , j 0 * ). The composition is exactly the unit of the adjunction ((j 0 • s U ) * , (j 0 • s U ) * ). Hence, the claim is proved as j 0 • s U ≃ s X • j. Now notice that s X * is conservative (s X is a closed morphism), i.e. the counit s * X s X * → 1 is a natural equivalence. If we compose the oblique arrow with it, we obtain the unit of (j * , j * ) evaluated in s * X Q ℓ,VX (β). The statement about the module structures is clear. Remark 5.2.3. Notice that our main theorem provides a generalization of the formula proved in [9]: assume that we are given a proper flat morphism p : X → S from a regular scheme to an excellent strictly henselian trait 8 . Let s X : X → A 1 X be the pullback of the section S → A 1 S given by the uniformizer π. Then we can consider the diagram where all squares are cartesian. The theorem we have just proved tells us that By the regularity assumption on X, Sing(X, s X ) ≃ Sing(X 0 ). If we apply p σ * to the formula above we find: which is exactly the content of [9,Theorem 4.39]. Also notice that in this case 2.17) as the first Chern class of the trivial line bundle is zero. This recovers the equivalence proved in loc.cit.
6 The ℓ-adic realization of the dg category of singularities of a twisted LG model of rank r In this section we will explain how, by means of a theorem due to D. Orlov and J. Burke -M. Walker, the main theorem we proved in the previous section allows us to compute the ℓ-adic realization of the dg category of singularities of the zero locus of a global section of any vector bundle on a regular scheme. This is the reason that led us to consider twisted LG model, as defined in §3.1, as the author was initially interested in computing the ℓ-adic realization of the zero locus of a multifunction X → A n S (with X regular).

Reduction of codimension
In this section we will provide an ∞-functorial lax monoidal enhancement of the "reduction of codimension" equivalence proved by D. Notice that all we said in section §3.1 can be generalised mutatis mutandis to the situation where line bundles L S are replaced by vector bundles of a fixed rank r E S . In particular, for a fixed vector bundle E S on S, we can define a symmetric monoidal (ordinary) category LG ⊞ (S,ES ) , analogous to the one we defined in section §3.1.
Notation 6.1.3. Let E S be a vector bundle of rank r over S. We will denote P(E S ) = P roj S (Sym OS (E ∨ S )) and π S : P(E S ) → S the associate projective bundle and projection. Moreover, we will denote by O(1) the twisting sheaf on P(E S ). following the lead of [11] and [42]. Let (X, s) ∈ LG (S,ES) . Consider W s ∈ Γ(P(E ∨ X ), O(1)) defined as the morphism induced by the morphism of modules Here (−) is the functor that associates a quasicoherent module on P(E ∨ X ) to a graded Sym OX (E X )-module. The assignment (X, s) → (P(E ∨ X ), W s ), together with the obvious law for morphisms defines a functor It is immediate to observe that (S, 0) → (P(E ∨ S ), 0), i.e. the functor is compatible with the units of the two symmetric monoidal structures. It remains to show that Ξ((X, s) ⊞ (Y, t)) ≃ Ξ((X, s)) ⊞ Ξ((Y, t)).
On the left hand side we have while on the right hand side we have Since The claim follows and thus we obtain the desired symmetric monoidal functor.
If we compose the symmetric monoidal functor (6.1.1) with the lax monoidal ∞-functor defined in Section 3. which, at the level of objects, corresponds to the assignment (X, s) → Sing(P(E ∨ X ), W s ).
We define the dg category of singularities of a twisted LG model (X, s) of rank r over (S, E S ) in the following way: consider the derived zero locus of s, defined as the homotopy pullback of s along the zero section As 0 : S → V(E S ) is a closed lci morphism (locally, is of the form U → A r S ), so is i : X 0 → X. Thus the pushforward induces a dg functor i * : Sing(X 0 ) → Sing(X).
. This is locally true (see for example [43,Remark 1.22] or [31]) and the global statement follows from the existence of a morphism of dg algebras 1.11) where N is the normalized Moore complex functor.
In the same way as in Construction 3.2.14 and in the following discussion, we can construct a lax monoidal ∞-functor LG op,⊞ (S,ES) → dgCat idm,⊗ S (6.1.14) Moreover, using the same arguments as in the discussion following Remark 1.27 in [43], we get the lax monoidal ∞-functor (6.1.9). At this point, the reader might complain that there are too many Sing's involved, and this may cause confusion. As a partial justification to the choice of notation we made, let us show that these different ∞-functors are closely related.
