The center of the categorified ring of differential operators

Let \Y be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study H(\Y), a monoidal DG category that might be regarded as a categorification of the ring of differential operators on \Y. When \Y = \LS_G is the derived stack of G-local systems on a smooth projective curve, we expect H(\LS_G) to act on both sides of the geometric Langlands correspondence, compatibly with the conjectural Langlands functor. Second, we construct a novel theory of D-modules on derived algebraic stacks. Contrarily to usual D-modules, this new theory, to be denoted by D^{der}, is sensitive to the derived structure. Third, we identify the Drinfeld center of H(\Y) with D^{der}(L\Y), the DG category of D^{der}-modules on the loop stack of \Y.

A natural invariant of Y is the differential graded (DG) algebra HC(A) := RHom A⊗ A (A, A) of Hochschild cochains of A. As explained later, the DG category of DG modules over HC(A), denoted H(Y ) in the sequel, carries a monoidal structure. This monoidal DG category is important in the theory of singular support of coherent sheaves on Y ; see [1,3] for the notion of singular support and [8,10,11] for the role of H(Y ). 0.1.2. Now, given any monoidal DG category A, it is natural to try to compute its Drinfeld center Z(A). In our situation, we show that the Drinfeld center of H(Y ) is equivalent to the DG category of D-modules on LY := Y × Y × Y Y , with the following two complications.
The first complication is that the fiber product computing LY must be taken in the derived sense. (In fact, the underived truncation of LY is isomorphic to Y .) Practically, this means that in the formula LY ≃ Spec(A ⊗ A⊗ A A) the tensor product over A ⊗ A must be derived. In the cases of interest, Y is (quasi-smooth but) not smooth. Then LY is extremely derived: technically speaking, LY is unbounded, which means that its DG algebra of functions A ⊗ A⊗ A A (the DG algebra of Hochschild chains of A) has infinitely many nonzero cohomology groups.
The second complication is that the theory of D-modules as understood up to now (e.g. in [18] and in [19,Volume 2,Chapter 4]) is insufficient to deal with unbounded derived schemes. To fix this, we introduce a new (and yet natural) theory of D-modules, denoted by D der , which makes the following statement true: Z(H(Y )) ≃ D der (LY ). 0.2. The main results. We now proceed to explain the above situation more organically and in higher generality.
theorem that relates the Hochschild (co)homology of the ring of differential operators on Y to the de Rham cohomology of Y , see e.g. [12]. 0.2.6. For Y quasi-smooth but not smooth, one checks that LY is unbounded (see below for the simplest example). This is the easiest situation for which the full content of Theorem A can be appreciated and it is, after all, our main case of interest. For instance, in the next section we will consider the quasi-smooth stack Y = LS G that parametrizes G-local systems over a smooth complete curve.
Example 0.2.7. Suppose Y = Spec( [ǫ]), with ǫ in cohomological degree −1. This is arguably the simplest truly derived affine scheme. It is quasi-smooth, but not smooth. By (5.8), its loop scheme LY is isomorphic to the spectrum of the graded algebra [ǫ, η], where ǫ is as before and η is a variable in (cohomological) degree −2. It follows that LY is unbounded. 0.3. The DG category H(Y) and its relatives. The origin of (relatives of) H(Y) can be traced back to the spectral gluing theorem occurring in geometric Langlands, where the categories IndCoh 0 ((LS G ) ∧ LSP ) play a crucial role, see [2]. 0.3.1. Let us explain the notations: • G denotes a connected reductive group over ; • P is one of its parabolic subgroups; • LS P (resp., LS G ) denotes the quasi-smooth stack of de Rham P -local systems (resp., G-local systems) on a smooth complete -curve X; • the map LS P → LS G used to construct the formal completion is the natural one, induced by the inclusion P ⊆ G.
Finally, and most importantly, the definition of the DG category IndCoh 0 ((LS G ) ∧ LSP ) is an instance of the following general construction, applied to the above map LS P → LS G . 0.3.2. Before explaining the construction, let us point the reader to Section 0.6 for our conventions regarding DG categories and higher category theory. The conventions regarding derived algebraic geometry, prestacks, formal completions and ind-coherent sheaves are explained in Section 2.
Definition 0.3.3. Let f : Y → Z be a map of perfect algebraic stacks with bounded Y. Recall that any quasismooth stack (e.g., LS P ) is bounded. Then we define the DG category IndCoh 0 (Z ∧ Y ) as the fiber product where the functor IndCoh(Z ∧ Y ) → IndCoh(Y) is the pull-back along the natural map ′ f : Y → Z ∧ Y , while the functor QCoh(Y) → IndCoh(Y) is the standard inclusion Υ Y , see [16]. The construction of IndCoh 0 will treated in more detail in Section 3.1.
Remark 0.3.4. The reader might have noticed an abuse of notation here: the definition of IndCoh 0 (Z ∧ Y ) really depends on the map Y → Z ∧ Y , not just on Z ∧ Y . We hope that no confusion will ever arise and refer to Section 3.1.4 for further discussion. To see this, recall first from [18] that D(Y) is defined either as QCoh(Y dR ) or as IndCoh(Y dR ), where Y dR ≃ pt ∧ Y . Also, thanks to [18,Proposition 2.4.4], we know that Υ Y dR : QCoh(Y dR ) → IndCoh(Y dR ) is an equivalence, the equivalence between left and right D-modules. Then the equivalence (0.2) follows from the fact that Υ intertwines * -pullbacks on the QCoh side with !-pullbacks on the IndCoh side. In general, IndCoh 0 (Z ∧ Y ) is the "correct" way to define the DG category of relative left D-modules with respect to Y → Z.
Remark 0.3.7. For bounded Y, the inclusion Υ Y admits a continuous right adjoint, denoted by Φ Y in this paper. 0. 3.8. Without any extra assumptions on the map f : Y → Z, it is very difficult to get a handle of IndCoh 0 (Z ∧ Y ): for example, it is not clear whether this DG category is compactly generated and it is very difficult to exhibit compact objects. As discussed in Proposition 2.5.2 and Section 3.1.6, the situation simplifies as soon as we restrict to perfect stacks that are locally of finite presentation (lfp), that is, with perfect cotangent complex. 4 Note that any quasi-smooth stack tautologically has this property. This condition guarantees that each IndCoh 0 (Z ∧ Y ) has pleasant features: it is compactly generated, self-dual and equipped with a monadic adjunction , Here, U(T Y/Z ) is the universal envelope of the Lie algebroid T Y/Z → T Y , which is, by definition, the monad Remark 0.3.9. In this paper, we use the notation U QCoh (T Y/Z ) for the monad Φ Y • U(T Y/Z ) • Υ Y . Note that parallel between (0.4) and the formula featuring in (2.8).
The tangent complex appears in this context as a consequence of (a variant of) the equivalence between formal moduli problems and DG Lie algebras. For the latter equivalence, see [21] and references therein. 0.3.10. The assignment IndCoh 0 (Z ∧ Y ) enjoys functorialities of two kinds: • (∞, 1)-categorical functorialities, where we consider IndCoh 0 (Z ∧ Y ) as a mere DG category; • (∞, 2)-categorical functorialities, where we consider IndCoh 0 (Z ∧ Y ) as a left module category for H(Z), see below.
In this paper we treat the first item, leaving the second item to [8]. However, we are tacitly preparing ourselves for the (∞, 2)-categorical part of the theory, as we will be very much concerned with the study of the monoidal DG category The monoidal structure on H(Y) is the one given by convolution, inherited by the standard convolution structure on IndCoh((Y × Y) ∧ Y ). Since Y is perfect, H(Y) is compactly generated and rigid 5 . By contrast, IndCoh((Y × Y) ∧ Y ) is not rigid in general (although it is compactly generated).
Remark 0.3.11. The adjuction (0.3) yields in this case a monoidal functor with continuous conservative right adjoint. In particular, H(Y) is the DG category of modules for the inertia  • it is easy to see that H(Y) acts by convolution on IndCoh(Y), on QCoh(Y) and on the category of singularities IndCoh(Y)/QCoh(Y); • more generally, for X → Y a map of stacks as above, H(Y) acts on IndCoh 0 (Y ∧ X ) and on IndCoh(Y ∧ X ), again by convolution; • If Y is quasi-smooth, the H(Y)-action on IndCoh(Y) preserves any subcategory of IndCoh(Y) cut out by a singular support condition, see [1]. 0.4.4. Digression on "geometric Langlands". A much less trivial example of an H(Y)-action is given by the following. Referring to Section 0.3.1 for the notation, let Bun G denote the stack of G-bundles on X and by G the Langlands dual group of G.
We claim that H(LSǦ) acts on D(Bun G ) via Hecke operators. This action and the explanation of the terminology, i.e. the connection with derived Satake, is addressed in [8,Section 1.4]. Here we just mention that the datum of such action proves almost immediately the conjecture about tempered D-modules formulated in [1,Section 12]. 5 See Section 0.6.4 or [14, Section 6.1] for the meaning of the adjective "rigid". 0.5. The center of the monoidal DG category H(Y). Let us come back to the contents of the present paper. After having studied the functoriality of the assignment IndCoh 0 , we turn to the computation of the Drinfeld center Z(H(Y)) of H(Y). By definition, Z(H(Y)) is the DG category In the analogy of Section 0.4.2, one may suggest that Z(H(Y)) is the categorifcation of the center of the ring of differential operators on Y.
Remark 0.5.1. As pointed out later in Remark 5.1.6, it turns out that the DG category underlying Z(H(Y)) is canonically equivalent to the trace of H(Y), namely the DG category defined by 0.5.2. We believe that the computation of Z(H(Y)) is interesting in its own right. However, we were brought to it by the need to make sure that the monoidal functor factors through Z(H(Y)). In other words, we wanted to construct a functor making the following diagram commutative: Here, the functor oblv L is the "left forgetful" functor from D-modules to quasi-coherent sheaves, see [18], while ev is the tautological functor that "forgets the central structure", that is, the evaluation functor where ⋆ denotes the monoidal structure of H(Y) and are the two projections forming the infinitesimal groupoid of Y. Hence, a "homotopically coherent" identification that is, a left crystal structure on Q, promotes ∆ * ,0 (Q) to an object of the center of H(Y). 0.5.5. Rather than turning this argument into a proof, we will first compute the full center Z(H(Y)) in geometric terms and then exhibit the natural map ζ from D(Y). In this paper we only perform the former task, leaving the latter to a sequel. Let us however anticipate that ζ is the pushforward functor D(Y) ≃ D der (Y) → D der (LY) in the theory of D der -modules along the inclusion ι : Y → LY.
0.6. Conventions. Our conventions on higher category theory and derived algebraic geometry follow those of [19]. Let us recall the most important ones. The reader might also consult [14] for a brief digest. 0.6.1. Throughout the paper the term "DG category" means "stable presentable -linear ∞-category", in the sense of [20]. Unless otherwise stated, our DG categories are cocomplete and functors between them are colimit preserving. In other words, we work within the ∞-category DGCat of cocomplete DG categories and continuous functors. Such ∞-category is symmetric monoidal when equipped with the Lurie tensor product defined in [20]. 0.6.2. Given a DG category C as above and two objects c, c ′ ∈ C, we denote by Hom C (c, c ′ ) the DG vector space of morphisms from c to c ′ .
If C ′ is another DG category, the symbol Fun(C, C ′ ) denotes the DG category of continuous functors from C to C ′ . We also set End(C) := Fun(C, C). 0.6.3. We assume familiarity with the notion of dualizability for objects in a symmetric monoidal ∞category, as well as with the notion of ind-completion and compact generation for DG categories. Recall that a compactly generated DG category is automatically dualizable. Given C ∈ DGCat, we denote by C cpt its non-cocomplete full subcategory of compact objects.  D-modules (that is, we never refer to the abelian categories). Accordingly, for A a DG algebra, the notation Amod stands for the DG category of DG modules over A.
The pushforward and pullback functors between DG categories of sheaves are always understood in the derived sense. Fiber products and tensor products are always derived, too. 0.6.8. The ∞-category of ∞-groupoids is denoted by Grpd ∞ . The other main notations of (derived) algebraic geometry are reviewed in Section 2 and especially in Section 2.1. 0.7. Contents of the paper. In the first section, we give an overview of the computation of Z(H(Y)) to explain how D der (LY) comes about. Then, in the second section, we review the bit of algebraic geometry that we need and discuss ind-coherent sheaves on our prestacks of interest: algebraic stacks and formal completions thereof. In the third section, we define the DG categories IndCoh 0 (Z ∧ Y ) in the bounded case (that is, when Y is bounded), and we extend the assignment [Y → Z] IndCoh 0 (Z ∧ Y ) to a functor out of a category of correspondences. In the fourth section, we extend the definition of IndCoh 0 (Z ∧ Y ) to the case when Y is possibly unbounded. In this context, IndCoh 0 (Z ∧ Y ) lacks some of the pleasant features present in the bounded case; we discuss the features that do generalize. Lastly, in the fifth section, we apply the theory developed in the previous sections to identify the center of H(Y) with D der (LY).

