1 Introduction↩︎

Cohomological Hall algebras. Cohomological Hall algebras (CoHA’s) are associative algebras that are modelled on the cohomology of moduli stacks. The procedure is as follows. Let \(k\) be a field and let \(\mathcal{A}\) be a \(k\)-linear abelian category. We set \(\underline{\mathcal{A}}\) to be the moduli stack of objects in \({\mathcal{A}}\) and we set \(\underline{\mathcal{A}}^{\operatorname{ext}}\) to be the moduli stack parametrizing extensions in \(\mathcal{A}\). With these data, we set up the convolution diagram \[\label{convdiag} \begin{tikzcd} & {\underline{\mathcal{A}}^{\operatorname{ext}} } &&& {0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0} & \\ {\underline{\mathcal{A}} \times \underline{\mathcal{A}}} && {\underline{\mathcal{A}}} & {(A', A'')} && A \arrow["{p}"', from=1-2, to=2-1] \arrow["{q}", from=1-2, to=2-3] \arrow[maps to, from=1-5, to=2-4] \arrow[maps to, from=1-5, to=2-6] \end{tikzcd}\tag{1}\] Let \(H^{BM}_*(\underline{\mathcal{A}})\) denote the Borel–Moore homology — more generally any motivic Borel–Moore cohomology theory — of the stack \(\underline{\mathcal{A}}\). If the map \(p\) is lci and the map \(q\) is proper, the above diagram induces a map \[\label{product} q_*p^* \colon H^{BM}_*(\underline{\mathcal{A}})\otimes H^{BM}_*(\underline{\mathcal{A}}) \longrightarrow H^{BM}_*(\underline{\mathcal{A}}) \;\tag{2}\] This makes the space \(H^{BM}_*(\underline{\mathcal{A}})\) into an associative algebra. We refer to such algebra as the CoHA associated to the category \(\mathcal{A}\), and we denote it as \(\operatorname{HA}_{\mathcal{A}}\). Now, consider a stack \(\underline{\mathcal{A}}^{\operatorname{st}} \rightarrow \underline{\mathcal{A}}\), e.g. by introducing a framing and/or stability condition on the objects of \(\mathcal{A}\). Then, replacing \(A\) and \(A''\) in 1 with stable objects in \(\underline{\mathcal{A}}^{\operatorname{st}}\) yields a correspondence \[\label{action} \begin{tikzcd} & (\underline{\mathcal{A}}^{\operatorname{ext}})^{\operatorname{st}} \arrow[dl, "{p}"'] \arrow[dr, "{q}"] & \\ \underline{\mathcal{A}} \times \underline{\mathcal{A}}^{\operatorname{st}} && \underline{\mathcal{A}}^{\operatorname{st}} \; \end{tikzcd}\tag{3}\] for a suitable stack of extensions \((\underline{\mathcal{A}}^{\operatorname{ext}})^{\operatorname{st}} \rightarrow \underline{\mathcal{A}}^{\operatorname{ext}}\). With a similar procedure as before, this endows the space \(H^{BM}_*(\underline{\mathcal{A}}^{\operatorname{st}})\) with the structure of a representation for the CoHA \(\operatorname{HA}_{\mathcal{A}}\). The geometric property encoding the associativity of a CoHA action is that of a Hecke pattern, in the sense of Definition 6.1 in [2]. This property consists in asking that stable objects in \(\mathcal{A}\) be preserved under extensions by objects in \(\mathcal{A}\) — see Example 1. However, Hecke patterns are very rare to find in the geometric setting. The few known examples of this procedure include [1]–[6], and they all involve the category of zero dimensional sheaves on a smooth (quasi) projective surface, or torsion sheaves on a curve [7], [8].