Definition 6.1.9. Let (X, s) ∈ LG (S,ES ) . We define the projective bundle over X 0 associated to E X0 = i * E X as the derived pullback Recall that, given a global section of a line bundle on a scheme, we define its (derived) zero locus as in Definition 3.2.1. In particular, for (X, s) ∈ LG (S,ES) we have a global section W s of the line bundle O(1) on P(E ∨ X ) and thus we have a (derived) pullback square Let (X, s) ∈ LG (S,ES ) and consider the following diagram, where both squares are homotopy cartesian: and it is functorial in (X, s). We will need to enhance the assignment (X, s) → Υ (X,s) with a lax monoidal structure. We will use strict models. Assume that S = Spec(A) is affine and let E be a projective A module of rank r. Since E is an A-module of finite type, there exists a surjection A n → E. (6.1.19) Denote by t i (i = 1, . . . , n) the images in E of the elements (1, 0, . . . , 0), . . . , (0, . . . , 0, 1) ∈ A n . We will define a lax monoidal ∞-natural transformation where Here Sym B (E ∨ B ) (ti) denotes the ring of degree 0 elements of the graded ring where U • is the Cech hypercover of {D + (t i )} n i=1 , it will suffice to define an homotopy coherent diagram of lax monoidal ∞-natural transformations and every i = 1, . . . , n, define a pseudo-functor where with the grading shifted by 1 and the second morphism is induced by multiplication. Notice that . , x r ], d ⊗ id, x 1 · h 1 + · · · + x r · h r ) (6.1.31) and W s B|D + (t i ) = x 1 · s B,1 + · · · + x r · s B,r .
It follows immediately from the definitions of the pseudo functorial structures of (B, Remark 6.1.11. Let X = Spec(B) and X 0 = Spec(K(B, E ∨ B , s B )). Notice that Υ i models the dg functor . Construction 6.1.12. We will now endow Υ i with a pseudo lax monoidal structure. If (B, s B ) and (C, s C ) are affine twisted LG models of rank r over (A, E), then is defined on objects by We need to verify that the two compositions It is clear that there is a canonical isomorphism of B ⊗ A C-dg modules Then we only need to show that  commutes. Here A is the dg category with one object * and End( * ) = A. The diagonal arrows are determined by * → A, The previous constructions provide us with pseudo lax monoidal natural transformations It is obvious that the Υ ⊗ i preserve quasi-isomorphisms and that they are compatible with one another. In other words, they can be used to define the homotopy coherent diagram of lax monoidal ∞-natural transformations (6.1.24).
Therefore, we get a lax monoidal ∞-natural transformation between the ∞-functors LG aff,⊠,op (A,E) → dgCat idm,⊗ S . (6.1.46) By Kan extension and descent, we extend Υ ⊗ to a lax monoidal ∞-natural transformation between (6.1.47) i.e. we extend to all twisted LG models of rank r over (A, E).
Finally, if S is not affine, we define the lax monoidal ∞-natural tranformation (this is an analogue of Lemma 3.1.9 for twisted LG models of rank r, which can be proved mutatis mutandis).
Remark 6.1.13. Let (X, s X ) be a twisted LG model of rank r over (S, E). Then ) , (6.1.52) where we have used the notation of diagram (6.1.17) Lemma 6.1.14. j : P(E ∨ X0 ) → V (W s ) is an lci morphism of derived schemes. Proof. Since the property of being lci is local, we can assume that E X ≃ O r X and s X = f ∈ O r X (X). Since j is clearly of finite presentation, we will only need to show that the relative cotangent complex is of Tor amplitude [−1, 0]. Recall that we have the fundamental fiber-cofiber sequence . We just need to identify the morphism . It coincides with the morphism (T 1 , . . . , T r ). In particular, the cofiber of this morphism, i.e.
Construction 6.1.15. Since j is an lci morphism, the dg functor (6.1.48) preserves perfect complexes. Therefore, for every (X, s X ) ∈ LG (S,ES ) , we have an induced dg functor Υ (X,sX ) := j * p * 0 : Sing(X, s X ) → Sing(P(E ∨ X ), W sX ) (6.1.54) Starting from (6.1.48), by the usual standard arguments we thus obtain a lax monoidal ∞-natural transformation Proof. This is an immediate consequence of [11,Theorem A.4]. Indeed, as the dg categories are triangulated, it suffices to show that the statement is true on the induced functor of triangulated categories. This coincides by construction with that of loc. cit.
Remark 6.1.17. We actually believe that the statement above remains true even if s X is not assumed to be regular, as long as one considers the derived zero loci instead of the classical ones.