Outline of the center computation
As anticipated in Theorem A, the center Z(H(Y)) is (slightly incorrectly!) equivalent to the category of D-modules on LY, the loop stack of Y. Such answer is literally correct whenever LY is bounded, but should otherwise be modified as explained below (from Section 1.2 on). In Section 1.1, we show how to guess this incorrect answer; this will also give hints as to how correct it, which we take up in Section 1.2.
1.1. Computing the center. Let us get acquainted with H(Y) by explaining a natural approach to computing its Drinfeld center. Recall that Y is a quasi-compact derived algebraic stack that is perfect, bounded and with perfect cotangent complex. We usually denote by m : H(Y) ⊗ H(Y) → H(Y) the multiplication functor and by m rev the reversed multiplication (obtained from m by swapping the two factors).
1.1.1. Since H(Y) is rigid, the conservative functor ev admits a left adjoint ev L , whence Z := Z(H(Y)) is equivalent to the category of modules for the monad ev • ev L acting on H(Y). Moreover, thanks to the pivotality of H(Y) discussed in Section 3.5, the functor underlying ev • ev L is naturally isomorphic to To understand the composition m rev • m R , we first need to understand the multiplication m more explicitly. For this, it is convenient to anticipate some of the functoriality on IndCoh 0 that we will develop. To start, note that the assignment where Stk perf ,lfp is the ∞-category of perfect stacks locally of finite presentation and Arr(Stk perf ,lfp ) ′ is the full subcategory of Arr(Stk perf ,lfp ) := Fun(∆ 1 , Stk perf ,lfp ) spanned by those arrows Y → Z with bounded Y.
1.1.3. We denote by φ !,0 the structure functor By the theory sketched above, both arrows possess continuous right adjoints, whence m R is the continuous 1.1.5. To compute m rev • m R , we will resort to the horocycle diagram (see [5]) for the map Y → Y dR . In general, the horocycle diagram attached to a map Y → Z is the following commutative diagram with cartesian squares: p23 × p12 p13 1.1.6. Applied to the case Z = Y dR , and using the tautological isomorphisms where we have set P := p 12 × p 23 and P ′ = p 23 × p 12 to save space. Observe that m rev is the composition 1.1.7. Notation. We have denoted by LY := Y × Y×Y Y the loop stack of Y: the fiber product of the diagonal There are two standard maps: the insertion of "constant loops" ι : Y → LY and the projection π : LY → Y.
1.1.9. As an application of the functoriality of IndCoh 0 (specifically, the base-change isomorphisms established in Sections 3 and 4), one easily proves that these four squares are commutative. It follows that the monad ev • ev L ≃ m rev • m R is isomorphic (as a plain functor) to the monad of the adjunction We emphasize again that IndCoh 0 (pt ∧ LY ) ≃ D(LY), with IndCoh 0 (pt ∧ LY ) being well-defined thanks to the boundedness of LY. Remark 1.1.11. The center of a monoidal DG category comes equipped with a monoidal structure. In the case at hand, the monoidal structure on D(LY) is the one induced by composition of loops, that is, by the correspondence We will not use such monoidal structure in this paper and therefore do not discuss it further.
Example 1.1.12. As an example of the above computation, consider the case where Y = BG, the classifying stack of an affine algebraic group G. Then LY is isomorphic to the adjoint quotient G/G, which is bounded (in fact, smooth). By [7, Section 2], we know that is the monoidal DG category of Harish-Chandra bimodules for G. The theorem states that its center is equivalent to D(G/G).