Doubles. One of the main challenges in the theory of Hall algebras is computing their doubles, as we now explain. Suppose that we have a left and a right action of an algebra \(\operatorname{A}\) on the same vector space \(V\) \[\label{leftright} \operatorname{A} \curvearrowright V \curvearrowleft \operatorname{A} \;.\tag{4}\] Denoting as \(\operatorname{A}^{op}\) the opposite algebra, we can ask whether we can place an algebra structure on \(\operatorname{DA} = \operatorname{A}\otimes \operatorname{A}^{op}\) such that we have a naturally induced action \[\label{double32action} \operatorname{DA \curvearrowright} V \;.\tag{5}\] Whenever this is possible, we refer to the algebra \(\operatorname{DA}\) as the double of \(\operatorname{A}\). This situation arises naturally in the context of Hall algebra actions. The most classical example involves the CoHA associated with the category of zero dimensional sheaves on a surface, as defined by [9], generalizing [3], [10], [11]. The original motivation for these actions were the correspondences defined by Nakajima [12] and Grojnowski [13], later generalized by Negut [4] and Mellit–Minets–Schiffmann–Vasserot [2].

In the classical case where \(\mathcal{A}\) is a hereditary and finitary category, the associated Hall algebra essentially encodes all the information regarding the structure of extensions in \(\mathcal{A}\). Moreover, in this case the theory of Drinfeld doubles provides an algebraic solution to the problem discussed in this Section [15]. Even more, if \(\mathcal{A}\) is the category of finite dimensional nilpotent representations of a quiver \(Q\) over a field, then the double \(D{HA}_\mathcal{A}\) recovers the quantum group associated with \(Q\), and its natural triangular decomposition. In the two dimensional case, one can understand preprojective CoHA’s as algebra of creating operators on the cohomology (and K–theory) of Nakajima quiver varieties [16]. It has further been shown that such CoHA can be identified with the positive half of the Maulik–Okounkov Yangian [17], [18]. In general, CoHA’s are expected to provide a geometric way of constructing positive halves of quantum groups and their representations. However we lack of a systematic way of understanding their doubles. Thus, one usually tries to construct left and right representations as in 4 , and glue them to the action of a bigger algebra.

Let \(X\) be a smooth complex projective curve, and let \(\beta\) be a rational number. The category \(\operatorname{Coh}^{ss}_{\beta}(X)\) of coherent semistable sheaves of fixed slope \(\beta\) on \(X\) has homological dimension \(1\), but is not finitary. In this paper, we discuss a left and right action of the categorified Hall algebra attached to the category \(\operatorname{Coh}^{ss}_{\beta}(X)\) on the derived category of certain Quot spaces over the curve, as we now explain.

Quot spaces in tilted hearts and categorified action↩︎

Algebra. In Section 2 we recall the construction of the categorified Hall algebra (CatHA) in our particular setting, following [19] and [1]. In order to carry out the subsequent constructions, we freely use the language of derived stacks. We consider the standard derived enhancement \(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\) of the stack of semistable sheaves of slope \(\beta\) on \(X\). This stack fits in a derived convolution diagram \[\label{eqn32:32conv32for32ss32in32intro} \begin{tikzcd} & {(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))^{\operatorname{ext}}} & \\ {\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\otimes \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)} && \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X) \arrow["p", from=1-2, to=2-1] \arrow["q"', from=1-2, to=2-3] \end{tikzcd}\tag{6}\] where (underived) points in the stack \({(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))^{\operatorname{ext}}}\) correspond to extensions of semistable sheaves of slope \(\beta\). As observed by Dyckeroff–Vasserot [20] and Diaconescu–Porta–Sala [1], such diagram has to be understood as the shadow of a more complicated object. More precisely, we recall in Subsection 2.3 the notion of 2–Segal space, which is a simplicial stack encoding the higher combinatorics of the Hall product. Since the map \(p\) is smooth and the map \(q\) is proper we can define the composition \[q_*p^* \colon \operatorname{Coh}^{\operatorname{b}}(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)) \otimes \operatorname{Coh}^{\operatorname{b}}(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)) \longrightarrow \operatorname{Coh}^{\operatorname{b}}(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))\] This endows the stable \(\infty\)–category \(\operatorname{Coh}^{\operatorname{b}}(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))\) with an \(\mathbb{E}_1\)–monoidal structure. We refer to the category \(\operatorname{Coh}^{\operatorname{b}}(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))\) together with this \(\mathbb{E}_1\)–monoidal structure as categorical Hall algebra of semistable sheaves of slope \(\beta\) on \(X\). We can extract different CoHA’s out of the CatHA. For example, the spaces \[G_0(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))\text{and}H^{\text{BM}}_*(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))\] inherit the structure of unital associative algebras, recovering the construction of [9]–[11] which agree with the formalism discussed in the previous Section. These algebra are the cohomological and K–theoretical Hall algebras associated with the category \(\operatorname{Coh}^{ss}_{\beta}(X)\).

