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Geometric Derived Hall Algebra 3 GEOMETRIC DERIVED HALL ALGEBRA SHINTAROU YANAGIDA Abstract. We give a geometric formulation of To¨en’s derived Hall algebra by constructing Grothendieck’s six operations for the derived category of lisse-´etale constructible sheaves on the derived stacks of complexes. Our formulation is based on an variant of Laszlo and Olsson’s theory of derived categories and six operations for algebraic stacks. We also give an ∞-theoretic explanation of the theory of derived stacks, which was originally constructed by To¨en and Vezzosi in terms of model theoretical language. Contents 0. Introduction 1 1. Notations on ∞-categories and ∞-topoi 3 2. Recollections on derived stacks 10 3. Sheaves on ringed ∞-topoi 21 4. Derived algebraic spaces and ´etale sheaves 34 5. Lisse-´etale sheaves on derived stacks 43 6. Derived functors of constructible sheaves with finite coefficients 54 7. Adic coefficient case 59 8. Perverse sheaves on derived stacks 64 9. Moduli stack of perfect dg-modules 68 10. Derived Hall algebra and its geometric formulation 73 Appendix A. Algebraic stacks 78 Appendix B. ∞-categories 82 Appendix C. ∞-topoi 89 Appendix D. Stable ∞-categories 92 Appendix E. Spectra and stable modules 98 References 101 0. Introduction 0.1. The derived Hall algebra introduced by To¨en [T1] is a version of Ringel-Hall algebra, and roughly speaking it is a “Hall algebra for complexes”. In the case of the ordinary Ringel-Hall algebra for the abelian category of representations of a quiver, Lusztig [Lus] established a geometric formulation using the theory of perverse sheaves in the equivariant derived category of ℓ-adic constructible sheaves on the moduli spaces of the representations, and his formulation gives rise to the theory of canonical bases for quantum groups. arXiv:1912.05442v1 [math.AG] 10 Dec 2019 As mentioned in [T1, §1, Related and future works], it is natural to expect a similar geometric formulation for derived Hall algebras using the moduli space of complexes of representations. At the moment when [T1] appears, the theory of derived stacks was just being under construction, which would realize the moduli space of complexes. Soon after, To¨en and Vezzosi [TVe1, TVe2] completed the works on derived stacks, and based on them To¨en and Vaqui´e[TVa] constructed the moduli space of dg- modules. In [TVa, §0.6], it was announced among several stuffs that a geometric formulation of derived Hall algebra was being studied. As far as we understand, it has not appeared yet. The purpose of this article is to give such a geometric formulation of derived Hall algebras. More precisely speaking, we want to give the following materials. • The theory of derived categories of lisse-´etale constructible sheaves on derived stacks and the con- struction of Grothendieck’s six operations on them. • The theory of perverse ℓ-adic constructible sheaves on derived stacks. • The geometric formulation of derived Hall algebra. Date: December 10, 2019. 1 2 SHINTAROU YANAGIDA Any such attempt will require derived algebraic geometry. Although the works [TVe1, TVe2, TVa] are based on the language of model categories so that it might be natural to work over such a language, we decided to work using the language of ∞-categories following Lurie’s derived algebraic geometry [Lur5] – [Lur14]. The reason is that today most papers on derived algebraic geometry uses the language of ∞-categories rather than that of model categories. A “byproduct” of our project is • A “translation” of model-categorical [TVe1, TVe2, TVa] into the language of ∞-categories. Such a translation is in fact a trivial one, and it seems that experts on derived algebraic geometry use both languages freely. We decided to give somewhat a textbook-like explanation on this point, and we will start the main text with some recollection on the theory of ∞-categories and ∞-topoi (§1). The theory of derived stacks in the sense of To¨en and Vezzosi will be explained in the language of ∞-categories (§2). We will construct the derived category of sheaves on derived stacks using the theory of stable ∞-category established by Lurie [Lur2]. For this purpose, we will give in §3 an exposition of the general theory of the ∞-category of sheaves on ∞-topoi and the associated stable ∞-category. There are many overlaps between our exposition and a (very small) part of Lurie’s derived algebraic geometry [Lur5]–[Lur14]. We decided to give a rather self-contained presentation for the completeness of this article. The main object of this article is the lisse-´etale sheaves on derived stacks, which will be introduced in §5. Our