1.2.
Beyond the bounded case. The issue with the above argument leading to Z(H(Y)) ≃ D(LY) is that boundedness of LY is rare (for instance, see Remark 1.2.7 below) and, for LY unbounded, the entire bottom-right square of (1.1) makes no sense.
To remedy this, we need to search for an extension of the definition of IndCoh 0 (Z ∧ X ) to the case of unbounded X. Such definition must come with functors making the four squares of (1.1) commutative: then the above argument would go through and would show that the center of H(Y) is equivalent to IndCoh 0 (pt ∧ LY ), whatever the latter means.

1.2.1.
To concoct this more general definition, we will try to adapt (0.4), that is, we will try to define IndCoh 0 (Z ∧ X ) as the DG category of modules over a monad acting on QCoh(X).
The most naive attempt is to take the same formula as in (0.4); indeed, the expression Φ X • U(T X/Z ) • Υ X still makes sense as a monad. This attempt fails, however, as such monad is discontinuous in general (indeed Φ X is continuous iff X is bounded).

1.2.2.
To fix such discontinuity, we could restrict the functor in question to Perf(X), and then ind-complete. Let us denote the resulting (continuous) functor by We claim that such definition is not the right one either. To see this, look at the result of this operation in the case where T X/Z is an abelian Lie algebra in IndCoh(X), so that In such simple case, we expect our monad to be the functor of tensoring with the symmetric algebra of T QCoh X/Z . What we get instead is the functor of tensoring with This object is the convergent renormalization 6 of Sym(T QCoh X/Z ), which is different from Sym(T QCoh X/Z ) as soon as the latter is not bounded above in the t-structure of QCoh(X). Working with such convergent renormalizations is not pleasant: in fact, all the base-change results that we need fail.

1.2.3.
We can however turn the above failure into a positive observation. Note that Sym(T QCoh X/Z ) and Sym(T X/Z ) are filtered by Sym ≤n (T QCoh X/Z ) and Sym ≤n (T X/Z ), respectively. Since Sym ≤n (T QCoh X/Z ) is perfect for each n (in particular, bounded above), the renormalization procedure (1.3) applied to yields precisely the functor We thus have The general situation is analogous, thanks to the existence of a canonical filtration of U(T X/Z ), the PBW filtration, which specializes to the above in the case of abelian Lie algebras. See [19, Volume 2, Chapter 9, Section 6]. Rather than renormalizing U(T X/Z ) itself, we renormalize each piece of the filtration and then put them together. In symbols, we define is the only continuous functor whose restriction to Perf(X) is given by Φ X U(T X/Z ) ≤n Υ X .
1.2.5. In Section 4.1, we will prove that U QCoh (T X/Z ) comes equipped with the structure of a monad on QCoh(X). This allows to extend the definition of IndCoh 0 to the case of X unbounded as see Definition 4.1.6 where this is done officially. In the later parts of Section 4, we show that the assignment possesses all the functorialities that we need for the computation of Z(H(Y)). Specifically, our main Theorems 5.2.3 and 5.3.4 will assert that sitting in a monadic adjunction H(Y) ⇄ Z(H(Y)) defined exactly as in (1.2).
1.2.6. Let us comment on the relationship between D(LY) and D der (LY) in the general case. As shown in Section 4.2.7, there always exists a tautological functor which can be regarded as a passage from left to right D-modules in our setting. Contrarily to the usual setting, this functor is not an equivalence in general (see Corollary 4.3.13 for an example).

Ind-coherent sheaves on formal completions
This section is devoted to recalling the theory of ind-coherent sheaves. We are particularly interested in ind-coherent sheaves on formal completions of perfect stacks. The main references are [16] and [19].
2.1. Some notions of derived algebraic geometry. We shall often consider affine schemes that are both bounded and almost of finite type: we denote by Aff <∞ aft the ∞-category they form. Denote by Sch aft the ∞-category of quasi-compact (DG) schemes almost of finite type. Such a scheme S ∈ Sch aft is bounded if it is Zariski locally so (hence we reduce to the definition for affine schemes).
Moreover, S is the colimit of its truncations S ≤n , as n → ∞.