Module. A natural way of considering framed sheaves is Grothendieck’s Quot scheme, which parametrizes flat families of quotients of a fixed coherent sheaf \(\mathcal{V}\). The Hilbert scheme in Example 1 is a particular case of Quot scheme, where \(\mathcal{V}= \mathcal{O}_S\). In order to set up an action diagram as in 3 , we study a generalization of such classical notion. In Section 3.2 we construct a family of hearts \[\label{tilted32hearts} \mathcal{A}^{\beta}\subset D^b(\operatorname{Coh}(X))\tag{7}\] by tilting the standard t–structure with respect to a torsion pair. This is a standard procedure coming from the study of Bridgeland stabiliy conditions [21] (see [22] for a review). We denote as \[\operatorname{Quot}^\beta(X,\mathcal{V})\] the punctual Quot space parametrizing quotients of an object \(\mathcal{V}\) in the heart \(\mathcal{A}^{\beta}\). If \(\mathcal{V}=\mathcal{L}[1]\), where \(\mathcal{L}\) is a line bundle on \(X\) of degree smaller than \(\beta\), then closed points in the scheme \(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1])\) can be described as short exact sequences \[\label{ses} 0\rightarrow \mathcal{L}\rightarrow F \rightarrow T \rightarrow 0\tag{8}\] of coherent sheaves where

• the slope of every nonzero subsheaf of \(F\) is smaller then or equal to \(\beta\)

• the slope of every nonzero quotient of \(T\) is strictly bigger then \(\beta\).

By using this explicit description, we relate the above Quot space with other relevant moduli spaces occurring in the literature.

1. On the one hand the above description allows us to conclude that the Quot space \(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1])\) coincides with the classical truncation of the derived stack of \(\beta\)–stable \(\mathcal{L}\)–pairs \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{L})\) in the sense of [1], Definition III.4.10. See Proposition 15.

2. On the other hand, we have semistable Bradlow pairs, which arise from a geometric PDE on the curve \(X\), called vortex equation [23], [24]. In this context, the constraints on the short exact sequence 8 come from the Kobayashi–Hitchin correspondence relating stability with the solvability of the aforementioned vortex equation. We prove in Proposition 16 that the above Quot space is an open subscheme in the (coarse) space of semistable Bradlow pairs. This investigation has overlappings with [25], and [26].

The latter identification allows us to construct a categorical action of the Cat–Ha associated with the category \(\operatorname{Coh}^{ss}_{\beta}(X)\) on \(\operatorname{Coh}^{\operatorname{b}}(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1]))\). We state our main result.

Our strategy to prove Theorem 1 is reminiscent of Example 1. We define in Section 4.1 a space \(\mathcal{S}_1^l\operatorname{Quot}^{\beta}(X,\mathcal{L}[1])\) of extensions of quotients 8 in the sense of Definition 12. This yields an action diagram \[\begin{tikzcd} & {\mathcal{S}_1^l\operatorname{Quot}^{\beta}(X,\mathcal{L}[1])} & \\ {\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\times \operatorname{Quot}^{\beta}(X,\mathcal{L}[1])} && \operatorname{Quot}^{\beta}(X,\mathcal{L}[1]) \arrow["{u^l_1 \times \omega_1}", from=1-2, to=2-1] \arrow["{\omega_0}"', from=1-2, to=2-3] \end{tikzcd}\] and the left action is given by the pull–push operation \[\label{intro32left32action} (\omega_0)_*(u^l_1 \times \omega_1)^* \colon \operatorname{Coh^b}(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)) \otimes \operatorname{Coh}^{\operatorname{b}}(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1])) \longrightarrow \operatorname{Coh}^{\operatorname{b}}(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1]))\tag{9}\] The analogous construction for the right action is presented in Section 5.1. In order for the function@eq:intro32left32action to be well–defined we need to prove that the morphisms \(\omega_0\) is proper and \(u^l_1 \times \omega_1\) is derived lci. The two main technical points in the proof of Theorem 1 are Lemma 8 and Lemma 11, where we prove that the morphism \(\omega_0\) and its right counterpart are proper. The proofs of such Lemmas rely in a substantial way on the properties of the tilted hearts 7 , as mentioned in Remark 18 and in the final point [itm32:32coh32ss32topo32prop] of Section 5.1.