definition is a simple analogue of the lisse-´etale sheaves on the algebraic stacks established by Laumon and Moret-Bailly [LM]. In the case of algebraic stacks, the lisse-´etale topos is constructed using the ´etale topos on algebraic spaces. Thus we need a derived analogue of algebraic spaces, which we call derived algebraic spaces and introduce in §4. In [LO1, LO2], Laszlo and Orson constructed the Grothendieck six operations on the derived category of constructible sheaves on algebraic stacks, based on the correction [O1] of a technical error on the lisse-´etale topos in [LM]. In the sequel [LO3] they also give the theory of perverse ℓ-adic sheaves and weights on them over algebraic stacks. Our construction of six operations (§6, §7) and definition of perverse ℓ-adic sheaves (§8) are just a copy of their argument. Fundamental materials will be established up to §8, and we will then turn to the derived Hall algebra. In §9 we review To¨en and Vaqui´e’s construction [TVa] of the moduli space of dg-modules over a dg-category via derived stack. In §10 we explain the definition of derived Hall albebras, and give its geometric formulation, the main purpose of this article. Let us sketch an outline of the construction here. Let D be a locally finite dg-category over Fq (see §9 for an account on dg-categories). Fact ([TVa]). We have the moduli stack Pf(D) of perfect dg-modules over Dop. It is a derived stack, locally geometric and locally of finite presentation. We can also construct the moduli stack of cofibrations X → Y of perfect-modules over Dop, denoted by E(D). Then there exist morphisms s,c,t : E(D) −→ Pf(D) of derived stacks which send u : X → Y to X s(u)= X, c(u)= Y, t(u)= Y 0. a where s,t are smooth and c is proper. Thus we have a square E(D) c / Pf(D) p:=s×t Pf(D) × Pf(D) of derived stacks with smooth p and proper c. Next let Λ := Qℓ be the field of ℓ-adic numbers where ℓ and q are assumed to be coprime. Then we have b the derived category Dc(X, Λ) of constructible lisse-´etale Λ-sheaves over a locally geometric derived stack X. We also have derived functors (§4). GEOMETRIC DERIVED HALL ALGEBRA 3 Applying the general theory to the present situation, we have b c! b Dc(E(D), Λ) / Dc(Pf(D), Λ) O p∗ b Dc(Pf(D) × Pf(D), Λ) Now we set b b ∗ µ : Dc(Pf(D) × Pf(D), Λ) −→ Dc(Pf(D), Λ), M 7−→ c!p (M)[dim p] Theorem (Theorem 10.3.5). µ is associative. We will also give a definition of derived Hall category, which will be the span of the Lusztig sheaves in the sense of [S2]. By restricting to the “abelian category part”, we can reconstruct the Hall category in [S2], and have a geometric formulation of the ordinary Ringel-Hall algebra via the derived category of lisse-´etale ℓ-adic constructible sheaves on the moduli stack of objects in the abelian category. In [S2, p.2], such a stacky construction is regarded as “(probably risky) project” and avoided. Thus our exposition will also give a new insight to the ordinal Ringel-Hall algebra. We postpone the study of the function-sheaf dictionary for the lisse-´etale constructible sheaves on derived stacks to a future article. 0.2. Conventions and notations. We will use some basic knowledge on • dg-categories [T2, T4], • model categories [H], • simplicial homotopy theory [GJ], • ∞-categories and ∞-topoi [Lur1], • algebraic stacks in the ordinary sense [LM, O2] • derived algebraic geometry in the sense of To¨en-Vezzosi [TVe2, T6]. We give a brief summary of the theory of ∞-categories and ∞-topoi in §1, and explanations on some topics in Appendices B–D. Some recollections on algebraic spaces and algebraic stacks is given in Appendix A. Here is a short list of our global conventions and notations. • N denotes the set of non-negative integers. • We fix two universes U ∈ V and work on them. All the mathematical objects considered, such as sets, groups, rings and so on, are elements of U unless otherwise stated. For example, Set denotes the category of sets in U. We will sometimes say an object is small if it belongs to U. • The word ‘ring’ means a unital and associative one unless otherwise stated, and the word ‘∞-category’ means the one in [Lur1]. • A poset means a partially ordered set. A poset (I, ≤) is filtered if I is non-empty and for any i, j ∈ I there exists k ∈ I such that i ≤ k and j ≤ k. • A category will be identified with its nerve [Lur1, p.9] which is an ∞-category. Basically we denote an ordinary category in a serif font and denote an ∞-category in a sans-serif font. For example, in §2.2.2 we will denote by sCom the category of simplicial commutative rings, and by sCom the ∞-category of simplicial commutative rings.
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