A prestack is an arbitrary functor
Denote by PreStk the ∞-category of prestacks. Important for us is the subcategory PreStk laft of prestacks that are locally almost of finite type (laft), see [19, Volume 1, Chapter 2, Section 1.7]. Rather than the actual definition, what we need to know about PreStk laft are its following properties: • it is closed under fiber products; • it is closed under the operation Y Y dR (the de Rham prestack of Y); • it contains all perfect stacks (see below).
Remark 2.1.4. The point of the condition laft is that PreStk laft is equivalent to the ∞-category of arbitrary functors from (Aff <∞ aft ) op to Grpd ∞ .
2.1.5. Algebraic stacks. We will be quite restrictive on the kinds of stacks that we deal with. Namely, we denote by Stk ⊂ PreStk the full subcategory consisting of those (quasi-compact) algebraic stacks with affine diagonal and with an atlas in Aff aft . We will just call them stacks.
2.1.6. We say that Y ∈ Stk is bounded if for some (equivalently: any) atlas U → Y, the affine scheme U is bounded. Denote by Stk <∞ ⊂ Stk the full subcategory of bounded stacks. It is closed under products, but not under fiber products. We say that a map Y → Z in Stk is bounded if, for any S ∈ (Aff <∞ aft ) /Z , the fiber product S × Z Y belongs to Stk <∞ .
Following [4], we say that Y ∈ Stk is perfect if the DG category QCoh(Y) is compactly generated by its subcategory Perf(Y) of perfect objects.
We say that Y ∈ Stk is locally finitely presented (lfp) if its cotangent complex L Y ∈ QCoh(Y) is perfect.
In that case, we denote by T QCoh Y ∈ Perf(Y) its monoidal dual.
We denote by Stk <∞ perf ,lfp ⊆ Stk the full subcategory of stacks that are perfect, bounded and locally of finite presentation. Similarly, the notations Stk perf ,lfp and Stk perf have the evident meaning. By [4,Proposition 3.24], Stk perf is closed under fiber products (this is because our stacks have affine diagonal by assumption).

2.2.
Ind-coherent sheaves on schemes. This section is a recapitulation of parts of [16], [19] and [13]. It is included for the reader's convenience and to fix the notation.
2.2.1. For a scheme S ∈ Sch aft , consider the non-cocomplete DG category Coh(S), the DG category of cohomologically bounded complexes with coherent cohomology. We define IndCoh(S) := Ind(Coh(S)) to be its ind-completion. The latter comes equipped with an action of QCoh(S) and a tautological QCoh(S)-linear functor Ψ S : IndCoh(S) → QCoh(S). Indeed, for Perf(S) to be contained in Coh(S), the structure sheaf has to be bounded. (When Ξ S exists, it is automatically QCoh(S)-linear.) Thus, for bounded schemes, IndCoh is an enlargement of QCoh; more precisely, Ψ is a colocalization.
Remark 2.2.4. For unbounded schemes, the situation is unwieldy. For instance, consider the scheme S = Spec(Sym V * [2]), with V a finite dimensional ordinary vector space over . In this case, Ψ S is fully faithful (but not an equivalence): indeed, the augmentation module Karoubi generates Coh(S) and it is perfect by Example 2.2.7 below.

The assignment S
IndCoh(S) underlies an (∞, 2)-functor where DGCat 2−Cat denotes the (∞, 2)-category of DG categories and the notation Corr(C) adm vert ;horiz is taken from [19, Volume 1, Chapter 7]. In any case, the above (∞, 2)-functor is a fancy way to encode the following data: • for any map f : S → T in Sch aft , we have a push-forward functor f IndCoh * The latter admits a continuous right adjoint if and only if ω S ∈ Coh(S), which in turn is equivalent to S being bounded. Since such right adjoint does not have a notation in the original paper [16], we shall call it , with V a nonzero finite dimensional ordinary vector space and n ≥ 1. Then tautologically. On the other hand, we have the Koszul duality equivalence Indeed, Coh(Y n ) is generated by a single object, the augmentation module, and a standard Koszul resolution yields Under (2.1) and (2.2), the functor Υ Yn is the tensor product with the augmentation , the latter viewed as 2.2.8. As in [1, Lemma F. 5.8], one shows that the composition Φ S • Υ S is the functor of convergent renormalization, computed explicitly as follows: where i n : S ≤n ֒→ S is the inclusion of the n-connective truncation of S. This shows that Υ S is fully faithful when S is bounded: indeed, in that case S ≃ S ≤n for some n and the limit stabilizes.
2.2.9. More generally, regardless of whether S is bounded or not, the unit of the adjunction M → M conv is the identity on Perf(S) and more generally on QCoh(S) − (the full subcategory of QCoh(S) consisting of objects bounded from above in the usual t-structure on QCoh(S)). In particular, whether S is bounded or not, we can consider Υ S (Perf(S)) and Υ S (QCoh(S) − ) as (non-cocomplete) full subcategories of IndCoh(S).

Consider instead the functor
It is shown in [16, Lemma 9.5.5] that the above yields an involutive equivalence which is the usual Serre duality. Such equivalence exhibits IndCoh(S) as its own dual.
2.2.14. For bounded S, it is easy to see that D Serre S exhanges the two subcategories Perf(S) ⊆ Coh(S) and where D QCoh S is the standard duality involution on Perf(S).
Remark 2.2.15. The Serre duality isomorphism holds true slightly more generally. Indeed, observe that D Serre S := Hom QCoh(S) (−, ω S ) extends to a contravariant functor of IndCoh(S) that sends colimits to limits. Then it is easy to check that (2.3) is valid for N ∈ Coh(S) and M ∈ IndCoh(S) arbitrary.

Recall that a laft prestacks are by definition presheaves of spaces on Aff <∞
aft . Ind-coherent sheaves are defined for arbitrary laft prestacks: one simply right-Kan extends the functor along Aff <∞ aft ֒→ PreStk laft . In particular, for any Y ∈ PreStk laft , the !-pullback along Y → pt yields a canonical object ω Y ∈ IndCoh(Y). Since, as in the case of schemes, IndCoh(Y) admits an action of QCoh(Y), we have the canonical functor corresponding to the action on ω Y .

2.3.2.
Let us now discuss ind-coherent sheaves on stacks (recall our convention of the term "stack": our stacks are all algebraic, quasi-compact, with affine diagonal, and laft).
Translated into plain language, the theorem states that: • ( * , IndCoh)-push-forwards are defined only for ind-inf-schematic maps and have base-change isomorphisms against !-pullbacks; • if f ind-inf-schematic and ind-proper, then f IndCoh * is left adjoint to f ! .
2.4.3. Luckily, in this paper we do not need such high level of generality (whence, we will not need to recall the definitions of those words). We only need to be aware of the following fact: if X → Y → Z is a string in Stk with X → Y schematic (and proper), then the resulting map Z ∧ X → Z ∧ Y is ind-inf-schematic (and ind-proper). This yields: in Arr(Stk) is said to be schematic (or proper) if so is the map Proof. The only thing to check is the 1-categorical composition of correspondences. This boils down to the following fact (whose proof, left to the reader, is a diagram chase): for a cospan in Arr(PreStk), the resulting map is an isomorphism. Proof. This (∞, 2)-functor is the composition of (2.5) with (2.4). 2.5. Nil-isomorphisms and self-duality.
2.5.1. A map of laft prestacks is said to be a nil-isomorphism if it is an isomorphism at the reduced level. If a map is a nil-isomorphism, then the resulting IndCoh-pullback is conservative, see [19, Volume 2, Chapter 3, Proposition 3.1.2].
As a main example consider the following: for f : Y → Z a map in Stk, the natural map ′ f : Y → Z ∧ Y is a nil-isomorphism. Thus, we obtain the following statement.
Recall that a compactly generated DG category is automatically dualizable (for a proof, see [14, Proposition 2.3.1]). The next result describes the dual of IndCoh(Z ∧ Y ).
Proposition 2.5.4. In the situation above, the DG category IndCoh(Z ∧ Y ), which is automatically dualizable by the above proposition, is self-dual.
Proof. We will exhibit two functors and prove they form a self-duality datum for IndCoh(Z ∧ Y ). We set: coev : Vect where the second functor is continuous as ∆ : (since Y has schematic diagonal) and the last equivalence holds because IndCoh(Z ∧ Y ) is dualizable. As for the functor going the opposite direction, we set: where π : Z ∧ Y → Y dR is the tautological inf-schematic map and Γ(Y dR , −) ren is the functor of renormalized de Rham global sections, see [13]. By definition, Γ(Y dR , −) ren is the dual of (p Y dR ) ! under the standard self-duality of D(Y) and Vect.
After a straightforward diagram chase, proving that these two functors yield a self-duality datum boils down to proving that the functor is the identity. It suffices to check this smooth-locally on Z. Then we can assume that Z = Z is a scheme, in which case and the assertion is obvious (the functor in question being dual to the identity). To prove (2.9), first use the dualizability of IndCoh(Z) as a D(Z)-module (proven in [