1.1 Relation to other work↩︎

Diaconescu–Porta–Sala constructed an action of the categorical Hall algebra associated with the category of zero dimensional sheaves on a surface \(S\) on the derived category of the stack of Pandharipande–Thomas stable pairs on \(S\) [27] — see Corollary 4.40 in [1]. In fact, it is shown in Proposition III.4.11 of [1] that Pandharipande–Thomas stable pairs can be described as stable pairs in the sense of Definition III.4.10 in [1]. On the other hand, Bridgeland [28] related Pandharipande–Thomas stable pairs with Quot spaces in a tilted heart to derive wall–crossing formulas for Donaldson–Thomas invariants and Pandharipande–Thomas invariants. In the case of curves, Bradlow pairs have been related to Quot schemes in a tilted heart by different authors [25] [26]. The main novelty in the present paper are the further connections between Quot spaces in tilted hearts, Bradlow pairs, and stable pairs on a curve. In turn, these connections are essential to the proof of Theorem 1.

1.2 Acknowledgements↩︎

First, I want to thank my advisor Andrei Negut for his continued support and effective advice. I wish to thank Francesco Sala for sharing valuable insights about tilting and stable pairs. I thank Mauro Porta for his feedback and discussions. I further thank Ana Pavlakovic for introducing me to the notion of Bradlow pair. Moreover, I thank Tommaso Botta, Ben Davison, Dragos Fratila, Shivang Jidal, Archi Kaushik, Aliaksandra Novik, Olivier Schiffmann and Sebastian Schlegel Mejia for many interesting discussions and suggestions.

1.3 Notations and conventions↩︎

We make a list of notations and conventions that are used in this paper. We refer to [29] for background material. We only work over the field \(\mathbb{C}\) of complex numbers.

• We denote as \(\mathcal{S}\) the \(\infty\)–category of spaces

• We denote as \(\operatorname{Cat_{\infty}}\) the \(\infty\)–category of \(\infty\)–categories.

• We denote as \(\operatorname{Cat^{st}_{\infty}}\) the \(\infty\)–category of stable \(\infty\)–categories with exact functors between them.

• Let \(\mathcal{C}\in\operatorname{Cat_{\infty}}\). We denote as \(\mathcal{C}^{\simeq}\) the maximal infinity groupoid contained in \(\mathcal{C}\).

• We denote as \(\operatorname{dAff}\) the \(\infty\)–category of derived affine schemes.

• We denote as \(\operatorname{dSt}\) the \(\infty\)–category of derived stacks. This is the hypercompletion of the \(\infty\) topos of sheaves on the site \(\operatorname{dAff}\) with the Grothendieck topology induced by the étale topology.

• We denote as \(\operatorname{dGeom}\) the \(\infty\)–category of derived geometric stacks as in [30].

• Let \(\operatorname{X} \in \operatorname{dSt}\). We denote as \(\operatorname{Perf(X)}\) the \(\infty\)–category of perfect complexes on \(\operatorname{X}\).

