D-Modules on Infinite Dimensional Varieties

D-MODULES ON INFINITE DIMENSIONAL VARIETIES

SAM RASKIN

Contents 1. Introduction 1 2. D-modules on prestacks 4 3. D-modules on schemes 7 4. Placidity 19 5. Holonomic D-modules 30 6. D-modules on indschemes 33 References 44

1. Introduction 1.1. The goal of this foundational note is to develop the 퐷-module formalism on indschemes of ind-inﬁnite type.

1.2. The basic feature that we struggle against is that there are two types of infinite dimensionality at play: pro-infinite dimensionality and ind-infinite dimensionality. That is, we could have an infinite dimensional variety 푆 that is the union 푆 “Y푖푆푖 “ colim푖푆푖 of finite dimensional varieties, or 푇 that is the projective limit 푇 “ lim푗 푇푗 of finite dimensional varieties, e.g., a scheme of infinite type. Any reasonable theory of 퐷-modules will produce produce some kinds of de Rham homology and cohomology groups. We postulate as a basic principle that these groups should take values in discrete vector spaces, that is, we wish to avoid projective limits. Then, in the ind-infinite dimensional case, the natural theory is the homology of 푆:

퐻˚p푆q :“ colim 퐻˚p푆푖q 푖 while in the pro-inﬁnite dimensional case, the natural theory is the cohomology of 푇 :

˚ ˚ 퐻 p푇 q :“ colim 퐻 p푇푗q. 푗 For indschemes that are inﬁnite dimensional in both the ind and the pro directions, one requires a semi-infinite homology theory that is homology in the ind direction and cohomology in the pro direction. Remark 1.2.1. Of course, such a theory requires some extra choices, as is immediately seen by considering the ﬁnite dimensional case. For example, for a smooth variety, we have a choice of normalization for the cohomological shifts. 1 2 SAM RASKIN

1.3. Theories of semi-infinite homology have appeared in many places in the literature. We do not pretend to survey the literature on the subject here, but note that in the case of the loop group, it is well-known that semi-infinite cohomology, in the sense above, may be defined using the semi-infinite cohomology of Lie algebras. We provide such a theory in large generality below. In fact, we develop two theories 퐷! and 퐷˚ of derived categories of 퐷-modules on indschemes of ind-infinite type. The theory 퐷! is contravariant, and therefore carries a natural dualizing complex, and the theory 퐷˚ is covariant, and therefore is the place where cohomology is defined. For placid indschemes (a technical condition defined below), the two categories are identified after a choice of dimension theory, and therefore allows us to define the renormalized or semi- infinite cohomology of the scheme. The extra choice of dimension theory here precisely reflects the numerical choice of cohomological shifts discussed above. Remark 1.3.1. The main difference between our approach and other approaches taken in the literature is that we work systematically with derived categories of 퐷-modules, rather than simply working with homology or with abelian categories. This is facilitated by our use of higher category theory, i.e., with the use of homotopy limits and colimits of DG categories. 1.4. Overview. In S2, we give very general definitions of our categories 퐷! and 퐷˚ of 퐷-modules: we define these categories for arbitrary prestacks. There’s not much one can say in this level of generality, but it is convenient to work in this setting in order to note how formal the definitions are. We note for reference below that 퐷! is contravariant, while 퐷˚ is covariant. Moreover, for a ! prestack 푆, the DG category 퐷!p푆q admits a tensor product b and acts on 퐷˚p푆q in a canonical way satisfying a version of the projection formula.

1.5. In S3, we develop the theory in the setting of (quasi-compact, quasi-separated) schemes. The key technique here is to use Noetherian approximation, as developed in [Gro] and [TT]. Note that this idea is already essentially present in [KV]; the authors of loc. cit. credit it to Drinfeld.

1.6. In S4 we will introduce the notion of placidity. One can understand this condition as saying that the singularities of a scheme are of ﬁnite type in a precise sense. The key point of placid schemes is that they admit a “renormalized dualizing complex” that lies in 퐷˚p푆q. This is notable because, as we recall, 퐷˚ is covariant: its natural functoriality (with respect to inﬁnite type morphisms) is through pushforwards. Moreover, the functor of action on the renormalized dualizing complex gives an equivalence 퐷!p푆q » 퐷˚p푆q. In particular, one obtains a covariant structure on 퐷! and a contravariant structure on 퐷˚ is the placid setting.

1.7. In S5, we discuss the holonomic theory for schemes of inﬁnite type. There is nothing terribly unsurprising here. Remark 1.7.1. This material could just as well be given for ℓ-adic sheaves. 1.8. In S6, we move to the setting of indschemes. The key part is again a theory of placid indschemes with properties similar to the setting of placid schemes described above. It is here that dimension theories enter the story.

1.9. There are many pushforward and pullback functors constructed in the text below. The bulk of this text is really devoted to checking when certain functors are deﬁned, when they coincide, when they are adjoint, when they satisfy base-change, etc. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 3

Such a state of aﬀairs can only lend itself to confusion for the reader, so to conclude this overview, we include a table describing which functors are deﬁned when and what their basic properties are. The reader should refer to the body of this text for more information; this table is simply meant to be available for convenient reference. We let 푓 : 푆 Ñ 푇 be a morphism of indschemes in this table.

Hypotheses Functor Adjunctions Base-change References properties ! ! ! None 푓 : 퐷 p푇 q Ñ 퐷 p푆q See 푓˚,!´푑푅, 푓˚,푟푒푛 See 푓˚,!´푑푅, 푓˚,푟푒푛 S3.6, S?? ˚ ˚ ¡ ˚,푟푒푛 ¡ ˚,푟푒푛 None 푓˚,푑푅 : 퐷 p푆q Ñ 퐷 p푇 q See 푓 , 푓 See 푓 , 푓 S3.18, S?? ! ! ! 푓 ind- 푓˚,!´푑푅 : 퐷 p푆q Ñ 퐷 p푇 q Left adjoint to 푓 for Always satisfies S3.9, S6.3 finitely 푓 ind-proper; right ad- base-change with presented joint to 푓 ! up to co- upper-! functors homological shift for 푓 smooth ¡ ˚ ˚ 푓 ind- 푓 : 퐷 p푇 q Ñ 퐷 p푆q Right adjoint to 푓˚,푑푅 Always satisfies S3.22, S6.3 finitely for 푓 proper; left ad- base-change with presented joint to 푓˚,푑푅 up to co- lower-* functors homological shift for 푓 smooth ! ! ! 푆 and 푇 푓˚,푟푒푛 : 퐷 p푆q Ñ 퐷 p푇 q Right adjoint to 푓 for 푓 For 푓 placid, S6.16, Prop. placid and placid and with the di- satisfies base- 6.18.1 equipped mension theory of 푆 in- change against with di- duced from that of 푇 by the upper-! func- mension Construction 6.12.6 tors of finitely theories presented morphisms !,푟푒푛 ˚ ˚ 푆 and 푇 푓 : 퐷 p푇 q Ñ 퐷 p푆q Left adjoint to 푓˚,푑푅 for For 푓 finitely S6.16, Prop. placid and 푓 placid and with the presented, satis- 6.18.1 equipped dimension theory of 푆 fies base-change with di- induced from that of 푇 against the lower- mension by Construction 6.12.6 * functors of theories placid morphisms Table 1. 퐷-module functors in infinite type

1.10. Mea culpa. This theory is inadequate in that it completely ignores that most characteristic feature of the theory of 퐷-modules: the forgetful functor to quasi-coherent sheaves. The reason is that one needs a theory of ind-coherent sheaves (c.f. [GR]) for (ind)schemes of (ind-)inﬁnite type. The methods used below are apparently inadequate for this purpose. The problem is that base- change between upper-! and lower-˚ functors does not hold for Cartesian squares in the category of classical schemes as it does for 퐷-modules: rather, one needs the square to be Cartesian in the category of derived schemes, and this immediate appearance of non-eventually coconnective derived schemes is in tension with our appeals to Noetherian approximation. However, at least in the placid case, the situation is okay: c.f. to [Gai3]. 4 SAM RASKIN

1.11. Categorical conventions. Our basic methodology in treating the above problems is the use of modern derived techniques. However, we (mostly) do not appeal to derived algebraic geometry. We spell this out further in what follows. 1.12. We appeal frequently to higher category theory, and it is convenient to use higher category theory as the basic building block of our terminology. Therefore, by category we mean p8, 1q- category, and similarly by groupoid we mean 8-groupoid. We let Cat and Gpd denote the corresponding categories. When we mean to specify that we are working with the classical notion of category, we use the term p1, 1q-category instead. 1.13. We work always over a field 푘 of characteristic 0. We let DGCat denote the category of DG categories (alias: stable categories enriched over 푘- modules). Let Vect “ Vect푘 denote the DG category of 푘-modules. 1 Let DGCat푐표푛푡 denote the category of cocomplete DG categories and continuous functors: i.e., DG categories admitting all colimits, and functors commuting with all colimits. We freely use the linear algebra of such categories (tensor products, duality and all that) from [Gai1]. 1.14. Let AffSch denote the p1, 1q-category of (classical, i.e., non-derived) affine schemes. Let PreStk denote the category of (classical) prestacks: by definition, this is the category of functors AffSch표푝 Ñ Gpd. Recall that e.g. the categories of schemes and indschemes are by definition full subcategories of PreStk. 1.15. For 푆 a scheme of finite type, we let 퐷p푆q denote the DG category of 퐷-modules on 푆. We refer to [GR], where this construction is given in detail in a format convenient for our purposes. 1.16. Acknowledgements. This material has been strongly influenced by [BD] S7, [Dri] and [KV]. We also thank Dennis Gaitsgory for many helpful discussions about this material; in particular, the idea of systematically distinguishing between 퐷! and 퐷˚, our very starting point, is due to him. Finally, we thank Dario Beraldo for helpful conversations on this material.

2. D-modules on prestacks 2.1. In this section, we deﬁne 퐷! and 퐷˚ for general prestacks. There is not much to say in this level of generality: we work in this generality because the deﬁnitions are most natural like this, and simply to point out that it can be done. The later sections of this text are then dedicated to studying the special cases of schemes and indschemes.

2.2. Let AffSch푓.푡. Ď AffSch denote the subcategory of finite type affine schemes. Note that AffSch » PropAffSch푓.푡.q: this is essentially the statement that a classical commutative algebra is the union of its finite type subalgebras. 2.3. Definition of 퐷!. We define:

! 표푝 퐷 : AffSch Ñ DGCat푐표푛푡 푓.푡.,표푝 as the left Kan extension of the functor 퐷 : AffSch Ñ DGCat푐표푛푡 attaching to a finite type affine scheme 푆 its category of 퐷-modules and attaches to a morphism 푓 : 푆 Ñ 푇 the corresponding upper-! functor. We extend this definition to:

1We freely ignore cardinality issues in what follows, but here cocomplete should be taken to mean presentable; we recall that the difference is a set-theoretic condition that is always satisfied in the examples usedbelow. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 5

! 표푝 퐷 : PreStk :“ HompAffSch, Gpdq Ñ DGCat푐표푛푡 by right Kan extension. Remark 2.3.1. Here is what the above deﬁnition says more concretely:

‚ For a classical commutative algebra 퐴, write 퐴 “Y푖퐴푖 with 퐴푖 ﬁnite type. Note that for 퐴푖 Ď 퐴푗, we have a map Specp퐴푗q Ñ Specp퐴푖q, and therefore a !-pullback functor for 퐷-modules. Then 퐷!pSpecp퐴qq is computed as:

! 퐷 pSpecp퐴qq :“ colim 퐷pSpecp퐴푖qq P DGCat푐표푛푡.

‚ For a nice enough stack 풴, choose a hypercovering by aﬃne schemes 푌푛 “ Specp퐴푛q, so 풴 “ |푌‚| “ colimr푛sPΔ표푝 푌푛. ! ! Then 퐷 p풴q “ limr푛sPΔ 퐷 p푌푛q. ‚ More generally, for any prestack 풴, we can formally write 풴 “ colim푖PI 푌푖 for some diagram 푖 ÞÑ 푌푖 P AffSch. We then have:

! ! 퐷 p풴q “ lim 퐷 p푌푖q. 푖PI표푝 Less precisely, one should think that a 퐷!-module F on 풴 is deﬁned by its compatible system of restrictions to aﬃne schemes 푌 mapping to 풴, and a “typical” 퐷!-module on such a 푌 is pulled back along some 푌 Ñ 푍 with 푍 P AffSch푓.푡.. Notation 2.3.2. For 푓 : 풴 Ñ 풵 a morphism of prestacks, we have a tautological map:

! ! 퐷 p풵q Ñ 퐷 p풴q P DGCat푐표푛푡 which we denote by 푓 !. Note that there is no risk for confusion in this notation, since if 푓 is a map between prestacks locally of ﬁnite type, this functor corresponds with the usual functor 푓 !. 2.4. For any prestack 풴, note that 퐷!p풴q is a symmetric monoidal category. Indeed, for any prestacks 풴 and 풵, there is a tautological functor:

퐷!p풴q b 퐷!p풵q ÝÝÝÑ´b´ 퐷!p풴 ˆ 풵q which is an equivalence if 풴, 풵 P AffSch (indeed: this claim immediately to the case 풴, 풵 P AffSch푓.푡.). ! Then pairing F b G :“ ∆!pF b Gq defines the desired symmetric monoidal structure, where ! ∆ : 풴 Ñ 풴 ˆ 풴 is the diagonal map. We remark that b commutes with colimits in each variable. 2.5. Definition of 퐷˚. We now define the second category of 퐷-modules on a prestack. The definition is formally dual to the definition of 퐷!. Namely, we define:

˚ 퐷 : AffSch Ñ DGCat푐표푛푡 푓.푡. as the right Kan extension of the functor 퐷 : AffSch Ñ DGCat푐표푛푡 attaching to a finite type affine scheme 푆 its category of 퐷-modules and attaches to a morphism 푓 : 푆 Ñ 푇 the corresponding lower-˚ functor. We extend this definition to: 6 SAM RASKIN

˚ 퐷 : PreStk :“ HompAffSch, Gpdq Ñ DGCat푐표푛푡 by left Kan extension. Remark 2.5.1. Again, here is the concrete interpretation of this deﬁnition:

‚ For a classical commutative algebra 퐴 “Y푖퐴푖, we have:

˚ 퐷 pSpecp퐴qq :“ lim 퐷pSpecp퐴푖qq P DGCat푐표푛푡 where the structure functors are lower-˚ functors. ‚ For any prestack 풴, we can write 풴 “ colim푖PI 푌푖 for some diagram 푖 ÞÑ 푌푖 P AffSch. We then have:

˚ ˚ 퐷 p풴q “ colim 퐷 p푌푖q. Less precisely, one should think that a typical 퐷˚-module on 풴 is pushed forward along some map 푌 Ñ 풴 with 푌 P AffSch, and a 퐷˚-module on such a 푌 is defined by the knowledge of its compatible system of push forwards along maps 푌 Ñ 푍 with 푍 P AffSch푓.푡.. 2.6. Locally finite type case. When 풴 is a prestack locally of finite type, we have canonical identifications 퐷!p풴q “ 퐷p풴q, where we recall that 퐷p풴q is defined in [GR]. For 풴 a scheme of finite type, it is easy to identify this category with 퐷˚p풴q as well (by a descent argument). 2.7. The projection formula. Next, we discuss the relationship between 퐷! and 퐷˚. By the projection formula in the finite type setting, 퐷!p풴q acts on 퐷˚p풴q for any prestack 풴. ˚ ! More precisely, 퐷 p풴q is a module for 퐷 p풴q in DGCat푐표푛푡. We discuss this a bit heuristically here, and give a more precise construction in S2.8. Indeed, in the case where 풴 “ 푆 P AffSch, this action is characterized by the formula:

! ! ! 푓˚,푑푅p푓 pFq b Gq “ F b 푓˚,푑푅pGq for 푓 : 푆 Ñ 푇 with 푇 P AffSch푓.푡., F P 퐷!p푇 q “ 퐷p푇 q, and G P 퐷˚p푆q. In the case of general 풴, this action is characterized by the formula:

! ! ! F b 푔˚,푑푅pGq “ 푔˚,푑푅p푔 pFq b Gq for 푔 : 푆 Ñ 풴 with 푆 P AffSch, F P 퐷!p풴q, and G P 퐷˚p푆q. ˚ ˚ Moreover, for 푓 : 풴 Ñ 풵 any morphism, 푓˚,푑푅 : 퐷 p풴q Ñ 퐷 p풵q is (canonically) a morphism of 퐷!p풴q-module categories. We remark that this structure encodes the projection formula. 2.8. Here is a more precise construction of the above structures. Let C (temporarily) denote the category whose objects are pairs A P ComAlgpDGCat푐표푛푡q and M a module for A in DGCat푐표푛푡, and where morphisms pA, Mq Ñ pB, Nq are pairs of a symmetric monoidal and continuous functor A Ñ B plus N Ñ M a continuous morphism of A-module categories (where the A-module category structure on N is induced by A Ñ B). Now observe that there is a functor 퐷 : Sch푓.푡.,표푝 Ñ C sending 푆 to p퐷p푆q, 퐷p푆qq equipped with upper-! functoriality in the first variable and lower-* functoriality in the second variable. Indeed, this follows from the formalism of correspondences from [GR]. Then we obtain a AffSch표푝 Ñ C sending 푆 to the pair p퐷!p푆q, 퐷˚p푆qq as the left Kan extension of the [GR] functor. Indeed, one computes filtered colimits in C as a colimit in the first variable and a limit in the second variable. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 7

This treats the projection formula in the case of schemes, and the same method works for general prestacks.