IndCoh 0 in the bounded case
We start this section by officially defining the DG category IndCoh 0 (Y → Z ∧ Y ) attached to a map of stacks Y → Z, with Y ∈ Stk <∞ . A crucial condition to make this DG category manageable is the perfection of the relative cotangent complex L Y/Z . Another useful condition to impose is the perfection of Y itself: as we show below, this makes IndCoh 0 (Y → Z ∧ Y ) compactly generated. Thus, for simplicity, we will restrict our attention to stacks that are bounded perfect and lfp. It will then be immediately clear that the assignment We shall extend such functor to an (∞, 2)-functor out of an (∞, 2)-category of correspondences, see (3.9).
We will also discuss descent for IndCoh 0 , as well as its behaviour under tensoring up over QCoh.
3.1. Definition and first properties. In this section, we define the DG category IndCoh 0 (Y → Z ∧ Y ) attached to the nil-isomorphism ′ f : Y → Z ∧ Y and discuss some generalities.
as the DG category sitting in the pull-back square The definition is taken from [2], with the proviso that [2] assumed quasi-smoothness of the stacks involved and used the functor Ξ Y in place of Υ Y (those two functors differ by a shifted line bundle in the quasi-smooth case).
. In other words, ι is the inclusion of a symmetric monoidal subcategory.
3.1.4. Warning. We will abuse notation and write IndCoh 0 (Z ∧ Y ) instead of the more precise IndCoh 0 (Y → Z ∧ Y ). Observe that the latter category really depends on the formal moduli problem Y → Z ∧ Y , and not just on Z ∧ Y : indeed, Z ∧ Y is insensitive to any derived or non-reduced structure on Y, while IndCoh 0 (Y → Z ∧ Y ) is not.
is actually commutative.
Proof. We just need to verify that the functor preserves the subcategory Υ Y (QCoh(Y)). We proceed as in [ suffices to check the assertion for each n th associated graded piece. Since L Y/Z is perfect, the latter is the functor of tensoring with 3.1.6. Let us assume, as in the above proposition, that f : Y → Z is a map in Stk with Y bounded and with L Y/Z perfect. Since ( ′ f ) !,0 is continuous and conservative, the monadic adjunction .
This monadic description implies the compact generation of IndCoh 0 (Z ∧ Y ) as follows.
Corollary 3.1.7. With the notation above, assume furthermore that Y is perfect. Then the DG category IndCoh 0 (Z ∧ Y ) is compactly generated by objects of the form 3) by changing the vertical arrows with their right adjoints, is commutative. Checking this boils down to proving that ι R sends ( ′ f ) IndCoh * . This is a simple computation, which uses the fact that Φ Y is QCoh(Y)-linear.

3.2.
Duality. Let Y → Z be a morphism in Stk, with Y bounded and perfect. Assume also that L Y/Z perfect. We show that the DG category IndCoh 0 (Z ∧ Y ) is naturally self-dual.

Denote by IndCoh
(Ξ) 0 (Z ∧ Y ) the DG category defined as in diagram (3.2), but with the inclusion Ξ Y in place of Υ Y . Reasoning as above, we see that there is a monadic adjunction .
Proof. Both categories are compactly generated, hence dualizable. Furthermore, they are retracts of the dualizable DG category IndCoh(Z ∧ Y ) by means of the right adjoints of the structure inclusions Observe that these right adjoints are continuous in view of Remark 3.1.8. Now, using the self-duality of IndCoh(Z ∧ Y ) proven in Proposition 2.5.4, we see that the dual of IndCoh 0 (Z ∧ Y ) is the full subcategory of IndCoh(Z ∧ Y ) consisting of those objects F for which the natural arrow (ι R ) ∨ ι ∨ (F) → F is an isomorphism. This happens if and only if is an isomorphism for any C ∈ Coh(Y), which in turn is equivalent to In other words, ( ′ f ) ! F must belong to Ξ Y (QCoh(Y)), which means precisely that F ∈ IndCoh Remark 3.2.3. One easily checks that the dual of the inclusion ι : Proposition 3.2.4. In the situation above, there exists an equivalence σ : IndCoh that renders the triangle commutative. In particular, IndCoh 0 (Z ∧ Y ) is self-dual in the only way that makes ( ′ f ) * ,0 and ( ′ f ) !,0 dual to each other.
Proof. The adjunction (3.6) is monadic, and the monad is easily seen to coincide with the one of the monadic adjunction (3.4): indeed, it suffices to show that each functor U ≤n (T Y/Z ) preserves the subcategory Ξ(Perf(Y)) ⊆ IndCoh(Y), which is immediately checked at the level of the associated graded.
This fact implies the existence of the equivalence σ fitting in the above triangle. As for the duality statement, let us compute the evaluation between f * ,0 (Q) and an arbitrary F ∈ IndCoh 0 (Z ∧ Y ). We have as claimed.
3.3. Functoriality. The results of the previous sections show that the DG category IndCoh 0 (Z ∧ Y ) is particularly well-behaved in the case Y ∈ Stk <∞ perf and the relative contangent complex L Y/Z is perfect. Hence, it makes sense to restrict IndCoh 0 to arrows Y → Z in Stk <∞ perf ,lfp . In this section, we upgrade the functor to a functor out a certain category of correspondences of Arr(Stk <∞ perf ,lfp ). To do so, we shall reduce the question to the functoriality of , which is known thanks to Theorem 2.4.2.
3.3.1. To simplify the notation, denote by Arr := Arr(Stk <∞ perf ,lfp ) the ∞-category of arrows in Stk <∞ perf ,lfp . A 1-morphism in Arr, say between [Y 1 → Z 1 ] and [Y 2 → Z 2 ], is represented by a commutative diagram where, by convention, objects of Arr are always drawn as vertical arrows. is well defined. We will show that the assignment [Y → Z] IndCoh 0 (Z ∧ Y ) can be upgraded to an (∞, 2)functor out of the above (∞, 2)-category of correspondences, with pushforward functor directly induced by the ( * , IndCoh)-pushforward. More precisely, we will establish the following theorem. Proof. It is clear that !-pullbacks always preserve the IndCoh 0 subcategories. It remains to check that, for a diagram (3.8) with schematic and bounded top arrow, the IndCoh-pushforward functor