2 Algebra↩︎

Let \(X\) be a smooth complex projective curve, and fix a number \(\beta \in \mathbb{Q}\sqcup\{+\infty\}\). In this Section we recall the construction of the Categorical Hall Algebras associated with the categories \(\operatorname{Coh}(X)\) of coherent sheaves and \(\operatorname{Coh}_{\beta}^{ss}(X)\) of semistable sheaves of slope \(\beta\) on \(X\), as defined by Diaconescu–Porta–Sala – see Sections II.1.4 and II.1.5 in [1].

2.1 Introduction to the Section↩︎

We recall the definition of the Cohomological Hall Algebra associated with the category \(\operatorname{Coh}^{ss}_{\beta}(X)\). The stack \(\underline{\operatorname{Coh}}(X)\) parametrizes flat families of coherent sheaves on \(X\), and it is a smooth algebraic stack locally of finite type over \(\operatorname{Spec}(\mathbb{C})\). As we mentioned in the introduction, the CoHA multiplication is given by the convolution diagram \[\label{eqn32:32coha32diagram} \begin{tikzcd} & {(\underline{\operatorname{Coh}}(X))^{\operatorname{ext}}} & \\ {\underline{\operatorname{Coh}}(X) \otimes \underline{\operatorname{Coh}}(X)} && \underline{\operatorname{Coh}}(X) \arrow["p", from=1-2, to=2-1] \arrow["q"', from=1-2, to=2-3] \end{tikzcd}\tag{10}\] where the stack \({(\underline{\operatorname{Coh}}(X))^{\operatorname{ext}}}\) parametrizes extensions \(0\rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0\) of coherent sheaves, and \[p \colon 0\rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0 \mapsto (F',F'') \quad q \colon 0\rightarrow F' \rightarrow F \rightarrow F'' \rightarrow 0 \mapsto F\] Since the category \(\operatorname{Coh}(X)\) has homological dimension \(1\), the map \(p\) is smooth. It is also easy to see that the map \(q\) is proper. In order to define a convolution product as in 2 , the authors of [1] consider the Borel–Moore homology — or, more generally, any motivic Borel–Moore homology theory as developped by A. Khan [31], [32] — of the above correspondence. We are interested in the subcategory \(\operatorname{Coh}^{ss}_{\beta}(X)\subset \operatorname{Coh}(X)\) of semistable sheaves of fixed slope \(\beta\), for a fixed \(\beta\). This category is closed under extensions. Moreover if \(F\in \operatorname{Coh}^{ss}_{\beta}(X)\), then every subsheaf \(F' \subset F\) of slope \(\beta\) is semistable. Similarly, any quotient \(F \twoheadrightarrow F''\) of slope \(\beta\) is semistable. It follows that the above construction extends verbatim to the open substack \(\underline{\operatorname{Coh}}_{\beta}^{ss}(X) \hookrightarrow \underline{\operatorname{Coh}}(X)\) thus delivering the CoHA of semistable sheaves of fixed slope. Porta–Sala [19] provided a categorification of this construction. That is, an \(\mathbb{E}_1\)–monoidal structure on the bounded derived category of the stack \(\underline{\operatorname{Coh}}(X)\) which recovers the above CoHA under decategorification. In order to get such a structure, we need a suitable enhancement of the convolution diagram 1 , which is provided by the Waldhausen construction in Section 2.3. We explain the categorification to the case of semistable sheaves in Subsection 2.4.

2.2 Derived stacks of objects in hearts of t–structures↩︎

In order to carry out the constructions in the following Sections, we introduce derived and more general versions of the classical stacks mentioned in the previous Subsection. Let \(\boldsymbol{\operatorname{Perf}}(X)\) be the derived moduli stack of perfect complexes on \(X\). This is defined as the functor \[\label{eqn32:32perf32functor} \boldsymbol{\operatorname{Perf}}(X)\;\colon \operatorname{dAff}^{\operatorname{op}} \longrightarrow \mathcal{S}\tag{11}\] which sends a derived affine scheme \(S\) to the maximal \(\infty\)–grupoid \(\operatorname{Perf}(X\times S)^{\simeq}\) contained in the stable \(\infty\)–category \(\operatorname{Perf}(X\times S)\) of perfect complexes of quasi–coherent sheaves on \(X\times S\) which have proper support over \(S\). See Construction 2.5 in [19], and example II.2.21 in [1]. This is the derived version of the stack of perfect complexes on \(X\).