3. D-modules on schemes 3.1. In this section, we treat 퐷! and 퐷˚ in the special case of (quasi-compact quasi-separated) schemes. The main idea is to use Noetherian approximation (c.f. S3.2) to reinterpret 퐷! and 퐷˚ on such schemes. This will give us a handle on (possibly non-aﬃne) morphisms of ﬁnite presentation, which allows us to increase the functoriality of this functors. 3.2. Noetherian approximation. We begin with a brief review of the theory of Noetherian approximation (alias: Noetherian descent). This theory is due to [Gro] and [TT].

푓.푝. 3.3. Let 푆 be a quasi-compact quasi-separated base scheme. Let Sch{푆 denote the category of schemes finitely presentated (in particular: quasi-separated) over 푆. If 푆 is Noetherian we will also 푓.푡. use the notation Sch{푆 because in this case finite type is equivalent to finite presentation. We say an 푆-scheme 푇 is almost affine if for every 푆1 Ñ 푆 of finite presentation every map 푇 Ñ 푆1 factors as 푇 Ñ 푇 1 Ñ 푆1 where 푇 Ñ 푇 1 is affine and 푇 1 Ñ 푆1 is finitely presented. Let al.aff Sch{푆 denote the category of almost affine 푆-schemes. aff 푓.푝. 푓.푝. Let Pro pSch{푆 q denote the full subcategory of PropSch{푆 q consisting of objects 푇 that arise as filtered limits 푇 “ lim 푇푖 of finitely presented 푆-schemes under affine structural morphisms 푇푗 Ñ 푇푖. We recall that projective limits of such systems exist and that if each 푇푖 is affine over 푆 then 푇 is as well. Clearly such limits commute with base-change. 3.4. The main result of Noetherian approximation is the following, due to [Gro] S8 and [TT] Appendix C. Theorem 3.4.1. (1) The right Kan extension:

aff 푓.푝. Pro pSch{푆 q Ñ Sch{푆 푓.푝. of the embedding Sch{푆 ãÑ Sch{푆 is defined and is fully-faithful. This right Kan extension 푎푙.푎푓푓 maps into Sch{푆 . If 푆 is Noetherian and affine, then the essential image of this functor is all schemes over 푆 that are quasi-compact and quasi-separated (in particular, quasi-compact quasi-separated 푘-schemes are almost affine). (2) Suppose 푇 “ lim 푇푖 is a filtered limit with each 푇푖 finitely presented over 푆 and 푇푗 Ñ 푇푖 1 affine. Then if 푇 is a finitely presented 푇 -scheme there exists an index 푖 and a 푇푖-scheme 1 1 1 푇푖 of finite presentation such that 푇 “ 푇푖 ˆ푇푖 푇 (as a 푇 -scheme). Moreover, if the map 푇 1 Ñ 푇 has any (finite) subset of the properties of being (e.g.) 1 smooth, flat, proper, or surjective, then 푇푖 Ñ 푇푖 may be taken to have the same properties. (3) Suppose 푇 “ lim푖PI표푝 푇푖 as in (2). Then if 푇 Ñ 푆 is an affine morphism, then there exists

푖0 P I such that for every 푖 P I푖0{, 푇푖 Ñ 푆 is affine. (4) Suppose that 푇 “ lim 푇푖 as in (2) and 푈 Ď 푇 is a quasi-compact open subscheme. Then for

some index 푖 P I and open 푈푖 Ď 푇푖 we have 푈 “ 푈푖 ˆ푇푖 푇 (as 푇 -schemes). Remark 3.4.2. We note that (3) appears in [TT] as Proposition C.6, where it is stated only in the case that 푆 is affine. However, this immediately generalizes, since 푆 is assumed quasi-compact and therefore admits a finite cover by affines. 8 SAM RASKIN

3.5. We will also use the following technical result.

Proposition 3.5.1. Suppose that 푇 “ lim푖PI 푇푖 is a filtered limit of schemes under affine structure maps. Let 훼푖 : 푇 Ñ 푇푖 denote the structure maps. Then passing to cotangent complexes, the canonical map:

˚ 1 1 ď0 colim 훼푖 pΩ푇 q Ñ Ω푇 P QCohp푇 q 푖PI 푖 1 is an equivalence. (Here e.g. Ω푇 denotes the whole cotangent complex). Proof. Let DGSch denote the category of DG schemes. Note that filtered limits of derived schemes under affine structural maps exists as well, and satisfy the same properties as in the non-derived case: namely, if 푇 “ lim 푇푖 in DGSch is a filtered limit under affine structural maps of affine 푆- schemes, then 푇 is affine over 푆 as well. In particular, we deduce that Sch Ď DGSch is closed under such limits. Now the result follows immediately from the description of the cotangent complex in terms of square-zero extensions in derived algebraic geometry. ! 3.6. 퐷 -modules. Let Sch푞푐푞푠 denote the category of quasi-compact quasi-separated 푘- schemes. Let 푆 P Sch푞푐푞푠 be fixed. 푓.푡. By Theorem 3.4.1, we can write 푆 “ lim 푆푖 with 푆푖 P Sch and all structure morphisms affine. Proposition 3.6.1. The canonical morphism:

! colim 퐷p푆푖q Ñ 퐷 p푆q P DGCat푐표푛푡 (3.6.1) 푖 is an equivalence. Remark 3.6.2. This claim is immediate from Theorem 3.4.1 (3) if 푆 is affine. The proof is deferred to S3.13. In the meantime, we will give some preliminary constructions on the left hand side of (3.6.1). 3.7. First, we claim that the left hand side of (3.6.1) is independent of the choice of way of writing 푆 “ lim 푆푖 with the above properties. 푓.푡. ! By Theorem 3.4.1, Sch푞푐푞푠 is a full subcategory of PropSch q. We define the functor 퐷 : 표푝 푓.푡.,표푝 Sch푞푐푞푠 Ñ DGCat푐표푛푡 as the left Kan extension of the functor 퐷 : Sch Ñ DGCat푐표푛푡. r Remark 3.7.1. Suppose that C0 is an (essentially small) category and C Ď IndpC0q is a full subcategory containing C0. Suppose that we are given 퐹 : C Ñ D a functor that is the left Kan 0 0 extension of its restriction to C . Then for any filtered colimit 푋 “ colim푖 푋푖 P C in IndpC q, we have 퐹 p푋q “ colim 퐹 p푋푖q. Indeed, by definition:

퐹 p푋q “ colim 퐹 p푋1q. 푋1Ñ푋,푋1PC0 But this also computes the left Kan extension from C0 to IndpC0q. Therefore, this claim reduces to the case C “ IndpC0q, where it is well-known. Applying this remark in our setting, we see that 퐷!p푆q computes the left hand side of (3.6.1). Therefore, we need to show that the map: r 퐷!p푆q Ñ 퐷!p푆q

r D-MODULES ON INFINITE DIMENSIONAL VARIETIES 9 is an equivalence.

! ! ! Notation 3.7.2. For 푓 : 푆 Ñ 푇 P Sch푞푐푞푠, we let 푓 : 퐷 p푇 q Ñ 퐷 p푆q denote the induced functor: this is hardly an abuse, since we will eventually be identifying this functor with 푓 ! : 퐷!p푇 q Ñ 퐷!p푆q. r r 3.8. Correspondences. Next, we extend the functoriality of 퐷!. 푓.푡. Let Sch푐표푟푟 be the (1,1)-category of ﬁnite type schemes under correspondences. By [GR], we have 푓.푡. the functor 퐷 : Sch푐표푟푟 Ñ DGCat푐표푛푡 that attaches to a ﬁnite typer scheme 푇 its category 퐷p푇 q of 훼 훽 퐷-modules and to a correspondence 푇o 퐻 / 푆 (i.e., a map 푇 Ñ 푆 in Sch푐표푟푟) attaches ! the functor 훽˚,푑푅훼 . Let Sch푞푐푞푠,푐표푟푟;푎푙푙,푓.푝. denote the category of quasi-compact quasi-separated schemes under correspondences of the form:

퐻 훼 훽 ~ 푇 푆 where 퐻 P Sch푞푐푞푠, 훽 is ﬁnitely presented and 훼 is arbitrary. Note that Sch푞푐푞푠,푐표푟푟;푎푙푙,푓.푝 contains 푓.푝. 표푝 Sch푐표푟푟 as a full subcategory. It also contains Sch푞푐푞푠 as a non-full subcategory where morphisms are correspondences where the right arrow is an isomorphism. We deﬁne the functor:

!,푒푛ℎ 퐷 : Sch푞푐푞푠,푐표푟푟;푎푙푙,푓.푝. Ñ DGCat푐표푛푡 by left Kan extension from Sch푓.푡. . r푐표푟푟 !,푒푛ℎ 표푝 Proposition 3.8.1. The restriction of 퐷 to Sch푞푐푞푠 canonically identifies with the functor ! 표푝 퐷 : Sch푞푐푞푠 Ñ DGCat푐표푛푡. r The proof will be given in S3.11. r 3.9. We assume Proposition 3.8.1 until S3.10 so that we can discuss its consequences. For 푓 : 푇 Ñ 푆 a map of quasi-compact quasi-separated schemes, the induced functor 퐷!,푒푛ℎp푆q “ 퐷!p푆q Ñ 퐷!,푒푛ℎp푇 q “ 퐷!p푇 q coincides with 푓 !. If 푓 is ﬁnitely presented we will denote the corre- ! ! ˚ sponding functor 퐷 p푇 q Ñ 퐷 p푆q by 푓˚,!´푑푅 (to avoid confusion with the functor 푓˚,푑푅 r: 퐷 p푇 q Ñ ˚ 퐷r p푆q deﬁnedr in S3.18r below). We refer to the functor 푓˚,!´푑푅 as the “!-dR ˚-pushforward functor.” Note that the formalismr ofr correspondences implies that we have base-change between ˚-pushforward and !-pullback for Cartesian squares.

Remark 3.9.1. Suppose that 푓 : 푇 Ñ 푆 is finitely presented. One can compute the functor 푓˚,!´푑푅 “algorithmically” as follows. Let 푓 be obtained by base-change from 푓 1 : 푇 1 Ñ 푆1 a map of schemes 1 of finite type via a map 푆 Ñ 푆 . Write 푆 “ lim 푆푖 where structure maps are affine and each 푆푖 is 1 1 a finite type 푆 -scheme. Then 푇 “ lim 푇푖 for 푇푖 :“ 푆푖 ˆ푆1 푇 . Let 훼푖 : 푆 Ñ 푆푖, 훽푖 : 푇 Ñ 푇푖 and 푓푖 : 푇푖 Ñ 푆푖 be the tautological maps. ! ! Then for ℱ P 퐷p푇푖q we have 푓˚,!´푑푅p훽푖pℱqq “ 훼푖푓푖,˚,푑푅pℱq, which completely determines the functor 푓˚,!´푑푅. One readily deduces the following result from [GR]. 10 SAM RASKIN

Proposition 3.9.2. If 푓 : 푆 Ñ 푇 is a proper (in particular, finitely presented) morphism of ! quasi-compact quasi-separated schemes, then 푓 is canonically the right adjoint to 푓˚,!´푑푅. This identification is compatible with the correspondence structure: e.g., given a Cartesian diagram:

휓 푆1 / 푆

푓 1 푓 휙 푇 1 / 푇 with 푓 proper, the identification:

! » ! 1 푓˚,푑푅휙 ÝÑ 휓 푓˚,!´푑푅 arising from the correspondence formalism is given by the adjunction morphism. Similarly, we have the following. Proposition 3.9.3. If 푓 : 푆 Ñ 푇 is a smooth finitely presented map of quasi-compact quasi- ! 1 separated schemes, then 푓 r´2 ¨ 푑푆{푇 s is left adjoint to 푓˚,!´푑푅. Here 푑푆{푇 is the rank of Ω푆{푇 regarded as a locally constant function on 푆.

Remark 3.9.4. By a locally constant function 푇 Ñ Z on a scheme 푇 , we mean a morphism of 푇 Ñ Z with considered as the indscheme Spec 푘 . Z 푛PZ p q If 푇 is quasi-compact quasi-separated and therefore a pro-finite type scheme 푇 “ lim 푇푖 (under affine structure maps), then, by Noetherianš approximation, any locally constant function on 푇 arises by pullback from one on some 푇푖. In other words, if we define 휋0p푇 q as the profinite set lim푖 휋0p푇푖q, then locally constant functions on 푇 are equivalent to continuous functions on 휋0p푇 q. Remark 3.9.5. Recall that there is an automatic projection formula given the correspondence framework. Indeed, for 푓 : 푆 Ñ 푇 a finitely presented map of quasi-compact quasi-separated schemes, F P 퐷!p푇 q and G P 퐷!p푆q, we have a canonical isomorphisms:

! ! r r ! 푓˚,!´푑푅 푓 pFq b G » F b 푓˚,!´푑푅pGq coming base-change for F b G P 퐷!p푇 ˆ` 푆q and the˘ Cartesian diagram:

r Γ푓 푆 / 푇 ˆ 푆

푓 id푇 ˆ푓

Δ푇 푇 / 푇 ˆ 푇 where Γ푓 is the graph of 푓 and ∆푇 is the diagonal. By the ﬁnite type case, these isomorphisms are given by the adjunctions of Proposition 3.9.2 and 3.9.3 when 푓 is proper or smooth. 3.10. In the proof of Proposition 3.8.1 we will need the following technical result. Let 푇 be a quasi-compact quasi-separated scheme. Consider the category 풞푇 of correspondences:

훼 훽 푓.푡. 풞푇 :“ t 푆o 퐻 / 푇 | 훽 ﬁnitely presented, 푆 P Sch and 퐻 P Sch푞푐푞푠u. Here, as usual, compositions are given by ﬁber products. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 11

푓.푡.,표푝 푓.푡. Note that 풞푇 contains as a non-full subcategory Sch푇 { of maps 7 : 푇 Ñ 푆 with 푆 P Sch , 7 id푇 where given such a map we attach the correspondence 푆o 푇 / 푇 .

푓.푡. Lemma 3.10.1. The embedding Sch푇 { Ñ 풞푇 is cofinal.

훼 훽 Proof. Fix a correspondence p 푆o 퐻 / 푇 q P 풞푇 . Translating Lurie’s 8-categorical Quillen Theorem A to this setting, we need to show the contractibility of the category C of commutative diagrams:

훿 퐻 / 퐻1 휖 훽 훽1 7 푇 / 푇 1 푆 such that the square on the left is Cartesian, 퐻1, 푇 1 P Sch푓.푡. and 휖 ˝ 훿 “ 훼. Here a morphism from one such diagram (denoted with subscripts “1”) to another such diagram (denoted with subscripts 1 1 1 1 “2”) is given by maps 푓 : 푇1 Ñ 푇2 and 푔 : 퐻1 Ñ 퐻2 such that the following diagram commutes and all squares are Cartesian:

훿2 1 ( 1 퐻 / 퐻1 / 퐻2 훿1 푔 휖 1 1 2 훽 훽1 훽2 푓 71 1 1 푇 / 푇1 /6 푇2 푆. 72 First, we observe that the category C is non-empty. Indeed, because 훽 is finitely presented we can find 푇 Ñ 푇 1 P Sch푓.푡. and 훽1 : 퐻1 Ñ 푇 1 so that 퐻 is obtained from 퐻1 by base-change. Noting that 퐻 can be written as a limit under affine transition maps of 퐻1 obtained in this way and 푆 is finite type, we see that 퐻 Ñ 푆 must factor though some 퐻1 obtained in this way. To see that C is contractible, note that C admits non-empty finite limits (because Sch admits finite limits) and therefore C표푝 is filtered.