6.2 The ℓ-adic realization of the dg category of singularities of a twisted LG model of rank r It is now easy to obtain the following computation: Theorem 6.2.1. Let (X, s X ) be a twisted LG model of rank r over (S, E S ).
Assume that X is a regular scheme and that s X is a regular global section of E X . The following equivalence holds in Mod R ℓ,∨ X (Sing(X,0)) (Shv Q ℓ (X)) is the closed embedding of the zero locus of W sX and p : P(E ∨ X ) → X is the canonical projection. Proof. As X is regular and s X is a regular section, we have an equivalence Sing(X, s X ) ≃ Sing(V (s X )). Notice that in this situation V (s X ) coincides with the underived zero locus of s X . As P(E ∨ X ) is regular, by Theorem 6.1.16, we have that Then R ℓ,∨ X (Sing(X, s X )) ≃ p * R ℓ,∨ If we pullback the first diagram along the morphism S → A 1 S given by an uniformizer, we recover the second one. It is not surprising that in this way we are only able to recover the so called tame vanishing cycles. Indeed, the wild inertia group is of arithmetic nature. One can define nearby cycles for schemes over A 1 S in the usual way: for f : • The ℓ-adic sheaf of nearby cycles of F is defined as Notice that, since µ n,S := O S [x]/(x n − 1) naturally acts on X Un , then i * j n * j * n F has a natural induced action. Then Ψ f (F) has a natural action of lim ← −n∈O × S µ n,S =: µ ∞,S .
• There is a natural morphism i * F → Ψ f (F), that we can see as a µ ∞,Sequivariant morphism if we endow i * F with the trivial action. The sheaf of vanishing cycles Φ f (F) of F is then defined as the cofiber of this morphism, with the induced µ ∞,S -action.

Tame vanishing cycles over A 1 S /G m,S
We can reproduce the situation described above for schemes over A 1 S /G m,S . In this case, the role of the zero section is played by BG m,S → A 1 S /G m,S , and that of the open complementary by S ≃ G m,S /G m,S → A 1 S /G m,S . We can consider elevation to the n th power in this case too: i 0,n (7.2.4) Notice that even if t 0,n : BG m,S → BG m,S × A 1 S /Gm,S A 1 S /G m,S is not an equivalence, it shows BG m,S × A 1 S /Gm,S A 1 S /G m,S as a nilpotent thickening of BG m,S . More explicitly, objects of BG m,S are pairs (p : X → S, L X ), while those of BG m,S × A 1 S /Gm,S A 1 S /G m,S are triplets (p : X → S, L X , s X ), where s X is a ntorsion global section of L X . What is important for us is that pullback along t 0,n induces an equivalence in étale cohomology. Moreover, Θ n is a finite morphism of stacks. In particular, the base change theorem should be valid for the cartesian squares above. However, even if the 6-functors formalism for the étale cohomology of stacks has been developed ( [37], [38], [39]), the proper base change theorem has been proved in the case of representable morphisms. The morphisms Θ n are not representable. For example, the pullback of Θ n along the canonical atlas A 1 n → A 1 S /G m,S is A 1 S /µ n . For any X-point (p : X → S, L X , s X ) of A 1 S /G m,S (e.g. for any twisted LG model over S), we can consider the following diagram, cartesian over (7.2.4)  We can then propose the following definition.
We can then consider the images of these morphisms in Shv(X 0 , Λ) Ω 0,n * (sp n : i * n Ω * n F → Ψ n (F)). (7.2.8) Consider the diagram ≃ i * n Ω n m * Ω n * m Ω * n F ≃ i * n j n * j * n Ω * n F. (7.2.10) We can then consider lim − → n∈N × Ω 0,n * i * n Ω * n F → i * 0 j 0 * j * 0 F ∈ Shv(X 0 , Λ). For Λ = Z/ℓ d Z one can then consider the induced morphism on ℓ-adic sheaves obtained by taking the limit over d and then tensor with Q ℓ . It is expected that in this way one is able to recover monodromy-invariant vanishing cycles. In order to explain way the first Chern class of the line bundle appears in the computation, it might be useful to look at the base diagram. The ℓ-adic cohomology of BG m,S is Q ℓ [c 1 ], where c 1 is the universal first Chern class and lies in degree 2. We expect that where L X0 is the line bundle which determines the morphism X 0 → BG m,S .

Some remarks on the regularity hypothesis
In the theorems about the ℓ-adic realizaiton of the dg category of a (twisted, n-dimensional) LG model the regularity assumption on the ambient scheme is crucial. Indeed, we are not able to compute the motivic realization (and hence, the ℓ-adic one) of Coh b (X 0 ) Perf (X) . However, this dg category sits in the following pullback diagram