Flatness and openness of t–structures. In order to define the derived stack \(\boldsymbol{\operatorname{Coh}}(X)\) we need to introduce the concept of flatness. We discuss a general notion of flatness with respect to a t–structure, as this will be useful in the following Sections. In the following we denote as \(X_s\) the fiber of a morphism \(X \times S \rightarrow S\).

As an example, a t–structure on \(\operatorname{Perf}(X)\) induces a constant t–structure \(\underline{\tau}\) on \(X\times S\).

Let \(\tau\) be a t–structure on \(\operatorname{Perf}(X)\). Then we have a subfunctor² of 11 \[\label{eqn32:32flatstack32def} \boldsymbol{\operatorname{Coh}}(X,\tau) \longrightarrow \boldsymbol{\operatorname{Perf}}(X)\tag{12}\] obtained by sending \(S\in \operatorname{dAff}\) to the full subspace \(\operatorname{Coh}_S(X\times S)^{\simeq}\) spanned by families of coherent sheaves which are flat with respect to the constant t–structure \(\underline{\tau}\). If we consider the standard t–structure \(\tau_{\operatorname{std}}\), flatness reduces to the usual notion and we get a derived stack \(\boldsymbol{\operatorname{Coh}}(X)\) of coherent sheaves on \(X\) as in [19]. More precisely, the stack \(\boldsymbol{\operatorname{Coh}}(X,\tau)\) is just the standard derived enhancement of the classical stack of families of \(\tau\)–flat sheaves defined in [33].

We have an important definition.

The following is going to be useful in many parts of the present paper. See Proposition II.2.56 in [1] for a non commutative (more general) version.

2.3 Waldhausen construction↩︎

We recall the construction of the categoricall Hall algebra associated with the stack \(\boldsymbol{\operatorname{Coh}}(X)\) from [19], building on [20] and [34]. We define the simplicial derived stacks encoding associativity of the Hall product. Let \[\operatorname{T} := \operatorname{Hom}_{\Delta}([1] ,-) \colon \Delta \longrightarrow \operatorname{Cat}_{\infty}\] where \(\Delta\) is the simplicial category. We denote \(\operatorname{T}_n = \operatorname{T}([n])\).

It is then straightforward to check that the above construction induces a simplicial object \[\label{eqn32:32waldhausen32C} \mathcal{S}_{\bullet}\mathcal{C} \colon \Delta^{op} \longrightarrow \operatorname{Cat}_{\infty}\tag{13}\] Theorem 7.3.3 in [20] states that 13 is a 2–Segal space, in the sense of Definition 2.3.1 therein. The importance of this notion in the context of the present paper is expressed by the following Theorem. We denote by \(\operatorname{Corr}^{\times}(\operatorname{dSt})\) the \((\infty,2)\) category of correspondences of derived stacks, equipped with the symmetric monoidal structure induced by the cartesian product on \(\operatorname{dSt}\) — see [34] \(\S 7.2.1\).

The stack \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{Coh}}(X,\tau)\). This is the combinatorial data encoding the algebra structure. Next, we transfer these data to \(\boldsymbol{\operatorname{Coh}}(X)\). We have \[\mathcal{S}_{\bullet}\boldsymbol{\operatorname{Perf}}(X)\;\colon \Delta \longrightarrow \operatorname{dSt}\] by applying the Waldhausen construction to \(\boldsymbol{\operatorname{Perf}}(X)\). This is a simplicial derived stack such that \[\mathcal{S}_0\boldsymbol{\operatorname{Perf}}(X)\cong \operatorname{Spec}(\mathbb{C}), \;\mathcal{S}_1\boldsymbol{\operatorname{Perf}}(X)\cong \boldsymbol{\operatorname{Perf}}(X),\;\mathcal{S}_2\boldsymbol{\operatorname{Perf}}(X)\cong \boldsymbol{\operatorname{Perf}}(X)^{\operatorname{ext}}\;,\] where \(\boldsymbol{\operatorname{Perf}}(X)^{\operatorname{ext}}\) parametrizes extensions in \(\operatorname{Perf}(X)\). The stack \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{Perf}}(X)\) is naturally a 2–Segal stack.