3.11. We now prove Proposition 3.8.1.

! !,푒푛ℎ Proof of Proposition 3.8.1. We have an obvious natural transformation 퐷 퐷 표푝 . It Ñ |Sch푞푐푞푠 suffices to see that this natural transformation is an equivalence when evaluated on any fixed 푇 P Sch푞푐푞푠. r r !,푒푛ℎ 훼 훽 With the notation of S3.10, 퐷 is by definition the colimit over p 푆o 퐻 / 푇 q P 풞푇 of the category 퐷p푆q. By Lemma 3.10.1, this coincides with the colimit over diagrams where 훽 is an isomorphism, as desired. r Remark 3.11.1. Neither Lemma 3.10.1 nor Proposition 3.8.1 is particular to schemes, but rather a general interaction between pro-objects in a category with finite limits and correspondences. 12 SAM RASKIN

3.12. Descent. Next, we discuss descent for 퐷!. For a map 푓 : 푆 Ñ 푇 of schemes and r푛s P Δ let Cech푛p푆{푇 q be defined as: r Cech푛p푆{푇 q :“ 푆 ˆ ... ˆ 푆. 푇 푇 푛 times 푛 Of course, r푛s ÞÑ Cech p푆{푇 q forms a simplicial schemelooooomooooon in the usual way. We use the terminology of Voevodsky’s ℎ-topology, developed in the infinite type setting in [Ryd]. We simply recall that ℎ-coverings are finitely presented2 and include both the classes of fppf coverings and proper3 coverings. Proposition 3.12.1. Let 푓 : 푆 Ñ 푇 be an ℎ-covering of quasi-compact quasi-separated schemes. Then the canonical functor (induced by pullback):

퐷!p푇 q Ñ lim 퐷!pCech푛p푆{푇 qq (3.12.1) r푛sPΔ is an equivalence. r r Recall from [Ryd] Theorem 8.4 that the ℎ-topology of Sch푞푐푞푠 is generated by finitely presented Zariski coverings4 and proper coverings. Therefore, it suffices to verify Lemmas 3.12.2 and 3.12.3 below. Lemma 3.12.2. 퐷! satisfies proper descent, i.e., for every 푓 : 푇 Ñ 푆 a proper (in particular, finitely presented) surjective morphism of quasi-compact quasi-separated schemes the morphism (3.12.1) is an equivalence.r Proof. We can find 푓 1 : 푆1 Ñ 푇 1 a proper covering between schemes of finite type and 푇 Ñ 푇 1 so 1 that 푓 is obtained by base-change. Let 푇 “ lim 푇푖 where each 푇푖 is a 푇 -scheme of finite type and 1 structure maps are affine. Let 푆푖 :“ 푇푖 ˆ푇 1 푆 . We now decompose the map (3.12.1) as:

! » 푛 퐷 p푇 q “ colim 퐷p푇푖q ÝÑ colim lim 퐷pCech p푆푖{푇푖qq Ñ 푖PI 푖PI r푛sPΔ 푛 푛 lim colim 퐷pCech p푆푖{푇푖qq “ lim 퐷pCech p푆{푇 qq. rr푛sPΔ 푖PI r푛sPΔ Here the isomorphism is by ℎ-descent in the ﬁnite type setting. Therefore, it suﬃces to see that the map:

푛 푛 colim lim 퐷pCech p푆푖{푇푖qq Ñ lim colim 퐷pCech p푆푖{푇푖qq 푖PI r푛sPΔ r푛sPΔ 푖PI is an isomorphism. It suﬃces to verify the Beck-Chevalley conditions in this case (c.f. [Lur] Propo- sition 6.2.3.19). For each 푖 P I and each map r푛s Ñ r푚s in I, the functor:

푚 푛 퐷pCech p푆푖{푇푖qq Ñ 퐷pCech p푆푗{푇푗qq

2More honestly: it seems there is a bit of disagreement in the literature whether ℎ-coverings are required to be finitely presented or merely finite type. We are using the convention that they are finitely presented. 3We include “finitely presented” in the definition of proper. 4We explicitly note that these are necessarily finitely presented because we work only with quasi-compact quasi- separated schemes. That is, any open embedding of quasi-compact quasi-separated schemes is necessarily of finite presentation: the only condition to check is that it is a quasi-compact morphism, and any morphism of quasi-compact schemes is itself quasi-compact. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 13 admits a left adjoint given by the !-dR ˚-pushforward as in Proposition 3.9.2. By base change between upper-! and !-dR ˚-pushfoward (Proposition 3.8.1), the Beck-Chevalley conditions are satisﬁed, since for every 푗 Ñ 푖 in I and r푛s Ñ r푚s in Δ the diagram:

푚 푛 Cech p푆푖{푇푖q / Cech p푆푗{푇푗q

푛 푚 Cech p푆푖{푇푖q / Cech p푆푗{푇푗q is Cartesian. ! 표푝 Lemma 3.12.3. 퐷 : Sch푞푐푞푠 Ñ DGCat푐표푛푡 satisfies quasi-compact Zariski descent. Proof. It suﬃces to show for every 푆 a quasi-compact quasi-separated scheme and 푆 “ 푈 Y 푉 a r Zariski open covering of 푆 by quasi-compact open subschemes that the canonical map:

퐷!p푆q Ñ 퐷!p푈q ˆ 퐷!p푉 q 퐷!p푈X푉 q is an equivalence. r r r r Let 푗푈 : 푈 Ñ 푆, 푗푉 : 푉 Ñ 푆 and 푗푈X푉 : 푈 X푉 Ñ 푆 denote the corresponding (ﬁnitely presented) open embeddings. Deﬁne a functor:

퐷!p푈q ˆ 퐷!p푉 q Ñ 퐷!p푆q 퐷!p푈X푉 q ! !r r r ! F푈 P 퐷 p푈q, F푉 P 퐷 p푉 q, Fr푈 |푈X푉 » F푉 |푈X푉 “: F푈X푉 P 퐷 p푈 X 푉 q ÞÑ ´ ¯ Kerr 푗푈,˚,!´푑푅rpF푈 q ‘ 푗푉,˚,!´푑푅pF푉 q Ñ 푗푈X푉,˚,!´푑푅pFr푈X푉 q . We claim that this functor´ is inverse to the above. ¯ ! ! Note that e.g. 푗푈,˚,!´푑푅 : 퐷 p푈q Ñ 퐷 p푆q is fully-faithful. Indeed, by Proposition 3.9.3 we have ! an adjunction between 푗푈 and 푗푈,˚,!´푑푅. The counit: r r ! 푗푈 푗푈,˚,!´푑푅 Ñ id퐷!p푈q is an equivalence by Remark 3.9.1 and the corresponding statement in the ﬁnite type setting. This r shows that the composition:

퐷!p푈q ˆ 퐷!p푉 q Ñ 퐷!p푆q Ñ 퐷!p푈q ˆ 퐷!p푉 q 퐷!p푈X푉 q 퐷!p푈X푉 q is the identity. r r r r r r r For the other direction, note that we have a canonical map:

! ! ! id퐷!p푆q Ñ Ker 푗푈,˚,!´푑푅 푗푈 ‘ 푗푉,˚,!´푑푅 푗푉 Ñ 푗푈X푉,˚,!´푑푅 푗푈X푉 and it suﬃces to see that this map is an equivalence. But this again follows by reduction to the r ` ˘ ﬁnite presentation case via Remark 3.9.1. 14 SAM RASKIN

3.13. By Lemma 3.12.3, to prove Proposition 3.6.1, it suﬃces to observe the following result.

Lemma 3.13.1. 퐷! satisfies descent with respect to quasi-compact Zariski coverings.

Proof. As in S3.8, for 푓 : 풴 Ñ 풵 an affine (in particular, schematic) and finitely presented morphism ! ! of prestacks, there is a functor 푓˚,!´푑푅 : 퐷 p풴q Ñ 퐷 p풵q characterized by the fact that it satisfies base-change with the upper-! functors, and coincides with the usual pushforward functor in the ! finite type setting. Moreover, if 푓 is an open embedding, then we have an adjunction p푓 , 푓˚,!´푑푅q, and 푓˚,!´푑푅 is fully-faithful. We prove the following two results by induction.

p퐴푛q: For 푆 P AffSch and 푗 : 푈 ãÑ 푆 a quasi-compact open subscheme admitting a cover by 푛 affine schemes, the restriction functor 푗! : 퐷!p푆q Ñ 퐷!p푈q admits a fully-faithful right adjoint 푗˚,!´푑푅. Formation of 푗˚,!´푑푅 commutes with base-change with respect to maps 푇 Ñ 푆 P AffSch. p퐵푛q: For 푆 P AffSch and a cover 푆 “ 푈 X 푉 by quasi-compact Zariski open subschemes with 푉 affine and 푈 admitting an open cover by 푛 affine schemes, the functor:

퐷!p푆q Ñ 퐷!p푈q ˆ 퐷!p푉 q 퐷!p푈X푉 q is an equivalence.

We have already observed that 퐴1 is true. Moreover, the statement 퐴푛 implies 퐵푛 by the same argument as in Lemma 3.12.3. Therefore, we should show that 퐴푛 and 퐵푛 imply 퐴푛`1. Chose a cover 푈1 Y 푈2 with 푈2 aﬃne and 푈1 admitting a covering by 푛 aﬃne schemes. By 퐵푛, we have:

! ! ! 퐷 p푈q » 퐷 p푈1q ˆ 퐷 p푈2q. ! 퐷 p푈1X푈2q

Noting that 푈1 X 푈2 ãÑ 푈1 is an aﬃne morphism and therefore this intersection admits a cover by 푛 aﬃnes, we can construct this right adjoint as:

! ! ! 퐷 p푈1q ˆ 퐷 p푈2q Ñ 퐷 p푆q ! 퐷 p푈1X푈2q ! ! ! F1 P 퐷 p푈1q, F2 P 퐷 p푈2q, F1|푈1X푈2 » F2|푈1X푈2 “: F12 P 퐷 p푈1 X 푈2q ÞÑ ´ ¯ Ker 푗1,˚,!´푑푅pF1q ‘ 푗2,˚,!´푑푅pF2q Ñ 푗12,˚,!´푑푅pF12q ´ ¯ where 푗1, 푗2 and 푗12 respectively denote the embeddings of 푈1, 푈2 and 푈12 into 푆. This completes the proof of our inductive statements. By base-change, we obtain that for any quasi-compactopen embedding 푗 : 풰 ãÑ 풴, 푗! : 퐷!p풴q Ñ ! 퐷 p풰q admits a fully-faithful right adjoint 푗˚,!´푑푅 characterized by its compatibility with base- change, and then the proof of Lemma 3.12.3 gives the desired descent claim.

3.14. Having proved Proposition 3.6.1, we no longer distinguish between 퐷! and 퐷!, denoting both by 퐷!. r D-MODULES ON INFINITE DIMENSIONAL VARIETIES 15

3.15. Here is a useful corollary of Proposition 3.6.1.

Corollary 3.15.1. For 푆, 푇 P Sch푞푐푞푠, the functor:

퐷!p푆q b 퐷!p푇 q Ñ 퐷!p푆 ˆ 푇 q is an equivalence. Proof. This is immediate from the 퐷! perspective and the corresponding result for finite type schemes. r 3.16. Correspondences for prestacks. We say that a morphism 푓 : 풴 Ñ 풵 is finitely presented if it is schematic, and if for any 푆 Ñ 풵 with 푆 P AffSch the map 풴 ˆ풵 푆 Ñ 푆 is finitely presented. As another corollary of Proposition 3.6.1, we obtain that there is a functor 푓˚,!´푑푅 compatible with base-change for any finitely presented morphism 푓 : 풴 Ñ 풵 of prestacks. Remark 3.16.1. We emphasize that by schematic, we mean schematic in the sense of classical (i.e., non-derived) algebraic geometry, which is a more forgiving notion than that of derived algebraic geometry. This is relevant, say, for considering the embedding of 0 inside of the indscheme associated with an infinite-dimensional 푘-vector space, which is a classically schematic embedding but not a DG schematic embedding. 3.17. Equivariant setting. Suppose that 푆 is a quasi-compact quasi-separated base scheme and 풢 Ñ 푆 is a quasi-separated quasi-compact group scheme over 푆. Suppose that 푃 is a quasi-compact quasi-separated 푆-scheme with an action of 풢. In this case, the semisimplicial bar complex:

/ / ... / 풢 ˆ 풢 ˆ 푃 / 풢 ˆ 푃 // 푃 / 푆 푆 / 푆 (3.17.1) induces the diagram:

! ! / ! / 퐷 p푃 q // 퐷 p풢 ˆ 푃 q / 퐷 p풢 ˆ 풢 ˆ 푃 q / .... 푆 / 푆 푆 / and we deﬁne the 풢-equivariant derived category 퐷!p푃 q풢 of 푃 to be the limit of this diagram.

Example 3.17.1. Suppose that 풢 is constant, i.e., 풢 “ 푆ˆ풢0 for some quasi-compact quasi- separated ! group scheme 풢0 over Specp푘q. Then, by Corollary 3.15.1, 퐷 p풢0q obtains a coalgebra structure in DGCat푐표푛푡 in the usual way (e.g. the comultiplication is !-pullback along the multiplication for 풢0). ! ! ! 풢 As such, 퐷 p풢0q coacts on 퐷 p푃 q and 퐷 p푃 q is the usual (strongly) 풢0-equivariant category, i.e., the limit of the diagram:

/ 퐷!p푃 q / 퐷!p풢 q b 퐷!p푃 q // 퐷!p풢 q b 퐷!p풢 q b 퐷!p푃 q / .... / 0 / 0 0 / 1 Let 풫풢 Ñ 푆 be a 풢-torsor, i.e., 풢 acts on 풫풢 and after an appropriate fppf base-change 푆 Ñ 푆 we have a 풢-equivariant identiﬁcation:

1 1 풫풢 ˆ 푆 “ 풢 ˆ 푆 . 푆 푆 We obtain a canonical functor:

! ! 풢 휙 : 퐷 p푆q Ñ 퐷 p풫풢q . 16 SAM RASKIN

Proposition 3.17.2. In the above setting the functor 휙 is an equivalence.

Proof. By fppf descent (Proposition 3.12.1), we reduce to the case there 풫풢 is a trivial 풢-bundle over 푇 , i.e., 풫풢 “ 풢 ˆ푆 푇 . Then the bar complex extends to a split simplicial object in the usual way from which we deduce the result.

Remark 3.17.3. If 풫풢 Ñ 푆 is a 풢-torsor, we will sometimes summarize the situation in writing 푆 “ 풫풢{풢. 3.18. 퐷˚-modules. Next, we discuss the ˚-theory of 퐷-modules. We have the following analogue of Proposition 3.6.1. 푓.푡. Proposition 3.18.1. For 푆 P Sch푞푐푞푠 written as 푆 “ lim푖 푆푖 with 푆푖 P Sch and all structure morphisms affine, the functor:

˚ ˚ 퐷 p푆q Ñ lim 퐷 p푆푖q P DGCat푐표푛푡 푆Ñ푆푖 is an equivalence. Proof. The proof is similar to the proof of Proposition 3.6.1: As before, we denote the right hand side of the above by 퐷˚p푆q and note that it is independent of the choice of presentation 푆 “ lim 푆푖. The statement again reduces to an appropriate descent typer statement. Namely, for every 푆 P Sch푞푐푞푠 with cover 푆 “ 푈 Y 푉 for 푈, 푉 quasi-compact open subschemes, we claim that the functors:

퐷˚p푈q ‘ 퐷˚p푉 q Ñ 퐷˚p푆q 퐷˚p푈X푉 q 퐷˚p푈q ‘ 퐷˚p푉 q Ñ 퐷˚p푆q 퐷˚p푈X푉 q are equivalences. r r r r ˚ ˚ ˚ The key point again is that e.g. the pushforward functor 푗˚,푑푅 : 퐷 p푈q Ñ 퐷 p푆q (resp. 퐷 p푈q Ñ 퐷˚p푆q) is fully-faithful and admits a right adjoint compatible with base-change. This is an easy reduction to the finite type case for 퐷˚, and is proved for 퐷˚ by the same method asr how the analogousr statement was proved for 퐷! (c.f. the proof of Proposition 3.13.1). r 3.19. Recall that for 푆 a finite type scheme the category 퐷p푆q is self-dual under Verdier duality ! and for a map 푓 : 푇 Ñ 푆 between finite type schemes the functor dual to 푓 is 푓˚,푑푅. Therefore, for 푆 a quasi-compact quasi-separated scheme we obtain the following from [Gai1]. Proposition 3.19.1. If 퐷!p푆q is a dualizable category, then its dual is canonically identified with 퐷˚p푆q.

! Note that in this case this is an identiﬁcation of p퐷!p푆q, bq-module categories. Moreover, the ! functor dual to 푓 continues to be 푓˚,푑푅. 3.20. Constant sheaf. For 푇 quasi-compact quasi-separated, there is a canonical constant sheaf ˚ 푘푇 P 퐷 p푇 q constructed as follows. 푓.푡. ˚ For any 푆 P Sch and 훼 : 푇 Ñ 푆, we deﬁne an object “훼˚,푑푅p푘푇 q” P 퐷p푆q “ 퐷 p푆q by the formula: D-MODULES ON INFINITE DIMENSIONAL VARIETIES 17

“훼˚,푑푅p푘푇 q” :“ colim 7˚,푑푅p푘푇 1 q. 훽 7 푇 ÝÑ푇 1ÝÑ푆 푇 1PSch푓.푡.,7˝훽“훼 For any triangle:

훼1 푇 / 푆1 ? 훼 푓 푆 with 푆 and 푆1 P Sch푓.푡., we have a canonical isomorphism:

1 » “훼˚,푑푅p푘푇 q” ÝÑ 푓˚,푑푅p“훼˚,푑푅p푘푇 q”q ˚ and therefore we obtain the object 푘푇 P 퐷 p푇 q (with each 훼˚,푑푅p푘푇 q “ “훼˚,푑푅p푘푇 q”) as desired. ˚,푑푅 ˚ Letting 푝푇 : 푇 Ñ Specp푘q denote the structure map, the continuous functor 푝푇 : Vect Ñ 퐷 p푇 q sending 푘 to 푘푇 is readily seen to be the left adjoint to 푝푇,˚,푑푅. 3.21. Correspondences. Next, we extend the functoriality of 퐷˚ as in S3.8. Let Sch푞푐푞푠,푐표푟푟;푓.푝.,푎푙푙 denote the category of quasi-compact quasi-separated schemes under correspondences of the form:

퐻 훼 훽 ~ 푇 푆 where 퐻 P Sch푞푐푞푠, 훼 is ﬁnitely presented and 훽 is arbitrary. Note that Sch푞푐푞푠,푐표푟푟;푓.푝.,푎푙푙 contains 푓.푡. Sch푐표푟푟 as a full subcategory. It also contains Sch푞푐푞푠 as a non-full subcategory where morphisms are correspondences where the left arrow is an isomorphism. We deﬁne the functor:

˚,푒푛ℎ 퐷 : Sch푞푐푞푠,푐표푟푟;푓.푝.,푎푙푙 Ñ DGCat푐표푛푡 푓.푡. by right Kan extension from Sch푐표푟푟. Like Proposition 3.8.1, the following is immediate from Lemma 3.10.1. ˚,푒푛ℎ Proposition 3.21.1. The restriction of 퐷 to Sch푞푐푞푠 canonically identifies with the functor ˚ 퐷 |Sch푞푐푞푠 : Sch푞푐푞푠 Ñ DGCat푐표푛푡. 3.22. For 푓 : 푇 Ñ 푆 a map of quasi-compact quasi-separated schemes, the induced functor:

퐷˚,푒푛ℎp푇 q “ 퐷˚p푇 q Ñ 퐷˚,푒푛ℎp푆q “ 퐷˚p푆q ˚ coincides with 푓˚,푑푅. If 푓 is ﬁnitely presented we will denote the corresponding functor 퐷 p푆q Ñ 퐷˚p푇 q by 푓¡ to avoid confusion with the functor 푓 ! : 퐷!p푇 q Ñ 퐷!p푆q. Note that the formalism of correspondences implies that we have base-change between ˚-pushforward and ¡-pullback for Cartesian squares. Remark 3.22.1. Suppose that 푓 : 푇 Ñ 푆 is ﬁnitely presented. One can compute the functor 푓¡ ¡ ! as follows. In the notation of Remark 3.9.1, for ℱ P 퐷p푆q we have 훼푖,˚,푑푅푓 pℱq “ 푓푖 훽푖,˚,푑푅pℱq by ¡ base-change, computing 푓 pℱq in 퐷p푇 q “ lim 퐷p푇푖q as promised. 18 SAM RASKIN

One deduces from Remark 3.22.1 the following result. Proposition 3.22.2. If 푓 : 푆 Ñ 푇 is a finitely presented proper morphism of quasi-compact ¡ quasi-separated schemes, then 푓 is canonically the right adjoint to 푓˚,푑푅. Similarly, we have: Proposition 3.22.3. If 푓 : 푆 Ñ 푇 is a finitely presented smooth map of quasi-compact quasi- ¡ separated schemes, then 푓 r´2푑푆{푇 s is left adjoint to 푓˚,푑푅, with 푑푆{푇 as in Proposition 3.9.3. 3.23. Descent. Next, we discuss descent for 퐷˚. Proposition 3.23.1. For 푓 : 푆 Ñ 푇 an ℎ-covering of quasi-compact quasi-separated schemes the functor:

퐷˚p푇 q Ñ lim 퐷˚pCech푛p푆{푇 qq 푛PΔ ¡ 푛 induced by the functors 푓푛 with 푓푛 : Cech p푆{푇 q Ñ 푇 the canonical map is an equivalence. Proof. Because 푓 is finite presentation we can apply Noetherian approximation to find 푓 1 : 푆1 Ñ 푇 1 an ℎ-covering between schemes of finite type and 푇 Ñ 푇 1 so that 푓 is obtained by base-change. 1 Let 푇 “ lim 푇푖 where each 푇푖 is a 푇 -scheme of finite type (and structure maps are affine) and let 1 푆푖 :“ 푇푖 ˆ푇 1 푆 . 푛 Then each 푆푖 Ñ 푇푖 is an ℎ-covering between finite type schemes. Note that Cech p푆{푇 q “ 푛 lim Cech p푆푖{푇푖q. Now we have:

˚ » 푛 푛 ˚ 푛 퐷 p푇 q “ lim 퐷p푇푖q ÝÑ lim lim 퐷pCech p푆푖{푇푖qq “ lim lim 퐷pCech p푆푖{푇푖qq “ lim 퐷 pCech p푆{푇 qq. 푖PI표푝 푖PI표푝r푛sPΔ r푛sPΔ푖PI표푝 r푛sPΔ Here the indicated isomorphism is by usual ℎ-descent for ﬁnite type schemes and Proposition 3.21.1. Variant 3.23.2. One can similarly show that the functor:

colim 퐷˚pCech푛p푆{푇 qq Ñ 퐷˚p푇 q r푛sPΔ deﬁned by de Rham pushforwards is an equivalence for 푆 Ñ 푇 an ℎ-covering. Indeed: it is easy to verify for Zariski coverings (the argument is basically the same as for Lemma 3.12.3), and for proper coverings, it follows automatically from Proposition 3.23.1. This is the statement that should properly be thought of as dual to Proposition 3.12.1. 3.24. Equivariant setting. Suppose that we are in the setting of S3.17, i.e., 풢 is a group scheme over 푆 that acts on an 푆-scheme 푃 . In this case, (3.17.1) deﬁnes the coequivariant derived category:

˚ / ˚ / ˚ ˚ 퐷 p푃 q풢 :“ colim .... / 퐷 p풢 ˆ 풢 ˆ 푃 q / 퐷 p풢 ˆ 푃 q // 퐷 p푃 q (3.24.1) / 푆 푆 / 푆 with the colimit computed` in DGCat푐표푛푡. ˘ The analogue of Proposition 3.17.2 holds in this setting: if 푃 Ñ 푆 is an 풢-torsor, we obtain a functor:

˚ ˚ 퐷 p푃 q풢 Ñ 퐷 p푆q D-MODULES ON INFINITE DIMENSIONAL VARIETIES 19 that is an equivalence by essentially the same argument as in loc. cit, but using Variant 3.23.2 of Proposition 3.23.1.

4. Placidity 4.1. In this section, we introduce the notion of placidity and discuss its consequences. Recall from S1.6 that placidity is a technical condition on the singularities of a scheme 푆 allowing us to identify 퐷!p푆q with 퐷˚p푆q. ! ! !,푟푒푛 ! For a morphism 푓 : 푆 Ñ 푇 of placid schemes, we let 푓˚,푟푒푛 : 퐷 p푆q Ñ 퐷 p푇 q and 푓 : 퐷 p푇 q Ñ 퐷!p푆q denote the corresponding functors, obtained through the above identification. ! In general, these renormalized functors are very badly behaved, e.g., the pairs p푓 , 푓˚,푟푒푛q and !,푟푒푛 p푓 , 푓˚,푑푅q do not satisfy base-change. In S4.10, we introduce a notion of placid morphism, which is something like a pro-smooth morphism. Proposition 4.11.1 (generalized to the indschematic setting by Proposition 6.18.1) says that ! !,푟푒푛 for placid morphism, 푓 is left adjoint to 푓˚,푟푒푛, and similarly, 푓 is left adjoint 푓˚,푑푅. Here the dimension shifts implicit in the infinite dimensional setting work out to eliminate the usual cohomological shifts needed to make such statements in the finite dimensional setting. Moreover, Proposition 4.11.1 implies that there are good base-change properties for placid morphisms. 4.2. Definition of placidity. We now give the definition of placidity.

Definition 4.2.1. For 푇 P Sch we say an expression 푇 “ lim푖PI표푝 푇푖 is a placid presentation of 푇 if: (1) The indexing category I is filtered. (2) Each 푇푖 is finite type over 푘. (3) For every 푖 Ñ 푗 in I the corresponding map 푇푗 Ñ 푇푖 is an affine smooth covering. We say that 푇 P Sch is placid if it admits a placid presentation. Example 4.2.2. As is well known from the theory of group schemes, any affine group scheme is placid (we need the characteristic zero assumption on 푘 here). Example 4.2.3. Suppose that 푆 is a finite type scheme and 풢 Ñ 푆 is a projective limit under smooth surjective affine maps of smooth 푆-group schemes. Suppose that 풫풢 Ñ 푆 is a 풢-torsor in the sense of S3.17. Then 풫풢 is placid. Example 4.2.4. For a Cartesian square:

푆2 / 푇2

푆1 / 푇1 with 푇1 ﬁnite type, 푆1 and 푇2 placid, the scheme 푆2 is necessarily placid. Indeed, for 푆1 “ lim푖PI표푝 푆1,푖 and 푇2 “ lim푗PJ표푝 푇2,푗 placid presentations by 푇1-schemes, we have:

푆2 “ lim 푆1,푖 ˆ 푇2,푗. 표푝 표푝 p푖,푗qPI ˆJ 푇1 Obviously all structure maps are smooth aﬃne covers, so this is a placid presentation of 푆2. Remark 4.2.5. By Noetherian descent, if 푆 is placid and 푇 Ñ 푆 is ﬁnite presentation, then 푇 is placid as well. Moreover, there always exist placid presentations 푆 “ lim푖PI표푝 푆푖, 푇 “ lim푖PI표푝 푇푖 and compatible morphisms 푇푖 Ñ 푆푖 inducing 푇 Ñ 푆, and such that, for every 푖 Ñ 푗 P I, the diagram: 20 SAM RASKIN

푇푗 / 푆푗

푇푖 / 푆푖 is Cartesian.

Remark 4.2.6. By [Gro] Corollary 8.3.7, given a placid presentation 푇 “ lim푖 푇푖, each structure morphism 푇 Ñ 푇푖 is surjective on schematic points. Remark 4.2.7. A placid scheme is automatically quasi-compact and quasi-separated. Remark 4.2.8. As explained in [Dri], many natural schemes are Nisnevich (in particular: ´etale) locally placid, but not themselves placid. Much of the material that follows generalizes without diﬃculty to the case of Nisnevich locally placid schemes. However, we do not pursue this level of generality here because for our later applications, only placid schemes (and indschemes) occur.

4.3. If 푇 is a placid scheme with placid presentation 푇 “ lim푖PI표푝 푇푖 then we have:

˚ 퐷 p푇 q “ colim 퐷p푇푖q (4.3.1) 푖PI where the structure functors are the ˚-pullback functors (deﬁned because the maps 푇푗 Ñ 푇푖 are 표푝 ˚,푑푅 smooth). For 푖 P I and 푓푖 : 푇 Ñ 푇푖 the corresponding structure map, we let 푓푖 denote the ˚ ˚ functor 퐷 p푇푖q Ñ 퐷 p푇 q left adjoint to 푓푖,˚,푑푅. In particular, we see that 퐷˚p푇 q is compactly generated and therefore canonically dual to 퐷!p푇 q, which is also compactly generated. (Note that in the 퐷!-case, compact objects are !-pullbacks of compact objects from ﬁnite type schemes, where for 퐷˚ they are ˚-pullbacks). Similarly, we obtain:

! 퐷 p푇 q “ lim 퐷p푇푖q (4.3.2) 푖PI표푝 ! where the structure functors are the right adjoints to the 푓푖 functors, i.e., shifted de Rham cohomology functors (again, these are adjoint by smoothness).

˚ Remark 4.3.1. It follows from the identiﬁcation of 퐷 as a colimit that for placid 푇 “ lim푖PI표푝 푇푖 as above and F P 퐷˚p푇 q, the canonical map:

˚,푑푅 colim 푓푖 푓푖,˚,푑푅pFq Ñ F (4.3.3) 푖PI is an equivalence. 4.4. Let 푇 be a quasi-compact quasi-separated scheme. Let Presp푇 q denote the 1-category whose objects are placid presentations pI, t푇푖u푖PIq of 푇 and 1 2 where morphisms pI, t푇푖 u푖PIq Ñ pJ, t푇푗 u푗PJq are given by a datum:

1 2 퐹 : I Ñ J and t푓푖 : 푇푖 Ñ 푇퐹 p푖qu푖PI compatible morphisms of schemes under 푇. One easily shows that Presp푇 q is ﬁltered. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 21

1 2 4.5. Fix two placid presentations pI, t푇푖 u푖PIq and pJ, t푇푗 u푗PJq of a scheme 푇 . We will make use of the following observation. 1 2 Lemma 4.5.1. For every 푗 P J and every factorization 푇 Ñ 푇푖 Ñ 푇푗 for 푖 P I, the morphism 1 2 푇푖 Ñ 푇푗 is smooth. 1 Proof. Suppose 푥 is a geometric point of 푇 . For each 푖 P I, let 푥푖1 denote the corresponding 1 geometric point of 푇푖1 . Applying Proposition 3.5.1, we obtain:

˚ 1 ˚ 1 ˚ 1 ˚ 1 ˚ 1 Coker 푥푖 pΩ푇 1{푇 2 q Ñ 푥 pΩ푇 {푇 2 q “ colim Coker 푥푖 pΩ푇 1{푇 2 q Ñ 푥푖1 pΩ푇 1 {푇 2 q “ colim 푥푖1 pΩ푇 1 {푇 1 q. 푖 푗 푗 푖1PI 푖 푗 푖1 푗 푖1PI 푖1 푖 ˆ ˙ 푖{ ˆ ˙ 푖{ Because the structure maps 푇푗 Ñ 푇푖 are smooth the right hand side is a ﬁltered limit of vector spaces concentrated in degree 0 and therefore is concentrated in degree 0 as well. On cohomology we obtain a long exact sequence with segments:

푖´1 ˚ 1 푖 ˚ 1 푖 ˚ 1 ... Ñ 퐻 colim 푥 1 pΩ 1 1 q Ñ 퐻 푥푖 pΩ 1 2 q Ñ 퐻 푥 pΩ 2 q Ñ .... 1 푖 푇 1 {푇 푇 {푇 푇 {푇 푖 PI푖{ 푖 푖 푖 푗 푗 ´ ¯ ´ ¯ ´ ˚ 1 ¯ The left term is zero for 푖 ‰ 1 and the right term is zero for 푖 ‰ 0. But 푥푖 pΩ 1 2 q is tautologically 푇푖 {푇푗 concentrated in degrees ď 0, so it is concentrated in degree 0 as desired. 4.6. Dimensions. We digress briefly to fix some terminology regarding dimensions. ě0 Let 푇 be a finite type scheme. We define the dimension function dim푇 : 푇 Ñ Z to be the locally constant function that on a connected component is constant with value the Krull dimension of that connected component (i.e., the maximal dimension of an irreducible component of this connected component). For 푓 : 푇 Ñ 푆 a map between finite type schemes, we let dim푇 {푆 : 푇 Ñ Z be the locally constant ˚ function dim푇 ´푓 pdim푆q.

Example 4.6.1. If 푓 : 푇 Ñ 푆 is a smooth dominant morphism, then dim푇 {푆 is the rank of the vector 1 bundle Ω푇 {푆. Therefore, for a Cartesian diagram of ﬁnite type schemes:

휓 푇 1 / 푇

푔 푓 휙 푆1 / 푆 ˚ with 휙 and 휓 both dominant smooth morphisms, dim푇 1{푆1 “ 휓 pdim푇 {푆q. In particular, this identity holds whenever 휙 is a smooth covering map. : 1 Counterexample 4.6.2. We need not have dim푇 {푆 “ 푑푆{푇 “ rankpΩ푇 {푆q if 푓 : 푇 Ñ 푆 is smooth but not dominant. 2 1 1 For example, let 푆 “ A 0 A be a line and a plane glued along a point, and let 푇 “ G푚 ˆ A mapping to 푆 via the composition: š 1 1 2 1 G푚 ˆ A Ñ G푚 ãÑ A ãÑ A A . 0 ž 22 SAM RASKIN

Then 푑푆{푇 the constant function 1, while dim푆{푇 is the constant function dim푇 ´ dim푆 “ 2´2 “ 0.

Remark 4.6.3. By Remark 4.2.5, we see from Example 4.6.1 that dim푇 {푆 can be defined as a locally constant function 푇 Ñ Z for any finitely presented morphism 푇 Ñ 푆 of placid schemes by Noetherian descent. 푓 Given a pair of finitely presented morphisms 푇 ÝÑ 푆 Ñ 푉 of placid schemes, this construction satisfies the basic compatibility:

˚ dim푇 {푉 “ dim푇 {푆 ´푓 pdim푆{푉 q. (4.6.1) 4.7. Renormalized dualizing sheaf. Suppose that 푇 is placid scheme. We will now deﬁne the 푟푒푛 ˚ renormalized dualizing sheaf 휔푇 P 퐷 p푇 q. Fix a placid presentation 푇 “ lim푖PI표푝 푇푖 of 푇 . Because each structure map 휙푖푗 : 푇푗 Ñ 푇푖 is a smooth covering, we have canonical identiﬁcations:

˚,푑푅 휙푖푗 p휔푇푖 r´2 ¨ dim푇푖 sq “ 휔푇푗 r´2 ¨ pdim푇푗 qs. 푟푒푛 Therefore we have a uniquely deﬁned sheaf 휔푇 characterized by the fact that it is the ˚-pullback of 휔푇푖 r´2 ¨ dim푇푖 s from any 푇푖 to 푇 . 푟푒푛 We claim that 휔푇 canonically does not depend on the choice of placid presentation. Indeed, this follows from Lemma 4.5.1 and by ﬁlteredness of Presp푇 q.

푟푒푛 ˚ Example 4.7.1. Let 푇 be ﬁnite type. Then 휔푇 P 퐷 p푇 q “ 퐷p푇 q identiﬁes with 휔푇 r´2 ¨ dim푇 s.

Example 4.7.2. Suppose 푇 admits a placid presentation 푇 “ lim 푇푖 with each 푇푖 smooth. Then 푟푒푛 휔푇 “ 푘푇 . 4.8. Suppose that 푇 is a placid scheme. We deﬁne the functor:

! ˚ 휂푇 : 퐷 p푇 q Ñ 퐷 p푇 q 푟푒푛 by action on 휔푇 .

Proposition 4.8.1. The functor 휂푇 is an equivalence.

Proof. Choose 푇 “ lim푖PI표푝 푇푖 a placid presentation. We claim that the functor:

! ˚ (4.3.1) 휂푇 : 퐷 p푇 q :“ colim 퐷p푇푖q Ñ 퐷 p푇 q “ colim 퐷p푇푖q. 푖PI 푖PI is the colimit of the shifted identity functors id퐷p푇푖qr´2¨dim푇푖 s. Indeed, the colimit of these functors ! ! 푟푒푛 ˚ is a morphism of 퐷 p푇 q-module categories and sends 휔푇 P 퐷 p푇 q to 휔푇 P 퐷 p푇 q. Now the result obviously follows from this identiﬁcation. ! ˚ Example 4.8.2. If 푇 is ﬁnite type then 휂푇 is the composite equivalence 퐷 p푇 q :“ 퐷p푇 q “: 퐷 p푇 q shifted by ´2 dim푇 . 4.9. Renormalized functors. Let 푓 : 푇 Ñ 푆 a map of placid schemes. ! ! We let 푓˚,푟푒푛 : 퐷 p푇 q Ñ 퐷 p푆q denote the induced functor so that we have the commutative diagram: D-MODULES ON INFINITE DIMENSIONAL VARIETIES 23

푓˚,푟푒푛 퐷!p푇 q / 퐷!p푆q

» 휂푇 » 휂푆

푓˚,푑푅 퐷˚p푇 q / 퐷˚p푆q.