Note that
preserves the IndCoh 0 -subcategories. We can write the map (Z 1 ) ∧ Y1 → (Z 2 ) ∧ Y2 as the composition and analyze the two resulting functors α IndCoh * and β IndCoh * separately. Let us show that α IndCoh * preserves the IndCoh 0 -subcategories. As we know by Corollary 3.1.7, the DG category IndCoh 0 ((Z 1 ) ∧ Y1 ) is generated under colimits by the essential image of the induction map and α IndCoh * obviously sends each such generator to an object of IndCoh 0 ((Z 2 ) ∧ Y1 ). It remains to discuss the pushforward β IndCoh * along the rightmost map in (3.10). The question is settled by the following more general result.
Proof. It suffices to check that the image of the functor Since the question is smooth local in Y, we may pullback to an atlas of Y, thereby reducing the assertion to the case of X = X and Y = Y schemes. We need to prove that the natural transformation is an isomorphism. Passing to duals, this is equivalent to showing that is an isomorphism for any F ∈ Coh(Y ). We now use [ The conclusion follows from the fully faithfulness of Ξ Y .

Let
be a morphism in Arr as in (3.8), which is schematic and bounded.
We denote by ) the push-forward functor of the above theorem. Such notation matches the usage of the ( * , 0)-pushforwards that appeared earlier in the text. Indeed, if ξ is proper, ξ * ,0 is left adjoint to ξ !,0 .
3.3.6. Let us spell out the base-change isomorphism for IndCoh 0 stated in Theorem 3.3.3. A pair of maps in Arr corresponds to a commutative diagram in Arr, provided that f is bounded (so that W× Y U is bounded too). The theorem states that, if f is moreover schematic, the diagram is naturally commutative.
3.3.7. We now use the above functoriality to prove descent of IndCoh 0 "in the second variable".
satisfies descent along any map.
Proof. Let W → Y → Z be a string in Stk <∞ lfp , giving rise to a "nil-isomorphism" ξ :
Proposition 3.4.1. In the above situation, assume furthermore that U × Z X is bounded. Then the exterior tensor product descends to an equivalence Proof. Both DG categories are modules for monads acting on QCoh(U × Z X). Note that QCoh(U × Z X) is generated by objects of the form p * P ⊗ q * Q for P ∈ QCoh(U) and Q ∈ QCoh(X), where p : U × Z X → U and q : U × Z X → X are the two projections. We will identify the values of the two monads acting on such generators.
The monad on the LHS is given by while the monad on the RHS by

Now, the elementary isomorphism
taking place in QCoh(U × Z X), yields the assertion upon dualization.
3.4.2. As a consequence of the above exterior product formula, we obtain another kind of functor, the ?-pushforward, for IndCoh 0 . To construct it, consider maps X → Z ← Y in Stk <∞ perf ,lfp , with the property that X × Z Y is also bounded. We view the resulting cartesian diagram in Arr = Arr(Stk <∞ perf ,lfp ). Then the equivalence of Proposition 3.4.1, together with the usual adjunction h * : QCoh(Z) ⇄ QCoh(X) : h * , yields the adjunction 3.5. The monoidal category H(Y). In this short section, we officially introduce the main object of this paper: the monoidal category H(Y) attached to Y ∈ Stk <∞ lfp .
3.5.1. Let Y be as above and recall the convolution monoidal structure on IndCoh(Y × Y dR Y) defined by pull-push along the correspondence It is clear that is the inclusion of a monoidal sub-category: indeed, it suffices to show that convolution preserves the sub- , which is a simple diagram chase.

The same reasoning shows that the functor
is monoidal: indeed, it can be written as the composition of monoidal functors.
with the monoidal structure given by convolution, that is, the product induced by the correspondence Proof. Compact generation and rigidity follow immediately from the fact that is monoidal and generates the target under colimits. As for pivotality, we need to show that left and right duals of compact objects can be functorially identified. Note that H(Y) is self-dual in two different looking ways. The first way is a consequence of rigidity: any rigid monoidal DG category is self-dual with evaluation given by u R • m. In our case, this reads as The second self-duality datum comes from the general theory of IndCoh 0 , as proven in Section 3.2: it determines a second evaluation functor that we will denote by ev ⊗ . In view of [20, Lemma 4.6.1.10], the two duality data are canonically identified: ev ⋆ ≃ ev ⊗ . Denote by D the contravariant involution on H(Y) cpt induced by ev ⊗ (explicitly, D is the composition of Serre duality for IndCoh(Y × Y dR Y) and σ −1 ). Then, for It follows that ∨ F ≃ D(F) naturally: this fact yields the desired pivotal structure.