In the same way we obtain the functor 푓 !,푟푒푛 : 퐷˚p푆q Ñ 퐷˚p푇 q ﬁtting into a commutative diagram:

푓 !,푟푒푛 퐷˚p푆q / 퐷˚p푇 q O O 휂푆 » 휂푇 » 푓 ! 퐷!p푆q / 퐷!p푇 q Note that we have a canonical isomorphism

!,푟푒푛 푟푒푛 푟푒푛 푓 p휔푆 q “ 휔푇 (4.9.1) because:

! ! !,푟푒푛 푟푒푛 ! 푟푒푛 푟푒푛 푟푒푛 푓 p휔푆 q “ 푓 p휔푆q b 휔푇 “ 휔푇 b 휔푇 “ 휔푇 . Example 4.9.1 (Renormalized functors in finite type: 퐷!). Suppose 푓 : 푇 Ñ 푆 is a map between finite type schemes. We identify 퐷!p푆q and 퐷!p푇 q with 퐷p푆q and 퐷p푇 q in the canonical way. Then the functor 푓˚,푟푒푛 : 퐷p푇 q Ñ 퐷p푆q identifies with 푓˚,푑푅r´2 ¨ dim푇 {푆s. In particular, if 푓 is ! smooth and dominant, then p푓 , 푓˚,푟푒푛q form an adjoint pair of functors. Note that in this setting the functor 푓˚,!´푑푅 coincides with the (non-renormalized) functor 푓˚,푑푅.

Warning 4.9.2. If 푓 : 푆 Ñ 푇 is a closed embedding of placid schemes, then 푓˚,푟푒푛 is not left adjoint ! to 푓 (c.f. Example 4.9.1). In fact, if 푓 is a closed embedding of inﬁnite codimension, then 푓˚,푟푒푛 does not preserve compact objects and therefore does not admit a continuous right adjoint at all. ! Moreover, the composition 푓 푓˚,푟푒푛 is typically zero in this case. Warning 4.9.3. Given a Cartesian diagram:

휓 푇1 / 푆1

휑 푓 푔 푇2 / 푆2 of ﬁnite type schemes, we ﬁnd that:

! ! ! 푓 푔˚,푟푒푛 “ 푓 푔˚,푑푅r´2 ¨ dim푇2{푆2 s “ 휓˚,푑푅휙 r´2 ¨ dim푇2{푆2 s ! ! while 휓˚,푟푒푛휙 “ 휓˚,푑푅휙 r´2 ¨ dim푇1{푆1 s. Since dimensions do not always behave well under base- change, we see that base-change does not always hold between renormalized pushforward and upper-!. 24 SAM RASKIN

Example 4.9.4 (Renormalized functors in finite type: 퐷˚). Suppose 푓 : 푇 Ñ 푆 is a map between finite type schemes. We identify 퐷˚p푆q and 퐷˚p푇 q with 퐷p푆q and 퐷p푇 q in the canonical way. !,푟푒푛 ! Then the functor 푓 : 퐷p푆q Ñ 퐷p푇 q identifies with 푓 p´qr´2 dim푇 {푆s. Note that if 푓 is smooth and dominant, then 푓 !,푟푒푛 identifies canonically with 푓 ˚,푑푅. The functor 푓¡ coincides with the (non-renormalized) functor 푓 !. Remark 4.9.5. We emphasize explicitly that the “renormalization” here has nothing to do with the renormalized de Rham cohomology functor from [DG]. Rather, the terminology is taken from [Dri] S6.8. 4.10. Placid morphisms. We will now further analyze the renormalized functors under certain very favorable circumstances. We say a morphism 푓 : 푆 Ñ 푇 of placid schemes is placid if, for any placid presentations 푆 “ lim푖PI표푝 푆푖, 푇 “ lim푗PJ표푝 푇푗, for every 푗 P J there exists 푖 P I with the morphism 푆 Ñ 푇 Ñ 푇푗 factoring as 푆 Ñ 푆푖 Ñ 푇푗 and with 푆푖 Ñ 푇푗 a smooth covering. Obviously, if this holds for one pair of placid presentations then it holds for any. Example 4.10.1. By Noetherian descent and Remark 4.2.6, a smooth morphism of finite presentation which is surjective on geometric points is placid.

Example 4.10.2. Suppose that 푆 “ lim푖PI표푝 푆푖 and 푇 “ lim푖PI표푝 푇푖 are placid presentations, and suppose that we are given compatible smooth coverings 푓푖 : 푆푖 Ñ 푇푖 inducing 푓 : 푆 Ñ 푇 . Then 푓 is a placid morphism. We emphasize that by compatible we mean that the relevant squares commute, not that they are Cartesian. Remark 4.10.3. For categorical arguments, it is convenient to use the following formulation of this deﬁnition. 푓.푡. Let Sch푠푚–푐표푣 denote the category of ﬁnite type schemes where we only allow smooth coverings as morphisms. Let:

aff 푓.푡. 푓.푡. Pro pSch푠푚–푐표푣q Ď PropSch푠푚–푐표푣q denote the full subcategory where we only allow objects obtained as projective limits under morphisms that are aﬃne (in addition to being a priori smooth coverings). Then the functor:

aff 푓.푡. aff 푓.푡. Pro pSch푠푚–푐표푣q Ñ Pro pSch q “ Sch푞푐푞푠 is a (non-full) embedding of categories. Indeed, this is a general feature: (non-full) embeddings of p1, 1q-categories induce embeddings on Ind or Pro categories, since filtered limits and colimits of injections in Set are still injections. Moreover, its essential image are placid schemes, and a morphism lies in this non-full subcategory if and only if it is placid. aff 푓.푡. Observe that Pro pSch푠푚–푐표푣q Ñ Sch푞푐푞푠 commutes with filtered projective limits with affine 푓.푡. structure maps, i.e., this functor is the right Kan extension of its restriction to Sch푠푚–푐표푣. In- 푓.푡. aff 푓.푡. deed, Sch푞푐푞푠 Ď PropSch q commutes with such filtered projective limits, and Pro pSch푠푚–푐표푣q Ď 푓.푡. 푓.푡. 푓.푡. PropSch푠푚–푐표푣q does too. Moreover, PropSch푠푚–푐표푣q Ñ PropSch q tautologically commutes with filtered limits, proving the claim. Warning 4.10.4. Against the usual conventions for terminology in algebraic geometry, placid morphisms are not intended as a relative form of placidity. Indeed, we can only speak about placid morphisms between between schemes already known to be placid. Moreover, for a placid scheme 푆, the structure map 푆 Ñ Specp푘q may not be placid. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 25

The terminology is rather taken by analogy with the deﬁnition of placid schemes, as in Remark 4.10.3. Counterexample 4.10.5. It may be tempting to think of placid morphisms as being analogous to being a smooth covering morphisms, since this condition is equivalent for ﬁnite type schemes. The following example is meant to show the geometric limitations of this line of thought. 1 푛 1 푛 1 Let A ˆ A Ñ A ˆ A ´ by:

휆, p푥1, . . . , 푥푛q ÞÑ 휆, p푥1 ´ 휆 ¨ 푥2, . . . , 푥푛´1 ´ 휆 ¨ 푥푛q . Each of these morphisms´ is a smooth¯ covering.´ Moreover, these morphisms¯ are compatible as 푛 varies, and therefore induce a placid morphism (of inﬁnite type):

1 8 1 8 A ˆ A Ñ A ˆ A p휆, 푥1, 푥2,...q ÞÑ p휆, 푥1 ´ 휆 ¨ 푥2, 푥2 ´ 휆 ¨ 푥3,...q. 8 푛 푚 푛 where we use the notation A “ lim푛 A , the limit taken under structure maps A Ñ A (푚 ě 푛) of projection onto the ﬁrst 푛-coordinates. 1 Then for 0 ‰ 휆 P 푘, the ﬁber of this map at p휆, 0, 0,..., 0q is a copy of A , realized as the loci of points:

´1 ´2 p휆, 푥1, 휆 ¨ 푥1, 휆 ¨ 푥1,...q 1 with 푥1 P A arbitrary. However, the ﬁber at p0, 0, 0,...q is just the point scheme Specp푘q, realized as the locus p0, 0, 0,...q. Lemma 4.10.6. Given a Cartesian diagram:

휙 푆2 / 푇2

휓 푔 푓 푆1 / 푇1 of placid schemes with 푔 finite presentation and 푓 a placid morphism, the morphism 휙 is placid as well.

Proof. Let 푆1 “ lim푖 푆1,푖 and 푇1 “ lim푗 푇1,푗 be placid presentations. We take a compatible placid presentation 푇2 “ lim푗 푇2,푗 as in Remark 4.2.5. Note that:

푆2 “ lim lim 푆1,푖 ˆ 푇2,푗 푗 푖 푇1,푗 where we really only take the limit under 푖 such that the map 푆1 Ñ 푇1,푗 factors (necessarily uniquely) through 푆1,푖. For a pair of morphisms p푖1 Ñ 푖2q and p푗1 Ñ 푗2q, we claim that the induced map:

푆1,푖2 ˆ 푇2,푗2 Ñ 푆1,푖1 ˆ 푇2,푗1 푇1,푗2 푇1,푗1 is an aﬃne smooth covering. Indeed, we have 푇 “ 푇 ˆ 푇 so that the left hand side 2,푗2 1,푗2 푇1,푗1 2,푗1 of the above is 푆1,푖2 ˆ푇1,푗1 푇2,푗1 . Because 푆1,푖2 Ñ 푆1,푖1 is an aﬃne smooth covering, we obtain the claim.

Therefore, the terms 푆1,푖 ˆ푇1,푗 푇2,푗 deﬁne a placid presentation of 푆2. But each map: 26 SAM RASKIN

푆1,푖 ˆ 푇2,푗 Ñ 푇2,푗 푇1,푗 is a smooth covering because each 푆1,푖 Ñ 푇1,푗 is assumed to be, completing the proof. The following results from the argument above. Corollary 4.10.7. Suppose that we have a Cartesian square:

휙 푆2 / 푇2

휓 푔 푓 푆1 / 푇1 ˚ of placid schemes with 푔 finite presentation and 푓 a placid morphism. Then dim푆2{푆1 “ 휙 pdim푇2{푇1 q.

Proof. Let 푆1 “ lim푖 푆1,푖, 푇1 “ lim푗 푇1,푗 and 푇2 “ lim푗 푇2,푗 be as in the proof of Lemma 4.10.6. As in loc. cit., we have a placid presentation of 푆2 with terms:

푆1,푖 ˆ 푇2,푗. 푇1,푗

Fixing and index 푗0, as in loc. cit., we have:

푆1,푖 ˆ 푇2,푗 “ 푆1,푖 ˆ 푇2,푗0 . 푇1,푗 푇1,푗0 for every morphism 푗0 Ñ 푗. Therefore, the morphisms 푆1,푖 ˆ 푇2,푗 Ñ 푆1,푖 are obtained one from 푇1,푗 another by base-change, so that dim푆2{푆1 is deﬁned as the pullback of the function:

dim푆1,푖 ˆ 푇2,푗 {푆1,푖 푇1,푗 for any choice of indices. But because our maps are smooth coverings, this function is the pullback of dim푇2,푗 {푇1,푗 , giving the result. 4.11. For our purposes, the key feature of placid morphisms is given by the following proposition. Proposition 4.11.1. (1) For a placid morphism 푓 : 푆 Ñ 푇 of placid schemes, the left adjoint ˚,푑푅 ˚ ˚ 푓 to 푓˚,푑푅 : 퐷 p푆q Ñ 퐷 p푇 q is defined. (2) For a placid morphism 푓 : 푆 Ñ 푇 of placid schemes, there is a canonical identification 푓 !,푟푒푛 » 푓 ˚,푑푅 : 퐷˚p푇 q Ñ 퐷˚p푆q. More precisely, with Sch푝푙 denoting the category of placid schemes under placid morphisms, there is a canonical identification of functors:

˚ ˚,푑푅 ˚ !,푟푒푛 표푝 p퐷 , 푓 q » p퐷 , 푓 q : Sch푝푙 Ñ DGCat푐표푛푡 표푝 inducing the identity over the maximal subgroupoid of Sch푝푙 . (3) For a placid morphism 푓 : 푆 Ñ 푇 of placid schemes, the functor 푓 ! : 퐷!p푇 q Ñ 퐷!p푆q admits a right adjoint, and this right adjoint is functorially identified with 푓˚,푟푒푛 in the sense above. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 27

(4) For a Cartesian square of placid schemes:

휙 푆2 / 푇2

휓 푔 (4.11.1) 푓 푆1 / 푇1 with 푓 placid and 푔 finitely presented, the canonical morphisms:

!,푟푒푛 !,푟푒푛 푓 푔˚,푑푅 Ñ 휓˚,푑푅휙 ! ! 푓 푔˚,푟푒푛 Ñ 휓˚,푟푒푛휙 arising from the adjunctions above are equivalences. We begin with the following general remarks. 푙푎푑푗 Let DGCat푐표푛푡 denote the category of cocomplete DG categories under 푘-linear functors that 푟푎푑푗 admit continuous right adjoints. Let DGCat푐표푛푡 denote the category of cocomplete DG categories under 푘-linear functors that admit left adjoints. 푙푎푑푗 푟푎푑푗,표푝 We have an obvious equivalence DGCat푐표푛푡 » DGCat푐표푛푡 given by passing to the adjoint functor. One easily veriﬁes: 푙푎푑푗 푙푎푑푗 Lemma 4.11.2. The category DGCat푐표푛푡 admits colimits, and the functor DGCat푐표푛푡 Ñ DGCat푐표푛푡 푟푎푑푗 푟푎푑푗 preserves these colimits. Similarly, DGCat푐표푛푡 admits limits, and the functor DGCat푐표푛푡 Ñ DGCat푐표푛푡 commutes with limits.

Proof. The content is that given a diagram 푖 ÞÑ C푖 of cocomplete DG categories under structure functors admitting continuous right adjoints, a functor C :“ colim푖 C푖 Ñ D admits a continuous right adjoint if and only if each C푖 Ñ D does. But this is obvious, since C is then also the limit of the C푖 under the right adjoint functors.

Proof of Proposition 4.11.1. Recall from Remark 4.10.3 that Sch푝푙 is the full subcategory:

aff 푓.푡. 푓.푡. Pro pSch푠푚–푐표푣q Ď PropSch푠푚–푐표푣q. 푓.푡. Moreover, because Sch푝푙 Ñ Sch푞푐푞푠 is the right Kan extension of its restriction to Sch푠푚–푐표푣, we ˚ ˚ see that 퐷 |Sch is the right Kan extension of 퐷 | 푓.푡. “ 퐷| 푓.푡. . 푝푙 Sch푠푚–푐표푣 Sch푠푚–푐표푣 ˚ 푟푎푑푗 Moreover, note that 퐷 | 푓.푡. factors through DGCat by smoothness. Sch푠푚–푐표푣 푐표푛푡 As in Example 4.9.4, the corresponding functor:

푟푎푑푗 푙푎푑푗,표푝 퐷| 푓.푡. Ñ DGCat » DGCat Sch푠푚–푐표푣 푐표푛푡 푐표푛푡 !,푟푒푛 identiﬁes with p퐷, 푓 q| 푓.푡. , i.e., the functor attaching to a scheme of ﬁnite type its category Sch푠푚–푐표푣 of 퐷-modules, and to a smooth surjective morphism of schemes the corresponding renormalized pullback functor.5 푟푎푑푗 By Lemma 4.11.2, the right Kan extension of this functor also factors through DGCat푐표푛푡, proving 표푝 푙푎푑푗 (1). Moreover, it follows that the corresponding functor to Sch푝푙 Ñ DGCat encoding the left !,푟푒푛 adjoints is the left Kan extension of p퐷, 푓 q| 푓.푡.,표푝 . Sch푠푚–푐표푣 We have an equivalence:

5This identification is treated formally in the homotopical setting in[GR]. 28 SAM RASKIN

!,푟푒푛 ! p퐷, 푓 q|Sch푓.푡.,표푝 » p퐷, 푓 q|Sch푓.푡.,표푝 ´1 ! ! computed termwise on a ﬁnite type scheme 푆 as 휂푆 . Moreover, p퐷 , 푓 q is the left Kan extension of the left hand side. For a placid scheme 푆 with placid presentation 푆 “ lim 푆푖, we have:

! ! ˚ 휂푆 “ colim 휂푆 : 퐷 p푆q “ colim 퐷 p푆푖q Ñ colim 퐷 p푆푖q “ 퐷p푆q 푖 푖 푖 푖 the colimit on the right taken under renormalized pullback functors (equivalently: ˚-dR pullback). Indeed, this was already observed in the proof of Proposition 4.8.1. ! !,푟푒푛 !,푟푒푛 Therefore, we see that p퐷 , 푓 q is the left Kan extension of p퐷, 푓 q| 푓.푡. , as desired. Sch푠푚–푐표푣 This completes the proof of (2). Note that (3) is a formal consequence of (2). Therefore, it remains to show (4). Suppose we are given a Cartesian square (4.11.1). It obviously suﬃces to show either of the !,푟푒푛 !,푟푒푛 base-change morphisms is an equivalence, so we treat the map 푓 푔˚,푑푅 Ñ 휓˚,푑푅휙 . First, suppose that 푇1 and 푇2 are ﬁnite type. We take a placid presentation 푆1 “ lim푖 푆1,푖. We can assume each 푆1,푖 is a 푇1-scheme by Noe- therian approximation.

Because 푆1 Ñ 푇1 is placid, each 푆1,푖 Ñ 푇1 is a smooth covering. Deﬁne 푆2,푖 “ 푆1,푖 ˆ푇1 푇2. We use the notation:

훽푖 휙푖 푆2 / 푆2,푖 / 푇2

휓 휓푖 푔 (4.11.2)

훼푖 푓푖 푆1 / 푆1,푖 / 푇1. We now have:

!,푟푒푛 !,푟푒푛 !,푟푒푛 푓˚,푑푅푓 푔˚,푑푅 “ colim 푓푖,˚,푑푅푓 푔˚,푑푅 “ colim 푓푖,˚,푑푅휓푖,˚,푑푅휙 “ 푖 푖 푖 푖 !,푟푒푛 !,푟푒푛 !,푟푒푛 colim 푔˚,푑푅휙푖,˚,푑푅휙 “ 푔˚,푑푅휙˚,푑푅휙 “ 푓˚,푑푅휓˚,푑푅휙 푖 푖 Here the first and fourth equalities follows from filteredness of our index category and the adjunctions. The base-change in our second equality follows from the usual smooth base-change theorem in the finite type setting. Applying the above argument to the left square of (4.11.2) and applying (finite dimensional) smooth base-change to the right square, we see that the map:

!,푟푒푛 !,푟푒푛 훼푖,˚,푑푅푓 푔˚,푑푅 Ñ 훼푖,˚,푑푅휓˚,푑푅휙 is always an equivalence. But this suffices to see our base-change by definition of 퐷˚. We now treat the case of general 푔 of finite presentation. Suppose that we have a diagram:

휙 휃 1 푆2 / 푇2 / 푇2

휓 푔 푔1 (4.11.3) 푓 휀 1 푆1 / 푇1 / 푇1 1 with both squares Cartesian, the schemes 푇푖 of ﬁnite type, and the maps 휃 and 휀 placid. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 29

Then we have base-change maps:

!,푟푒푛 !,푟푒푛 1 !,푟푒푛 !,푟푒푛 !,푟푒푛 !,푟푒푛 푓 휀 푔˚,푑푅 Ñ 푓 푔˚,푑푅휃 Ñ 휓˚,푑푅휙 휃 . By our earlier analysis, the ﬁrst map is an equivalence by considering the right square of (4.11.3), and the composite map is also an equivalence by considering the outer square of (4.11.3). Therefore, we see that the map:

!,푟푒푛 !,푟푒푛 !,푟푒푛 !,푟푒푛 푓 푔˚,푑푅휃 Ñ 푓 푔˚,푟푒푛휃 1 !,푟푒푛 is an equivalence. Varying 푇1 over some placid presentation of 푇1, the corresponding functors 휃 ˚ generate 퐷 p푇2q, so this suﬃces. 4.12. As a consequence of Proposition 4.11.1, we show that some features from Examples 4.9.1 and 4.9.4 survive to greater generality. Proposition 4.12.1. For 푓 : 푇 Ñ 푆 a finitely presented morphism of placid schemes, wehave canonical identifications:

¡ !,푟푒푛 ˚ ˚ 푓 r´2 ¨ dim푇 {푆s “ 푓 : 퐷 p푆q Ñ 퐷 p푇 q ! ! 푓˚,!´푑푅r´2 ¨ dim푇 {푆s “ 푓˚,푟푒푛 : 퐷 p푇 q Ñ 퐷 p푆q. where dim푇 {푆 is defined as in S4.6.

Proof. Let 푆 “ lim 푆푖 be a placid presentation, and by Remark 4.2.5, we may assume we have a placid presentation 푇 “ lim 푇푖 so that we have maps 푓푖 : 푇푖 Ñ 푆푖 with each 푖 Ñ 푗 inducing a Cartesian diagram, and with 푓 obtained by base-change from each of the 푓푖. Note that dim푇 {푆 is then obtained by pullback from each dim푇푖{푆푖 . We use the notation:

휓푖 푇 / 푇푖

푓 푓푖

휙푖 푆 / 푆푖. For the ﬁrst part, note that by (4.3.3) and Example 4.9.4, we have:

¡ ˚,푑푅 ¡ ˚,푑푅 ¡ ˚,푑푅 !,푟푒푛 푓 “ colim 휓 휓푖,˚,푑푅푓 “ colim 휓 푓 휙푖,˚,푑푅 “ colim 휓 푓 휙푖,˚,푑푅r2 ¨ dim푇 {푆s. 푖 푖 푖 푖 푖 푖 푖 푖 ˚,푑푅 !,푟푒푛 By Proposition 4.11.1, 휓푖 “ 휓푖 . Therefore, we compute the above as:

!,푟푒푛 !,푟푒푛 !,푟푒푛 !,푟푒푛 !,푟푒푛 colim 휓 푓 휙푖,˚,푑푅r2 ¨ dim푇 {푆s “ colim 푓 휙 휙푖,˚,푑푅r2 ¨ dim푇 {푆s “ 푓 r2 ¨ dim푇 {푆s 푖 푖 푖 푖 푖 !,푟푒푛 ˚,푑푅 by again applying (4.3.3) and the identiﬁcation 휙푖 “ 휙푖 . For the second part, note that we have functorial base change isomorphisms:

! ! 휙푖푓푖,˚,푟푒푛 » 푓˚,푟푒푛휓푖 by Proposition 4.11.1. By Example 4.9.1, 푓푖,˚,!´푑푅r´2 ¨ dim푇 {푆s “ 푓푖,˚,푟푒푛. Moreover, these cohomological shifts are compatible with varying 푖, so we obtain the result by deﬁnition of 푓˚,!´푑푅. 30 SAM RASKIN

Corollary 4.12.2. Suppose we are given a Cartesian square:

휙 푆2 / 푇2

휓 푔 푓 푆1 / 푇1 with 푆1 and 푇2 placid schemes, 푓 and 푔 placid morphisms, and 푇1 finite type. Then the canonical morphisms:

!,푟푒푛 !,푟푒푛 푓 푔˚,푑푅 Ñ 휓˚,푑푅휙 ! ! 푓 푔˚,푟푒푛 Ñ 휓˚,푟푒푛휙 are equivalences.

Proof. Note that we have already seen in Example 4.2.4 that 푆2 is actually a placid scheme. It tautologically suﬃces to prove that the ﬁrst base-change morphism is an equivalence. We form the diagram:

푖 푝2 푆2 / 푆1 ˆ 푇2 / 푇2

푔 휓 id푆1 ˆ푔 Δ푓 푝2 푆1 / 푆1 ˆ 푇1 / 푇1.

Here ∆푓 is the graph of 푓. Note that each of these squares is Cartesian. In particular, 푖 is a ﬁnitely presented morphism. We are reduced to proving the base-change result for each of these squares separately. For the right square, the result is essentially obvious: it follows from the compatibility of pushforward with products of schemes. For the left square, note that the base-change result holds with the upper-¡ functor in place of the renormalized upper-! functor by the correspondence formalism. Therefore, the result follows from Proposition 4.12.1.

5. Holonomic D-modules 5.1. In this section, we discuss the holonomic theory in infinite type. Remark 5.1.1. Since many “standard” 퐷-modules in infinite type are not compact (e.g., the delta 퐷-module concentrated at a point in an infinite type placid scheme), it is convenient to break conventions with the usual 퐷-module theory and allow some non-compact 퐷-modules to be counted as holonomic.

5.2. Holonomic 퐷-modules. Let 푆 be a scheme of ﬁnite type. Let 퐷푐표ℎ,ℎ표푙p푆q denote the full subcategory of 퐷푐표ℎp푆q (the compact objects in 퐷p푆q) composed of those coherent complexes with holonomic cohomologies, deﬁned in the usual way. Let 퐷ℎ표푙p푆q Ď 퐷p푆q denote the full subcategory:

퐷ℎ표푙p푆q :“ Indp퐷푐표ℎ,ℎ표푙p푆qq Ď 퐷p푆q. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 31

6 We refer to objects of 퐷ℎ표푙p푆q simply as holonomic objects. For 푓 : 푆 Ñ 푇 a map of ﬁnite type schemes, the usual theory of 퐷-modules implies that the ! functors 푓˚,푑푅 and 푓 preserve the subcategories of holonomic objects. For 푆 a quasi-compact quasi-separated scheme, we obtain the categories:

! ˚ 퐷ℎ표푙p푆q and 퐷ℎ표푙p푆q ! ˚ ! ! deﬁned by a Kan extension, as in the case of 퐷 and 퐷 . We have obvious functors 퐷ℎ표푙p푆q Ñ 퐷 p푆q ˚ ˚ ˚ and 퐷ℎ표푙p푆q Ñ 퐷 p푆q, the latter being fully-faithful. For 푆 placid, we can express 퐷ℎ표푙p푆q as a ˚ ˚ ˚ limit as for 퐷 p푆q, and therefore we see that 퐷ℎ표푙p푆q Ñ 퐷 p푆q is fully-faithful in this case as well. ˚ ˚ ! We refer to subobjects of 퐷 p푆q lying in 퐷ℎ표푙p푆q as holonomic objects, and similarly for 퐷 when 푆 is placid. ! ˚ We have upper-! and lower-* functors for 퐷ℎ표푙p푆q and 퐷ℎ표푙p푆q respectively, compatible with the forgetful functors.

Proposition 5.2.1. For 푓 : 푆 Ñ 푇 a morphism of quasi-compact quasi-separated schemes, the ˚ ˚ ˚,푑푅 morphism 푓˚,푑푅 : 퐷ℎ표푙p푆q Ñ 퐷ℎ표푙p푇 q admits a left adjoint 푓 . ! ! ! If 푇 is placid and 푓 is finitely presented, then the morphism 푓 : 퐷ℎ표푙p푇 q Ñ 퐷ℎ표푙p푆q admits a left adjoint 푓!. Moreover, in each of the above settings, these left adjoints are well-behaved with respect to maps ˚ ˚ to non-holonomic objects as well, i.e., the partially-defined left adjoints to 푓˚,푑푅 : 퐷 p푆q Ñ 퐷 p푇 q and 푓 ! : 퐷!p푇 q Ñ 퐷!p푆q are defined on holonomic objects, and these left adjoints preserve the holonomic subcategories (and therefore are computed by the above functors). Of course, we are assuming 푓 finitely presented and 푇 placid when discussing 푓!. We will prove this in S5.4 below.

5.3. We digress to prove the following lemma.

표푝 Lemma 5.3.1. Let I be an indexing category with I filtered. Let p푖 ÞÑ C푖q and p푖 ÞÑ D푖q are two I-shaped diagrams of cocomplete categories under continuous functors, with structure functors:

휓훼 : C푖 Ñ C푗 휓푖 : C :“ lim C푗 Ñ C푖 푗PI 휙훼 : D푖 Ñ D푗 휙푖 : D :“ lim D푗 Ñ D푖 푗PI for 훼 : 푖 Ñ 푗 in I and for 푖 P I. Suppose 퐺푖 : C푖 Ñ D푖 are compatible functors with induced functor:

퐺 : C Ñ D.

If each functor 퐺푖 admits a left adjoint 퐹푖, then 퐺 admits a left adjoint 퐹 : D Ñ C such that, for every 푗 P I, we have:

휓푗퐹 “ colim 휓훼퐹푖휙푖. 표푝 p훼:푖Ñ푗qPpI{푗 q

6We note that, of course, this condition completely ruins all the nice finiteness conditions that “usual” (coherent) holonomic complexes satisfy, e.g., finiteness of de Rham cohomology. This loss is obvious necessary for the infinite dimensional setting. 32 SAM RASKIN

Proof. For 푗 P I ﬁxed, note that for any diagram:

훽 훼 푖1 ÝÑ 푖 ÝÑ 푗 we have the natural map:

휙훽 Ñ 휙훽퐺푖1 퐹푖1 Ñ 퐺푖휓훽퐹푖1 . By adjunction, this gives rise to a map:

퐹푖휙훽 Ñ 휓훽퐹푖1 .

Composing on the left with 휓훼 and on the right with 휙푖1 , we obtain the map:

휓훼퐹푖휙훽휙푖1 “ 휓훼퐹푖휙푖 Ñ 휓훼˝훽퐹푖1 휙푖1 “ 휓훼휓훽퐹푖1 휙푖1 . Expressing this in the obvious homotopy-compatible way, we obtain a diagram of functors:

표푝 p훼 : 푖 Ñ 푗q P pI{푗q ÞÑ 휓훼퐹푖휙푖. Deﬁne the functor:

“휓푗퐹 ” :“ colim 휓훼퐹푖휙푖. 표푝 p훼:푖Ñ푗qPpI{푗 q As 푗 varies, we see by ﬁlteredness that these functors are homotopy compatible, and therefore we obtain a functor 퐹 : D Ñ C with the property that we have functorial identiﬁcations:

휓푗퐹 “ “휓푗퐹 ” with “휓푗퐹 ” as above. For every 푗 P I, we have the map:

휓푗퐹 퐺 “ colim 휓훼퐹푖휙푖퐺 “ 휓훼퐹푖퐺푖휓푖 Ñ 휓훼휓푖 “ 휓푗. 표푝 p훼:푖Ñ푗qPpI{푗 q As 푗 P I varies, these maps are homotopy compatible and therefore we obtain the counit map:

퐹 퐺 Ñ idC . Similarly, for every 푗 P I, we have the map:

휙푗 “ colim 휙푗 “ colim 휙훼휙푖 Ñ colim 휙훼퐺푖퐹푖휙푖 “ 표푝 표푝 표푝 p훼:푖Ñ푗qPpI{푗 q p훼:푖Ñ푗qPpI{푗 q p훼:푖Ñ푗qPpI{푗 q

colim 퐺푗휓훼퐹푖휙푖 “ 퐺푗휓푗퐹 “ 휙푗퐺퐹. 표푝 p훼:푖Ñ푗qPpI{푗 q As 푗 varies, these maps are homotopy compatible and therefore give the unit map:

idD Ñ 퐺퐹. One readily checks that the unit and counit maps constructed above actually deﬁne an adjunction. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 33

5.4. We now prove the proposition above. Proof of Proposition 5.2.1. For any map 푓 : 푆 Ñ 푇 , it is easy to see that we can arrange to have 푆 “ lim푖PI표푝 푆푖, 푇 “ lim푖PI표푝 filtered systems of finite type schemes under affine maps and with compatible maps 푓푖 : 푆푖 Ñ 푇푖 inducing 푓 in the limit (note that we do not assume any diagrams are Cartesian). Therefore, the existence of the left adjoint 푓 ˚,푑푅 follows immediately from Lemma 5.3.1. Let us see that these objects map in the obvious way to non-holonomic objects. For 훼 : 푖 Ñ 푗, let 휙푖 : 푆 Ñ 푆푖, 휙훼 : 푆푗 Ñ 푆푖, 휓푖 : 푇 Ñ 푇푖, 휓훼 : 푇푗 Ñ 푇푖 denote the structure maps. ˚ ˚ ˚ ˚ Note that e.g. 퐷ℎ표푙p푇 q Ñ 퐷 p푇 q is continuous. Therefore, for F P 퐷ℎ표푙p푇 q and G P 퐷 p푆q, we have:

˚,푑푅 ˚,푑푅 ˚ Hom퐷 p푇 qp푓 pFq, Gq “ lim lim Hom퐷p푇푖qp휓훼,˚,푑푅푓푗 휙푗,˚,푑푅pFq, 휓푖,˚,푑푅pGqq “ 푖 훼:푖Ñ푗 ˚,푑푅 lim Hom퐷p푇 qp푓 휙푖,˚,푑푅pFq, 휓푖,˚,푑푅pGqq “ Hom퐷p푇 qp휙푖,˚,푑푅pFq, 푓푖,˚,푑푅휓푖,˚,푑푅pGqq “ 푖 푖 푖 푖

˚ Hom퐷p푇푖qp휙푖,˚,푑푅pFq, 휙푖,˚,푑푅푓˚,푑푅pGqq “ Hom퐷 p푇 qpF, 푓˚,푑푅pGqq

For 푓 ﬁnite presentation, we can take placid presentations 푆 “ lim 푆푖 and 푇 “ lim 푇푖 as in Remark 4.2.5: by base-change, the upper-! functors are compatible with the shifted lower-* functors expressing 퐷˚ as a limit (using placidity), so Lemma 5.3.1 again applies. The same argument as above treats maps to non-holonomic objects. 5.5. We also have the following observation. ! ˚ Proposition 5.5.1. If 푆 is placid, then 휂푆 identifies the subcategories 퐷ℎ표푙p푆q and 퐷ℎ표푙p푆q. ! ! ˚ Proof. Suppose F P 퐷 p푆q. We will show that F P 퐷ℎ표푙p푆q if and only if 휂푆pFq P 퐷ℎ표푙pFq. Let 푆 “ lim푖 푆푖 be a placid presentation of 푆 and let 훼푖 : 푆 Ñ 푆푖 denote the structure maps. ˚ By deﬁnition, 휂푆pFq is in 퐷ℎ표푙pFq if and only if 훼푖,˚,푟푒푛pFq P 퐷ℎ표푙p푆푖q for every 푖. By (4.3.3) and Proposition 4.11.1, we have:

! F “ colim 훼푖훼푖,˚,푟푒푛pFq 푖 giving the result. ! ! To see that for F “ 퐷ℎ표푙p푆q we have 훼푖,˚,푟푒푛pFq P 퐷ℎ표푙p푆푖q, note that 퐷ℎ표푙p푆q is tautologically ! generated under colimits by objects 훼푗pF푗q, for F푗 P 퐷ℎ표푙p푆푗q. By ﬁlteredness of our indexing ! category, we can compute 훼푖,˚,푟푒푛훼푗pF푗q as a colimit of objects obtained by pushing and pulling along correspondences 푆푖 Ð 푆푘 Ñ 푆푗 (coming from correspondences 푖 Ñ 푘 Ð 푗 in the indexing category). !,푟푒푛 Corollary 5.5.2. For 푓 : 푆 Ñ 푇 a morphism of placid schemes, the functors 푓˚,푟푒푛 and 푓 preserve holonomic objects in 퐷! and 퐷˚ respectively.