Beyond the bounded case
The computation of Z(H(Y)) sketched in Section 1 showed the need to extend the definition of IndCoh 0 (Z ∧ Y ) to the case where Y ∈ Stk lfp is not necessarily bounded, but still perfect. This task is the goal of the present section.
Inspired by the equivalence (0.4), we will define IndCoh 0 (Z ∧ Y ) as the DG category of modules for a monad U QCoh (T Y/Z ) acting on QCoh(Y), where U QCoh (T Y/Z ) is defined so that: • its meaning coincides with the already established one when Y is bounded; • it is equivalent to U QCoh (T QCoh Y/Z ) ⊗ −, when T Y/Z is a Lie algebra. To define such monad, we will use the PBW filtration of the universal envelope of a Lie algebroid.
After this is done, we will discuss the functoriality of IndCoh 0 in the unbounded context. Such functoriality is not as rich as the one discussed in the previous chapter, the issue being that we can no longer rely on the functoriality of ind-coherent sheaves on formal completions. For instance, in the present context, the ( * , 0)-pushforward will be defined only for maps in Arr(Stk perf ,lfp ) coming from diagrams of the form Similarly, the (!, 0)-pullback will be defined only for maps with cartesian associated square. These two kinds of functors are "good" because they preserve compact objects, whence they admit continuous right adjoints: the so-called ?-pullback and ?-pushforward, respectively.
We also discuss descent and tensoring up with QCoh. The material of this section is essential to carry out the computation Z(H(Y)) ≃ D der (LY), performed in Section 5.
4.1. The definition. Consider a Lie algebroid L on Y and its universal envelope U(L), which is a monad acting on IndCoh(Y). By [19, Volume 2, Chapter 9, Section 6.1], the assignment L U(L) upgrades to a functor The target ∞-category will be referred to as the ∞-category of monads (acting on IndCoh(Y)) with nonnegative filtration.
We will assume familiarity with the Rees-Simpson point of view that filtered objects are objects that lie over the stack A 1 /G m : we refer to [19, Volume 2, Chapter 9, Section 1.3] for the main results in the subject and for the notation. 4.1.1. Let L ∈ Lie-algbd(Y) be such that its underlying ind-coherent sheaf oblv Lie-algbd (L) belongs to the subcategory Υ Y (Perf(Y)) ⊂ IndCoh(Y). In this situation, we will show that the monad U(L) induces a canonical monad acting on QCoh(Y), denoted U QCoh (L). We need the following paradigm, whch goes under the slogan: the filtered renormalization of a filtered monad is also a filtered monad. Lemma 4.1.2. Let µ be a monad with non-negative filtration acting on a (cocomplete) DG category C. Let C 0 ⊂ C be a non-cocomplete subcategory with the property that, for each n ≥ 0, the n th piece of the filtration µ ≤n preserves C 0 . 8 Let Ind(C 0 ) be the ind-completion of C 0 and µ ≤n the ind-completion of the functor µ ≤n : C 0 → C 0 . Then the non-negatively filtered functor admits a canonical structure of monad with filtration.
Proof. Consider the DG category End(C) Fil := Fun(Z, End(C)), 8 Note that we do not require that µ have this property. equipped with the monoidal structure is given by Day convolution. A monad with filtration on C (for instance, µ) is an algebra object of the above DG category. Let us express End(C) Fil using the stack A 1 /G m . By [19, Volume 2, Chapter 9, Section 1.3], Since QCoh(A 1 /G m ) is dualizable (and self-dual), we further obtain that in such a way that the Day convolution on the LHS corresponds to the obvious monoidal structure on the RHS. Now, denote by C ′ 0 the Karoubi completion of C 0 . The assumption on µ means that its restriction along C ′ 0 ⊗ Perf(A 1 /G m ) gives rise to an (automatically algebra) object µ ′ of End non-cocomplete In the above formula, contrarily to our usage, we have considered exact endofunctors of the non-cocomplete DG category C ′ 0 ⊗ Perf(A 1 /G m ). By ind-extending µ ′ , we obtain an object Since Ind(C 0 ) = Ind(C ′ 0 ), the latter is the monad with filtration we were looking for. and oblv Lie-algbd (L) belongs to Υ Y (Perf(Y)) by assumption.
4.1.4. Through the above construction, the functor (4.1) yields a functor Explicitly, the functor underlying the monad U QCoh (L) is the unique endofunctor of QCoh(Y) whose restriction to Perf(Y) is given by The construction of U QCoh allows us to extend the definition of IndCoh 0 to the unbounded case as follows.
Definition 4.1.6 (IndCoh 0 in the unbounded case). For Y → Z in Stk lfp , with Y perfect but not necessarily bounded, we define Clearly, such DG category is compactly generated and hence dualizable.

Basic functoriality.
4.2.1. By construction, the assignment IndCoh 0 (− ∧ W ) underlies the structure of a covariant functor Stk W/ → DGCat. Explicitly, a string W → X → Y in Stk lfp gives a canonical map of Lie algebroids T W/X → T W/Z on W (equivalently, a map of ξ : X ∧ W → Y ∧ W of nil-isomorphisms under W). Induction along the resulting algebra arrow

4.2.2.
Since the functor ξ * ,0 preserves compact objects, it admits a continuous right adjoint that we shall denote by . This is just the forgetful functor along the above maps of universal enveloping algebras.
Proof. If suffices to construct a filtered isomorphism between the restrictions of the above functors to Perf(Y). By definition, this amounts to giving a compatible N-family of isomorphisms These isomorphisms are manifest since U ≤n (T Y/Z ) preserves Υ Y (Perf(Y)).

4.2.7.
This lemma shows that Υ Y : QCoh(Y) → IndCoh(Y) upgrades to a functor In view of the formulas the functor Υ D Y/Z might regarded as the passage from left to right relative D-modules. When Z = pt, we write Υ D Y rather than Υ D Y/pt .
Remark 4.2.10. The notation D der is in place to distinguish D der (Y) from the usual DG category of D-modules on Y.

D-modules and derived D-modules.
In this section, we compute the DG category D der (Y ) for Y = Spec(A) an affine DG scheme locally of finite presentation. We will first perform the computation in general and then apply it to the affine schemes of Example 2.2.7.
4.3.1. Since Y is affine, D der (Y ) is the DG category of modules over the DG algebra To describe this DG algebra, we need to compute the monad U(T Y ), understand its PBW filtration, and then do the filtered renormalization. Thus, to compute U QCoh (T Y ), we need to compute the filtered monad

For any
Similarly, for the filtered algebra Γ(Y, U QCoh (T Y )(O Y )), we need to understand the filtered monad structure In all cases, such structure is induced via base-change by the structure maps where we recall that s m : ∆ m → Y is the first of the two structure maps. It is clear that s m is proper and in fact finite. Before continuing, we need to establish another property of s m .
By passing to dual functors, we deduce that (s m ) * admits a left adjoint, to be denoted which gets intertwined to (s m ) IndCoh * by the Υ functors. In other words, 9 We are just making explicit the fact that the functor m ∆m is an algebra object in the (Day convolution) monoidal ∞-category of sequences of groupoids over Y .

We obtain that
where the last step used the fact that (s m ) ! (O ∆m ) is automatically perfect and that Υ Y is fully-faithful on Perf(Y ) even if Y is not bounded (see Section 2.2.9).

The naive duality on Perf(Y ) further yields
with algebra structure induced by the A-linear duals of the system of maps Let us declare that s m is the map corresponding to the natural map [v] → A m . Then, changing variables   Proof. When n is odd, Y n is bounded and thus the functor in question is an equivalence by the general theory. In particular this implies that the Weyl algebra on an odd vector space is Morita equivalent to .
If n = 2m is even, then we claim that the functor is not an equivalence. While this can proven directly, we prefer to give a quick argument that uses the shift of grading trick introduced in [1, Appendix A]. 10 Since φ n is G m -equivariant, we can apply the shift of grading m times to cancel the shifts. This shows that φ n is an equivalence if and only if so is φ 0 . But the latter is the functor D(A 1 ) → Vect of !-restriction at 0, which is obviously not an equivalence. 4.3.14. Let us return to the case of a general affine scheme Y = Spec(A). In regard to the formula D der (Y ) ≃ W Amod established above, we always expect W A to be a Weyl algebra. For instance, if A = ( [x i ], d) is a quasi-free commutative DG algebra with finitely many variables, we expect W A to look as follows: • as graded vector space, • the algebra structure is determined by the super-commutation relations [ We defer this computation to a future work.  Proof. Let W → Y → Z be a map in (Stk perf ,lfp ) W/ . Consider the Cech complex (Y × Z (•+1) ) ∧ W of the map Y ∧ W → Z ∧ W and the resulting pullback functor We need to show that such functor is an equivalence. By passing to left adjoints, this amounts to showing that the arrow is an equivalence, where now the structure maps forming the colimit are given by the ( * , 0)-pushforward functors (that is, induction along the maps between the universal envelopes). Hence, it suffices to show that the natural arrow taking place in Alg(End(QCoh(W)) Fil,≥0 ), is an isomorphism. Forgetting the monad structure is conservative, whence we will just prove that the arrow above is an isomorphism in End(QCoh(W)) Fil,≥0 (i.e., that it is an isomorphism of filtered endofunctors).
Since the filtrations in questions are non-negative, it is enough to prove the isomorphism separately for each component of the associated graded. Recall that the j th -associated graded of U QCoh (L) is the functor Sym j (Φ W (L)) ⊗ − : QCoh(W) → QCoh(W). Thus, we are to prove that the natural map is an isomorphism in QCoh(W) for each j ≥ 0. Since Sym commutes with colimits, it suffices to show that is an isomorphism. We will show that the cone of such map is zero. First, with no loss of generality, we may assume that W = Y. Then we compute the cone in question as and this expression is manifestly isomorphic to the zero object of QCoh(Y): indeed, the simplicial object in question is the Cech nerve of the map T QCoh Y/Z → 0 in QCoh(Y).