6. D-modules on indschemes 6.1. In this section, we generalize the earlier material to treat the theory of 퐷-modules on indschemes, and especially on placid indschemes. The principal new feature is that for treating renormalized functors, we need a choice of dimension theory, which was only implicit in the discussion in the schemes case. 34 SAM RASKIN

6.2. Indschemes. We say that 푇 P PreStk is a (classical) indscheme if 푇 “ colim푖Pℐ 푇푖 in PreStk where ℐ is filtered, 푇푖 P Sch푞푐푞푠 Ď PreStk and each structure map 푇푖 Ñ 푇푗 is a closed embedding (recall that in this case 푇 P Stk Ď PreStk). 6.3. Correspondences. We say a morphism 푓 : 푇 Ñ 푆 of indschemes is finitely presented if 푓 is schematic and its base-change by any scheme is a finitely presented morphism. Exactly parallel to Propositions 3.8.1 and 3.21.1 one shows that 퐷! and 퐷˚ upgrade (via Kan extensions) to functors 퐷!,푒푛ℎ and 퐷˚,푒푛ℎ from the categories of indschemes under correspondences where the “right” (resp. “left”) map is finitely presented. ! ! For 푓 : 푆 Ñ 푇 finitely presented we have the corresponding functors 푓˚,!´푑푅 : 퐷 p푆q Ñ 퐷 p푇 q and 푓¡ : 퐷˚p푇 q Ñ 퐷˚p푆q. If 푓 is additionally assumed proper or smooth, we again have the usual adjunctions. 6.4. Reasonable indschemes. The following definition is taken from [BD] S7. Definition 6.4.1. A subscheme 푆 Ď 푇 is a reasonable subscheme of 푇 if 푆 is a quasi-compact quasi- separated closed subscheme such that, for every closed subscheme 푆1 of 푇 containing 푆, the closed embedding 푆 ãÑ 푆1 is finitely presented. 푇 is a reasonable indscheme if 푇 is the colimit of its reasonable subschemes. Example 6.4.2. Every quasi-compact quasi-separated scheme is reasonable when regarded as an indscheme. Example 6.4.3. Every indscheme of ind-finite type is reasonable. Example 6.4.4. For an ind-pro finite set 푇 , considered as an indscheme in the obvious way, a subset 푆 Ď 푇 is reasonable if and only if it is compact and open in the ind-pro topology. Terminology 6.4.5. Because of Example 6.4.4, we sometimes refer to reasonable subschemes as “compact open” subschemes. We especially use this terminology in the group setting, where we speak of compact open subgroups, meaning group subschemes that are reasonable as subschemes. Lemma 6.4.6. Suppose 푇 is a reasonable indscheme and 푓 : 푆 Ñ 푇 a finitely presented morphism of indschemes. Then 푆 is a reasonable indscheme, and for every reasonable subscheme 푇 1 Ď 푇 , 푓 ´1p푇 1q Ď 푆 is a reasonable subscheme. Proof. Fix a reasonable subscheme 푇 1 Ď 푇 . It suffices to show that 푓 ´1p푇 1q Ď 푆 is a reasonable subscheme. First, suppose that 푇 1 Ď 푇 2 Ď 푇 is a reasonable subscheme of 푇 . We will show that 푓 ´1p푇 1q ãÑ 푓 ´1p푇 2q is a finitely presented closed embedding. Note that 푓 ´1p푇 1q Ñ 푇 1 is finitely presented because 푓 is, and similarly for 푇 2. Moreover, 푓 ´1p푇 1q Ñ 푇 2 is finitely presented, since it factors as 푓 ´1p푇 1q Ñ 푇 1 Ñ 푇 2 with the latter morphism being finitely presented because 푇 1 is reasonable. Therefore, since 푓 ´1p푇 1q Ñ 푓 ´1p푇 2q sits in the diagram:

푓 ´1p푇 1q Ñ 푓 ´1p푇 2q Ñ 푇 2 with the composite morphism and the right morphism finitely presented, the morphism 푓 ´1p푇 1q Ñ 푓 ´1p푇 2q is finitely presented as well (the relevant “two out of three” principle appears in [Gro] Proposition 1.6.2). To see that this suffices: suppose that 푓 ´1p푇 1q Ď 푆1 Ď 푇 is closed subscheme. We can take 푇 2 as above we 푆1 Ñ 푇 factoring through 푇 2. Therefore, we have: D-MODULES ON INFINITE DIMENSIONAL VARIETIES 35

푓 ´1p푇 1q Ď 푆1 Ď 푓 ´1p푇 2q. That 푓 ´1p푇 1q Ñ 푓 ´1p푇 2q is finite presentation means that the ideal sheaf of 푓 ´1p푇 1q is finitely generated over the structure sheaf of 푓 ´1p푇 2q. Therefore, we see that it is finitely generated over the structure sheaf of 푆1 as well, so that our closed embedding 푓 ´1p푇 1q Ď 푆1 is itself finitely presented.

6.5. The key feature of reasonable indschemes is the following. Suppose 푇 “ colim푖Pℐ 푇푖 as in the deﬁnition. ! ! Then every 훼 : 푇푖 Ñ 푇푗 is a ﬁnitely presented closed embedding and therefore 훼 : 퐷 p푇푗q Ñ ! ˚ ˚ ¡ 퐷 p푇푖q admits the left adjoint 훼˚,!´푑푅 and 훼˚,푑푅 : 퐷 p푇푖q Ñ 퐷 p푇푗q admits the right adjoint 훼 . Therefore, we have:

! ! 퐷 p푇 q “ colim 퐷 p푇푖q 푖Pℐ ˚ ˚ (6.5.1) 퐷 p푇 q “ lim 퐷 p푇푖q 푖Pℐ표푝 ¡ where on the left we use functors 훼˚,!´푑푅 and on the right we use functors 훼 . We deduce that for 푇 and 푆 reasonable indschemes we have canonical equivalences:

퐷!p푇 ˆ 푆q “ 퐷!p푇 q b 퐷!p푆q. (6.5.2)

6.6. Descent. We say a morphism 푓 : 푇 Ñ 푆 of indschemes is an ℎ-covering if its base-change by any aﬃne scheme is an ℎ-covering.

Proposition 6.6.1. Let 푓 : 푆 Ñ 푇 be an ℎ-covering of indschemes. Then the canonical functor:

퐷!p푇 q Ñ lim 퐷!pCech푛p푆{푇 qq r푛sPΔ given by !-pullback is an equivalence.

Proof. This is obvious from Proposition 3.12.1: it just amounts to commuting limits with limits. More generally, it holds for any ℎ-covering (in the above sense) of prestacks. Similarly, we have the following result under more restrictive hypotheses.

Proposition 6.6.2. Let 푓 : 푆 Ñ 푇 be an ℎ-covering of reasonable indschemes. Then the canonical functor:

퐷˚p푇 q Ñ lim 퐷˚pCech푛p푆{푇 qq r푛sPΔ given by ¡-pullback is an equivalence.

Proof. As above, this follows from Proposition 3.23.1 by commuting limits with limits, using the presentation (6.5.1) of 퐷˚. 36 SAM RASKIN

6.7. Equivariant setting. We now render the material of S3.17 and S3.24 to the indscheme setting. Suppose that 푆 is an indscheme and 풢 Ñ 푆 is a group indscheme over 푆. Suppose 푃 is an indscheme with a morphism 푃 Ñ 푆 and an action of 풢. We deﬁne the equivariant derived category 퐷!p푃 q풢 as the limit of the diagram formed using (3.17.1):

! 풢 ! / ! / ! / 퐷 p푃 q :“ lim 퐷 p푃 q / 퐷 p풢 ˆ 푃 q / 퐷 p풢 ˆ 풢 ˆ 푃 q / .... ˜ 푆 푆 푆 / ¸ Similarly, we deﬁne the coequivariant derived category by (3.24.1). Now suppose that 풫풢 Ñ 푆 is an indscheme with a 풢-action as above and that 풫풢 is a 풢-torsor 1 1 1 in the sense that, for every closed subscheme 푆 of 푆, the ﬁber product 풫풢 ˆ푆 푆 is a 풢 ˆ푆 푆 -torsor 1 1 1 in the sense of S3.17: after an fppf base-change in 푆 , 풫풢 ˆ푆 푆 Ñ 푆 is 풢-equivariantly isomorphic to 풢. Proposition 6.7.1. The pullback functor:

! ! 풢 퐷 p푆q Ñ 퐷 p풫풢q is an equivalence. The pushforward functor:

˚ ˚ 퐷 p풫풢q풢 Ñ 퐷 p푆q is an equivalence if 푆 is reasonable, and 풢 is a union 풢 “Y풢푖 where the 풢푖 are closed group 1 1 indschemes in 풢 with the property that 풢푖 ˆ푆 푆 Ñ 풢 ˆ푆 푆 is a reasonable subscheme for every reasonable subscheme 푆1 Ď 푆. Proof. For the ﬁrst functor, we commute limits with limits to d´evissage to the case where 푆 is a quasi-compact quasi-separated scheme. Then the result follows as in Proposition 3.17.2: by Propo- sition 6.6.1 we reduce to the case of a trivial 풢-bundle where it follows by using split simplicial objects. The second functor is analyzed similarly: commuting colimits with colimits, we reduce to the case where 푆 is a quasi-compact quasi-separated scheme. Note that 풫풢 must be induced as a torsor from some 풢푖-torsor for some 푖0. Therefore, 풫풢 is reasonable: it is a union of the induced 풢푖-torsors for 푖 Ñ 푖0, and these are obviously reasonable subschemes. Therefore, we can apply Proposition 6.6.2 to again reduce to the case of a trivial torsor. Remark 6.7.2. When our indschemes are reasonable, Example 3.17.1 translates verbatim to the present setting by using (6.5.2). Remark 6.7.3. We will sometimes use the notational convention of Remark 3.17.3 in the above setting as well. 6.8. Placidity. We now give an indscheme analogue of the notion of placidity. Definition 6.8.1. We say that 푇 P IndSch is a placid indscheme if 푇 is reasonable and every reasonable subscheme of 푇 is placid.

Remark 6.8.2. By Remark 4.2.5, we see that 푇 is placid if and only if we can write 푇 “ colim푖Pℐ 푇푖 as in the deﬁnition of indscheme so that each 푇푖 is placid and a reasonable subscheme of 푇 . Remark 6.8.3. By (6.5.1) and S4.3, for 푇 placid the categories 퐷!p푇 q and 퐷˚p푇 q are compactly generated and canonically dual. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 37

The following is the indscheme analogue of Example 4.2.3. Example 6.8.4. Suppose that 푆 is a placid indscheme and 풢 Ñ 푆 is a group indscheme over 푆. 1 1 1 Suppose moreover that for every closed subscheme 푆 of 푆 the ﬁber product 풢ˆ푆푆 Ñ 푆 is a group scheme that can be written as a projective limit under smooth maps of group schemes 풢푖 smooth and aﬃne over 푆1. Then 풢 is a placid indscheme. More generally, if 풫풢 Ñ 푆 is a 풢-torsor over 푆 in the sense of S6.7 then 풫풢 is a placid indscheme. Indeed, we reduce to showing that if 푆 as above is actually a placid scheme, then 풫풢 Ñ 푆 is a placid morphism. But 풫풢 is the projective limit of the induced 풢푖-torsors, giving the result. 6.9. Fiber products. We digress somewhat to give the following technical result, which we will need in [Ras].

Proposition 6.9.1. Let 푆1 Ñ 푆2 and 푇 Ñ 푆2 be morphisms of indschemes. (1) If 푆1 and 푆2 are finite type schemes, then the canonical morphisms:

! ! 퐷 p푇 q b 퐷p푆1q Ñ 퐷 p푇 ˆ 푆1q 퐷p푆2q 푆2 ˚ ˚ 퐷 p푇 q b 퐷p푆1q Ñ 퐷 p푇 ˆ 푆1q 퐷p푆2q 푆2 of ! and ¡-pullback respectively are equivalences. (2) If 푆1 is a placid indscheme and 푆2 is a finite type scheme and 푇 is an arbitrary indscheme, then:

! ! ! 퐷 p푇 q b 퐷 p푆1q Ñ 퐷 p푇 ˆ 푆1q 퐷p푆2q 푆2 is an equivalence. We will deduce Proposition 6.9.1 from the following two lemmas from the ﬁnite dimensional setting.

Lemma 6.9.2. Let 푆1 Ñ 푆2 and 푇 Ñ 푆2 morphisms of finite type schemes, the canonical morphism:

퐷p푇 q b 퐷p푆1q Ñ 퐷p푇 ˆ 푆1q 퐷p푆2q 푆2 is an equivalence.

This result is well-known, and follows easily e.g. from the 1-aﬃneness of the prestacks 푆푑푅 for 푆 ﬁnite type: see [Gai2] for the terminology and for this result. Lemma 6.9.3. For 푓 : 푆 Ñ 푇 a morphism of finite type schemes, 퐷p푆q is dualizable as a 퐷p푇 q- module category. Proof. We will show that 퐷p푆q is self-dual as a 퐷p푇 q-module category. Let ∆푓 denote the diagonal embedding 푆 Ñ 푆 ˆ푇 푆. We have the evaluation:

Δ! 푓 퐷p푆q b 퐷p푆q » 퐷p푆 ˆ 푆q ÝÝÑ푓 퐷p푆q ÝÝÝÑ˚,푑푅 퐷p푇 q 퐷p푇 q 푇 and coevaluation: 38 SAM RASKIN

푓 ! Δ 퐷p푇 q ÝÑ 퐷p푆q ÝÝÝÝÝÑ푓,˚,푑푅 퐷p푆 ˆ 푆q » 퐷p푆q b 퐷p푆q. 푇 퐷p푇 q One readily checks by base-change that these deﬁne a duality datum as required.

Proof of Proposition 6.9.1. For (1): the category 퐷p푆1q is dualizable as a 퐷p푆2q-module category. Therefore, tensoring over 퐷p푆2q with 퐷p푆1q commutes with limits of categories. Applying the definition of 퐷!, the result then immediately follows from the finite type case. ! Similarly, to prove (2) it suffices to show that 퐷 p푆1q is dualizable as a 퐷p푆2q-module category. Using the methods of [Gai1], this follows from the finite type case combined with (6.5.1).

6.10. Dimension theories. Let 푇 be a placid indscheme. We use the notation of S4.6 here.

Definition 6.10.1. A dimension theory 휏 “ 휏 푇 on 푇 is a rule that assigns to every reasonable subscheme 푆 of 푇 a locally constant function:

휏푆 : 푆 Ñ Z such that for any pair of reasonable subschemes 푆1 Ď 푆 Ď 푇 we have:

휏푆1 “ 휏푆|푆1 ` dim푆1{푆 . (6.10.1) Example 6.10.2. By Remark 4.6.3, every placid scheme 푇 carries a canonical dimension theory normalized by the condition that dim푇 be identically zero. Example 6.10.3. Let 푇 be an indscheme of ind-ﬁnite type. Then a reasonable subscheme of 푇 is just a closed ﬁnite type subscheme 푆, and the rule 휏푆 :“ dim푆 is a dimension theory on 푇 .

Remark 6.10.4. If 푇 “Y푖푆푖 is written as a union of reasonable subschemes, it suﬃces to deﬁne the

휏푆푖 satisfying the compatibility (6.10.1). Indeed, this again follows from Remark 4.6.3.

Example 6.10.5. By Remark 6.10.4, the product 푇1 ˆ 푇2 of indschemes 푇푖 equipped with dimension 푇 푇 ˆ푇 theories 휏 푖 inherits a canonical dimension theory 휏 1 2 such that, for every pair 푆푖 Ď 푇푖, 푖 “ 1, 2 of reasonable subschemes, we have:

휏 푇1ˆ푇2 푝˚ 휏 푇1 푝˚ 휏 푇2 푆1ˆ푆2 “ 1 p 푆1 q ` 2 p 푆2 q ˚ with 푝푖 denoting the restriction of a function along the projection. Remark 6.10.6. Dimension theories are ´etalelocal.

Remark 6.10.7. For 푇 a group indscheme, the choice of dimension theory may be seen as analogous to the choice of a Haar measure in the 푝-adic setting.

Remark 6.10.8. See [Dri] for relevant material on dimension theories. In particular, questions of existence (and non-existence) are treated in some detail. D-MODULES ON INFINITE DIMENSIONAL VARIETIES 39

6.11. We now give something of a classiﬁcation of the set of dimension theories.

Definition 6.11.1. A locally constant function 푇 Ñ Z on an indscheme 푇 is a morphism of indschemes 푇 Spec 푘 . Ñ Z “ 푛PZ p q

Remark 6.11.2. Forš푇 “ colim 푇푖, a locally constant function on 푇 is equivalent to a compatible system of locally constant functions on the 푇푖. As in Remark 3.9.4, we can make sense of 휋0p푇 q as an ind-profinite set, and a locally constant function on 푇 is equivalent to a continuous function 휋0p푇 q Ñ Z, with 휋0 equipped with its natural topology as an ind-profinite set. Clearly locally constant functions form an abelian group under addition. Moreover, they obviously act on the set of dimension theories on 푇 : given 푑 : 푇 Ñ Z and 휏 a dimension theory on 푇 , we obtain a new dimension theory 푑 ` 휏 with p푑 ` 휏q푆 “ 푑|푆 ` 휏푆 for every reasonable subscheme 푆 of 푇 . Proposition 6.11.3. Suppose that 푆 is a placid indscheme that admits a dimension theory. Then the set of dimension theories for 푆 is a torsor for the set of locally constant functions 푆 Ñ Z, i.e., the above action of locally constant functions on dimension theories is a simply transitive action. Proof. The difference between two dimension theories obviously defines a locally constant function on 푆.

6.12. The following construction of dimension theories is useful in many situations.

Definition 6.12.1. A morphism 푓 : 푇 Ñ 푆 of placid indschemes is healthy if there exists a reasonable subscheme 푆1 Ď 푆 such that: (1) The inverse image of any closed subscheme 푆1 Ď 푆2 Ď 푆 is a reasonable subscheme of 푇 . (2) For every closed subscheme 푆1 Ď 푆2 Ď 푆, we have:

1,˚ dim푇 1{푇 2 “ 푓 pdim푆1{푆2 q with 푓 1 : 푇 1 Ñ 푆1 the fiber product of 푓 along 푆1 and 푇 2 the fiber along 푆2. We say a subscheme 푆1 Ď 푆 is 푓-healthy if it is reasonable and satisfies the above conditions (so 푓 is healthy if and only if there exists an 푓-healthy subscheme of 푆).

Example 6.12.2. Every morphism 푓 : 푇 Ñ 푆 of placid schemes is healthy: 푆 itself is 푓-healthy.

Counterexample 6.12.3. For 푛 ě 0, let 푆푛 be the union of a line, a plane, up to an affine 푛-space all glued together along 0. Let 푆 “ colim 푆푛. Let 푇푛 be the union of 푛 (ordered) lines glued along 0, 푟 mapping to 푆푛 by embedding the 푟th irreducible component into A as a line into a vector space. Let 푇 “ colim푛 푇푛. Then the resulting map 푇 Ñ 푆 is not healthy. Example 6.12.4. In S6.17, we will give a definition of placid morphism of placid indschemes such that every placid morphism is healthy. Remark 6.12.5. Any reasonable subscheme containing an 푓-healthy subscheme is itself 푓-healthy. 1 1 In particular, we see that given two choices 푆1, 푆2 of 푓-healthy subschemes of 푆, there is always 1 a third 푆3 containing both. Our key use of this definition is the following construction. 40 SAM RASKIN

Construction 6.12.6. For 푓 : 푇 Ñ 푆 a healthy morphism of placid indschemes, any dimension theory 휏 푆 on 푆 induces a unique dimension theory 휏 푇 on 푇 such that for any 푓-healthy reasonable 1 푇 1,˚ 푆 1 1 1 1 subscheme 푆 Ď 푆, we have 휏푇 1 “ 푓 p휏푆1 q for 푓 : 푇 Ñ 푆 the base-change of 푓 along 푆 ãÑ 푆. Indeed, that this construction can be performed follows immediately from Remarks 6.10.4 and 6.12.5. Remark 6.12.7. Healthy morphisms are obviously preserved under compositions, and Construction 6.12.6 is obviously compatible with compositions. 6.13. As S6.12 generalizes Example 6.10.2, we now generalize Example 6.10.3. We say a morphism 푓 : 푇 Ñ 푆 of reasonable indschemes is ind-finitely presented if 푇 “ colim 푇푖 with each 푇푖 Ñ 푇 a reasonable subscheme such that 푇푖 Ñ 푆 factors through a reasonable subscheme 푆푖 of 푆 with 푇푖 Ñ 푆푖 ﬁnite presentation. We claim under this hypothesis that 푇 inherits a canonical dimension theory 휏 푇 from a dimension theory 휏 푆 of 푆. Indeed, for 푇 1 Ď 푇 a reasonable subscheme, the morphism 푇 1 Ñ 푆 factors through some reasonable subscheme 푆1 Ď 푆, and 푓 1 : 푇 1 Ñ 푆1 is ﬁnite presentation by assumption. We take:

푇 1,˚ 푆 휏푇 1 :“ dim푇 1{푆1 `푓 p휏푆1 q. 푖 To simultaneously show that 휏 푇 is well-deﬁned and actually deﬁnes a dimension theory, take 푇 1 ãÑ1 푖 푇 2 Ď 푇 reasonable subschemes mapping via 푓 1 and 푓 2 to reasonable subschemes 푆1 ãÑ2 푆2 Ď 푆 respectively, and compute:

푇 ˚ 푇 ˚ 1,˚ 푆 1,˚ ˚ 푆 휏푇 1 ´ 푖1 p휏푇 2 q :“ dim푇 1{푆1 ´푖1 pdim푇 2{푆2 q ` 푓 p휏푆1 q ´ 푓 푖2 p휏푆2 q “ 1,˚ 1,˚ “ ´푓 pdim푆1{푆2 q ` dim푇 1{푇 2 `푓 pdim푆1{푆2 q “ dim푇 1{푇 2 as desired, where we have used the expansions:

1,˚ dim푇 1{푆1 “ dim푇 1{푆2 ´푓 pdim푆1{푆2 q ˚ 푖1 pdim푇 2{푆2 q “ dim푇 1{푆2 ´ dim푇 1{푇 2 of (4.6.1). Example 6.13.1. If 푇 is a reasonable subscheme of a placid indscheme 푆, then the embedding 푇 ãÑ 푆 satisfies the hypotheses of this section. If 휏 푆 is a dimension theory on 푆, the induced dimension theory 휏 푇 on 푇 constructed above is the “obvious” one, which to a reasonable subscheme 푇 1 Ď 푇 푇 푆 assigns the function 휏푇 1 :“ 휏푇 1 . Warning 6.13.2. If 푓 : 푇 Ñ 푆 is a finitely presented morphism of placid schemes, the pullback constructed above of the dimension theory 휏 푆 given in Example 6.10.2 is not (generally) the dimension theory on 푇 constructed in Example 6.10.2: they differ by dim푇 {푆. 6.14. Renormalization. Let 푇 be a placid indscheme and let 휏 be a dimension theory on 푇 . We 휏 ˚ will define the “휏-renormalized dualizing sheaf” 휔푇 P 퐷 p푇 q below. Let 푖 : 푆 ãÑ 푇 be a reasonable subscheme. We formally define:

¡ 휏 푟푒푛 ˚ “푖 p휔푇 q” :“ 휔푆 r2휏푆s P 퐷 p푆q. Suppose that for 푆 as above 휄 : 푆1 Ñ 푆 is a reasonable subscheme (equivalently: of 푆 or of 푇 , or equivalently 휄 is a ﬁnitely presented closed embedding). Then we have canonical isomorphisms: D-MODULES ON INFINITE DIMENSIONAL VARIETIES 41

¡ ¡ 휏 ¡ 푟푒푛 !,푟푒푛 푟푒푛 푟푒푛 ¡ 휏 휄 p“푖 p휔푇 q”q “ 휄 p휔푆 qr2휏푆s “ 휄 p휔푆 qr2¨p휏푆`dim푆1{푆qs “ p휔푆1 qr2¨p휏푆`dim푆1{푆qs “:“p푖˝휄q p휔푇 q” where the second equality is Proposition 4.12.1 and the third equality is (4.9.1). 휏 ˚ These identiﬁcations are readily made homotopy compatible and therefore deﬁne 휔푇 in 퐷 p푇 q ¡ 휏 ¡ 휏 so that 휄 p휔푇 q “ “휄 p휔푇 q” for all 휄 : 푆 ãÑ 푇 as above.

6.15. Let 푇 and 휏 be as in S6.14. 휏 ! ˚ 휏 Let 휂푇 : 퐷 p푇 q Ñ 퐷 p푇 q denote the functor of action on 휔푇 . We immediately deduce from 휏 Proposition 4.8.1 that 휂푇 is an equivalence.

6.16. Let 푓 : 푇 Ñ 푆 a morphism of placid indschemes equipped with dimension theories 휏 푇 and 휏 푆. ! ! !,휏 ! ! Then as in S4.9 we obtain functors 푓˚,휏 : 퐷 p푇 q Ñ 퐷 p푆q and 푓 : 퐷 p푆q Ñ 퐷 p푇 q so that we have the commuting diagram:

!,휏 푓˚,휏 푓 퐷!p푇 q / 퐷!p푆q 퐷˚p푆q / 퐷˚p푇 q

» 휏푇 » 휏푆 » 휏푆 » 휏푇 휂푇 휂푆 휂푆 휂푇 ! 푓˚,푑푅 푓 퐷˚p푇 q / 퐷˚p푆q 퐷!p푆q / 퐷!p푇 q.

Example 6.16.1. If 푓 : 푇 Ñ 푆 is a map of placid schemes, each equipped with their canonical dimension theories (see Example 6.10.2), then the functors constructed above are the renormalized functors of S4.9. Notation 6.16.2. In light of Example 6.16.1, when the relative dimension theory 휏 is implicit we !,푟푒푛 !,푟푒푛 denote the functors 푓휏,푟푒푛 and 푓 above simply by 푓˚,푟푒푛 and 푓 . Fixing a map 푓 : 푇 Ñ 푆 of placid indschemes, we obtain a pullback map for locally constant functions and therefore an induced diagonal action of locally constant functions on 푆 on the set of pairs p휏 푇 , 휏 푆q of dimension theories for 푇 and 푆:

푇 푆 푇 푆 푑 : 푆 Ñ Z, p휏 , 휏 q ÞÑ p휏 ` 푑 ˝ 푓, 휏 ` 푑q. ´ ¯ Definition 6.16.3. A relative dimension theory for 푇 and 푆 is an equivalence class of pairs p휏 푇 , 휏 푆q of dimension theories for 푇 and for 푆 modulo the above action of locally constant functions on 푆.

!,푟푒푛 Clearly the functors 푓 and 푓˚,푟푒푛 only depend on the relative dimension theory deﬁned by the pair p휏 푇 , 휏 푆q. Example 6.16.4. Let 푓 : 푇 Ñ 푆 be an ind-ﬁnitely presented morphism of placid indschemes with 푆 equipped with dimension theory. By S6.13, we obtain a dimension theory on 푇 and therefore a relative dimension theory for 푓. !,푟푒푛 As in Examples 4.9.1 and 4.9.4, the functors 푓˚,푟푒푛 and 푓 canonically identify with 푓˚,!´푑푅 and 푓¡ respectively.7

7Unlike Example 4.9.1, there are no cohomological shifts in this formula. There is no real discrepancy because of Warning 6.13.2. 42 SAM RASKIN

6.17. Next, we extend the notion of placid morphism from S4.10 to the indscheme framework. Definition 6.17.1. A morphism 푓 : 푇 Ñ 푆 of placid indschemes is placid if there exists a reasonable subscheme 푆1 Ď 푆 such that: (1) The inverse image of any closed subscheme 푆1 Ď 푆2 Ď 푆 is a reasonable subscheme of 푇 . 1 2 2 2 2 (2) For every closed subscheme 푆 Ď 푆 Ď 푆, the morphism 푇 :“ 푆 ˆ푆 푇 Ñ 푆 is placid. Remark 6.17.2. By Corollary 4.10.7, we immediately see that any placid morphism is healthy. Example 6.17.3. If 푓 is ﬁnitely presented, smooth and surjective on geometric points, then 푓 is placid. Example 6.17.4. Suppose that 푆 is a placid indscheme and 풢 Ñ 푆 is a group indscheme satisfying the hypotheses of Example 6.8.4. Suppose 풫풢 Ñ 푆 is a 풢-torsor on 푆. Then 풫풢 Ñ 푆 is placid. In particular, this morphism is healthy. Indeed, this follows by Example 6.8.4. 6.18. We have the following indschematic version of Proposition 4.11.1. Proposition 6.18.1. Let 푓 : 푇 Ñ 푆 be placid and suppose that 푆 is equipped with a dimension theory. By Construction 6.12.6, this choice induces a dimension theory on 푇 . (1) The functors:

˚ ˚ 푓˚,푑푅 : 퐷 p푇 q Ñ 퐷 p푆q ! ! 푓˚,푟푒푛 : 퐷 p푇 q Ñ 퐷 p푆q admit left adjoints. Moreover, these left adjoints are canonically identified with 푓 !,푟푒푛 and 푓 ! respectively. (2) Suppose that we are given a Cartesian diagram:

휙 푇 1 / 푆1

휓 푔 푓 푇 / 푆 of placid indschemes with 푓 placid and 푔 finitely presented. Then 휙 is also placid, and the natural transformations:

!,푟푒푛 !,푟푒푛 푓 푔˚,푑푅 Ñ 휓˚,푑푅휙 ! ! 푓 푔˚,푟푒푛 Ñ 휓˚,푟푒푛휙 are equivalences. Here we have equipped 푆1 and 푇 1 with the dimension theories of S6.13 using the finitely presented maps 푔 and 휓. Proof. It suffices to prove each of these statements in the 퐷!-setting. Then (1) then follows immediately Proposition 4.11.1 (say, by applying a simplified version of Lemma 5.3.1). So it remains to show (2). Let 푆0 be a reasonable subscheme of 푆 satisfying the hypotheses of the definition of placid morphism for 푓. Then combining Lemmas 4.10.6 and 6.4.6., we find that its pullback to 푆1 satisfies the same conditions for 휙. In particular, we see that 휙 is placid. We form the commutative cube: D-MODULES ON INFINITE DIMENSIONAL VARIETIES 43

휙 1 0 1 푇0 / 푆0 휄1 푖1

푔0 1 1 휓0 푇 / 푆

푓0 푇0 / 푆0 휄 푖

푇 / 푆 where all faces are taken to be Cartesian squares. We equip these new schemes with the dimension theories obtained using Example 6.13.1. Note that the dimension theories on the back square are not (necessarily) the canonical ones on placid schemes from Example 6.10.2. 1 1 Still, the relative dimension theories of 푇0{푆0 and 푇0{푆0 are the same, so renormalized functors for these dimension theories coincide with those of S4.9. 1 Moreover, the dimension theories for 푆 {푆 diﬀers from the “canonical” one by dim 1 , and 0 0 푆0{푆0 1 1 similarly for 푇 {푇 . Note that this error term dim 1 pulls back to 푇 as dim 1 by Corollary 0 0 푆0{푆0 0 푇0{푇0 4.10.7. We will use the notation e.g. 푔0,˚,푟푒푛 here for the renormalized functor corresponding to our given dimension theory, therefore diﬀering by cohomological shifts from the so-named functor in S4.9. In this notation, we see from the above discussion that we can apply Proposition 4.11.1 to deduce:

! » ! 푓0푔0,˚,푟푒푛ÝÑ휓0,˚,푟푒푛휙0.

! 1 1 1 Because 퐷 p푆 q is generated under colimits by 퐷-modules of the form 푖˚,!´푑푅pFq “ 푖˚,푟푒푛pFq as we increase 푆0, it suﬃces to show that the natural transformation:

! 1 ! 1 푓 푔˚,푟푒푛푖˚,푟푒푛 Ñ 휓˚,푟푒푛휙 푖˚,푟푒푛 is an equivalence. Similarly, since 푇 is a union of the schemes 푇0 as 푆0 varies, it suﬃces to show that the natural transformation:

! ! 1 ! ! 1 휄 푓 푔˚,푟푒푛푖˚,푟푒푛 Ñ 휄 휓˚,푟푒푛휙 푖˚,푟푒푛 is an equivalence. Now we compute:

! ! 1 ! ! ! » ! ! ! 휄 푓 푔˚,푟푒푛푖˚,푟푒푛 “ 푓0푖 푖˚,푟푒푛푔0,˚,푟푒푛 “ 푓0푔0,˚,푟푒푛 ÝÑ 휓0,˚,푟푒푛휙0 “ 휄 휄˚,푟푒푛휓0,˚,푟푒푛휙0 “ ! 1 ! 1,! 1 ! 1 1,! ! 1 ! ! 1 휄 휓˚,푟푒푛휄˚,푟푒푛휙0푖 푖˚,푟푒푛 “ 휄 휓˚,푟푒푛휄˚,푟푒푛휄 휙 푖˚,푟푒푛 “ 휄 휓˚,푟푒푛휙 푖˚,푟푒푛 as desired. 44 SAM RASKIN

! ˚ 6.19. Holonomic 퐷-modules. For 푇 an indscheme, we deﬁne 퐷ℎ표푙p푆q and 퐷ℎ표푙p푆q by Kan extension, as in the deﬁnition of 퐷! and 퐷˚. We have canonical forgetful functors:

! ! ˚ ˚ 퐷ℎ표푙p푆q Ñ 퐷 p푆q and 퐷ℎ표푙p푆q Ñ 퐷 p푆q and compatible upper-! and lower-* functoriality, respectively. For 푆 reasonable (resp. placid), ˚ ˚ ! ! 퐷ℎ표푙p푆q Ñ 퐷 p푆q (resp. 퐷ℎ표푙p푆q Ñ 퐷ℎ표푙p푆q) is fully-faithful. Definition 6.19.1. A morphism 푓 : 푆 Ñ 푇 of reasonable indschemes is a reasonable morphism if ´1 there exists coﬁnal system 푇 “Y푇푖 of reasonable subschemes such that 푓 p푇푖q is a reasonable subscheme in 푆 (in particular: 푓 is schematic). Proposition 6.19.2. If 푓 : 푆 Ñ 푇 is a reasonable morphism of reasonable indschemes, then the ˚,푑푅 ˚ partially-defined left adjoint 푓 to 푓˚,푑푅 is defined on holonomic objects in 퐷 p푇 q. Similarly, if 푓 is a morphism of ind-finite presentation of placid indschemes, then the partially- ! ! ! defined left adjoint 푓! to 푓 : 퐷 p푇 q Ñ 퐷 p푆q is defined on holonomic objects. Proof. Follows from the combination of Proposition 5.2.1 and Lemma 5.3.1 by the same argument as in Proposition 5.2.1. We have the following counterparts to Proposition 5.5.1 and its Corollary 5.5.2, proved by the same arguments.

휏 ! Proposition 6.19.3. For 푆 a placid indscheme with a dimension theory 휏, 휂푆 identifies 퐷ℎ표푙p푆q ˚ with 퐷ℎ표푙p푆q. Corollary 6.19.4. For 푆 and 푇 placid indschemes with a dimension theories and 푓 : 푆 Ñ 푇 a !,푟푒푛 ! ˚ morphism, 푓˚,푟푒푛 and 푓 preserve holonomic objects in 퐷 and 퐷 respectively.

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