4.5.
Tensor products of IndCoh 0 over QCoh. In this section, we show that formation of IndCoh 0 behaves well with respect to fiber products.
There is a natural isomorphism Proof. We need to exhibit a compatible N-family of isomorphisms By the continuity of these functors and perfection of Y, it suffices to exhibit a compatible N-family of . When restricted to Perf(Y) ⊂ QCoh(Y), the LHS can be rewritten as It then suffices to give a compatible N-family of isomorphisms By ([19, Volume 2, Chapter 9, Section 6.5]), for a Lie algebroid L in IndCoh(Y), the functor U(L) ≤n can be written using the n th infinitesimal neighbourhood of the formal groupoid associated to L. In our case, let V (n) the n th infinitesimal neighbourhood attached to T X× Z Y/X → T X× Z Y , equipped with its two structure maps p s , p t : Similarly, let p s , p t : By the very construction of n th infinitesimal neighbourhoods, we have canonical isomorphisms which are compatible with varying n. Hence, the compatible isomorphisms (4.6) come from base-change for ind-coherent sheaves.
Corollary 4.5.2. With the notation of the above lemma, assume furthermore that at least one of the following two requirements is satisfied: • the map X → Z is affine (more generally, we just need that p * : QCoh(X × Z Y) → QCoh(Y) be right t-exact up to a finite shift); • Y is bounded.
Then the arrow obtained from (4.5) by adjunction, is an isomorphism of filtered functors from QCoh(X × Z Y) to QCoh(Y).
Proof. As the arrow in question is the colimit of the N-family it suffices to prove the assertion separately for each piece of the associated graded. For each n ≥ 0, the map in question is Let us now finish the proof in the situation of the first assumption, the argument for the second one is easier.
It suffices to check the isomorphism after restricting both sides to Perf(X × Z Y), in which case we are dealing with the arrow . Now the assertion follows from the projection formula and the fact (see Section 2.2.9) that Υ Y is fully faithful on the full subcategory of QCoh(Y) consisting of eventually connective objects. Proposition 4.5.5. Let Y → Z ← V ← U be a diagram in Stk <∞ perf ,lfp . Note that we do not assume that Y × Z U be bounded. Then the exterior product yields an equivalence Proof. Without loss of generality, we may assume that Z = V. Thus, for a diagram Y → Z ← U, we need to construct a QCoh(Y)-linear equivalence Both categories are modules for monads acting on QCoh(U) (this is true thanks to the hypothesis of affineness), so it suffices to construct a map between those monads and check it is an isomorphism.
Let p : U × Z Y → U denote the obvious projection. The two monads in questions are Note that the monad structure on the former functor has been discussed in Corollary 4.5.3.
By assumption, QCoh(U × Z Y) is compactly generated by objects of the form p * P for P ∈ Perf(U). Now the assertion follows from Lemma 4.5.1.

4.6.
The exceptional pull-back and push-forward functors. Let us now generalize Section 3.4.2.
4.6.1. Given maps X → Z ← Y in Stk perf ,lfp , we regard the resulting cartesian diagram in Arr(Stk perf ,lfp ). We emphasize that none of the stacks in question is required to be bounded. In this situation, we define the adjunction exactly as in Section 3.4.2, using the equivalence proven in Proposition 4.5.5.
4.6.2. Tautologically, the functors η !,0 and η ? fit in the commutative diagrams Example 4.6.3. Let us illustrate the adjuction (η !,0 , η ? ) in the simple example where X = Y = pt, both mapping to a marked point of Z. We further assume that Z = Z is a bounded affine scheme locally of finite presentation. In this case, X × Z Y = ΩZ := pt × Z pt and the adjunction takes the form Let us call nil-isomorphisms the arrows of the first type and cartesian the arrows of the second type. It is straightforward to check that the associated ∞-category Corr(Arr(Stk perf ,lfp )) niliso;cart of correspondences is well-defined. IndCoh 0 : Corr(Arr(Stk perf ,lfp )) niliso;cart −→ DGCat.
with all stacks in Stk perf ,lfp , gives rise to a correspondence and to a square IndCoh 0 (Y ∧ X ) IndCoh 0 (Z ∧ X ).
The latter is canonically commutative: to see this, use Proposition 4.5.5 to rewrite the two DG categories on the top as relative tensor products.

The center of H(Y)
The goal of this final section is to compute the center of the monoidal DG category H(Y) associated to Y ∈ Stk <∞ perf ,lfp .
5.1. The center of a monoidal DG category. In this preliminary section, we will recall some general facts on the center of a monoidal DG category A. For instance, we will recall why, for A rigid and pivotal, f (a 1 , . . . , a i ⋆ a i+1 , . . . , a n+1 ), if 1 ≤ i ≤ n f (a 1 , . . . , a n ) ⋆ a n+1 , if i = n + 1, with the obvious reinterpretation in the case n = 0.

5.1.4.
Assume from now on that A is rigid and pivotal (in particular, A is compactly generated), see Sections 0.6.4 and 0.6.5 for the definitions. Any rigid monoidal category is self-dual is such a way that m ∨ ≃ m R . Pivotality further implies that such self-duality A ∨ ≃ A is an equivalence of (A, A)-bimodules. Using this and [17, Proposition D.2.2], we can turn the limit of (5.1) into a colimit by taking left adjoints. We obtain that Z A (M ) can also be computed as the colimit of the simplicial DG category Remark 5.1.5. If A is rigid but not pivotal, then the equivalence A ∨ ≃ A is (A, A)-bilinear provided that we twist one of the two actions on A by the monoidal automorphism of A induced by a → (a ∨ ) ∨ at the level of compact objects. We are grateful to Lin Chen for pointing out this issue, which is discussed in [ Combining this with Proposition 4.5.5, we obtain It is a routine exercise (left to the reader) to unravel the structure functors.
We can now state and prove the first part of our main theorem.
Proof. The above proposition yields the equivalence where the totalization is taken with respect to the ?-pullbacks. It remains to apply descent on the RHS: we use Proposition 4.4.1 to obtain as claimed.

5.3.
Relation between H(Y) and its center. In the previous section, we have constructed an equivalence Z ≃ IndCoh 0 (pt ∧ LY ). Our next task is to describe what the adjunction H ⇄ Z becomes under such equivalence.

Let
be the map in Arr(Stk perf ,lfp ) defined by the cartesian diagram Y Y × Y.

LY Y
By its very construction, the functor s !,0 : By adjuction, s ? is the continuous functor induced by the QCoh-pushforward ∆ * . Since ∆ is affine, the functor s ? is monadic. where ind L is the induction functor for left D-modules (which makes sense as G is bounded).