Let \(\boldsymbol{\operatorname{T}}\) be a derived stack equipped with a morphism \[\boldsymbol{\operatorname{T}} \longrightarrow \boldsymbol{\operatorname{Perf}}(X)\] We can define a simplicial stack \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{T}}\) associated with the stack \(\boldsymbol{\operatorname{T}}\) by setting

\[\begin{tikzcd} {\mathcal{S}_n\boldsymbol{\operatorname{T}}} & {\mathcal{S}_n\boldsymbol{\operatorname{Perf}}(X)} \\ \boldsymbol{\operatorname{T}} & \boldsymbol{\operatorname{Perf}}(X) \arrow[from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow[from=1-2, to=2-2] \arrow[""{name=0, anchor=center, inner sep=0}, from=2-1, to=2-2] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=0] \end{tikzcd}\]

In particular, we can carry out this construction with respect to the embedding 12 . The following is a direct application of Proposition II.3.6 in [1].

2.4 Categorical Hall algebra of semistable sheaves of fixed slope↩︎

Fix a number \(\beta \in \mathbb{Q}\sqcup\{+\infty\}\). We have a substack \[\label{eqn32:32cohssbeta32embedding} \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\hookrightarrow \boldsymbol{\operatorname{Coh}}(X)\tag{14}\] of semistable coherent sheaves of slope \(\beta\). This embedding is open thanks to Lemma 21.12 in [33]. Proceeeding as in Subsection 2.3 we can define the derived simplicial stack \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\). In particular, under the isomorphism \(\mathcal{S}_2\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\cong (\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))^{\operatorname{ext}}\) we recover the convolution diagram 6 from the introduction \[\begin{tikzcd} & {(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))^{\operatorname{ext}}} & \\ {\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\otimes \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)} && \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X) \arrow["\partial_0 \times \partial_2", from=1-2, to=2-1] \arrow["\partial_1"', from=1-2, to=2-3] \end{tikzcd}\] where the map \(\partial_i\) is induced by the \(i\)-th face map. Explicitely, we have \[\partial_0 \times\partial_2 \colon 0\rightarrow F'\rightarrow F \rightarrow F'' \rightarrow 0 \mapsto (F', F'') \quad \text{and} \quad \partial_1 \colon 0\rightarrow F'\rightarrow F \rightarrow F'' \rightarrow 0 \mapsto F\;.\] We extract a stable \(\infty\)–category with a \(\mathbb{E}_1\)–monoidal structure out of the above correspondence.

We can define a refined pullback \(f^*\) for any derived lci map, and we can define pushforward \(f_*\) for any locally rpas map. Let \[\operatorname{dGeom}^{\operatorname{qc}} \hookrightarrow \operatorname{dSt}\] be the subcategory of quasi–compact geometric derived stack, and consider the \((\infty,2)\) category \[\operatorname{Corr}^{\times}(\operatorname{dGeom}^{\operatorname{qc}})_{\operatorname{rpas,lci}} \hookrightarrow \operatorname{Corr}^{\times}(\operatorname{dSt})\] of correspondences of quasi–compact geometric derived stacks where the horizontal morphisms are ind–derived lci and the vertical morphisms are locally rpas. Building on Gaitsgory–Rozenblyum’s work [34], Porta–Sala — see the appendix of [19] — defined a right–lax monoidal functor \[\label{eqn32:32coh32for32qc}\operatorname{Coh}^{\operatorname{b}} \colon \operatorname{Corr}^{\times}(\operatorname{dGeom}^{\operatorname{qc}})_{\operatorname{rpas,lci}} \longrightarrow \operatorname{Cat}^{\operatorname{st}}_{\infty}\tag{15}\] In particular \(\mathbb{E}_1\)–monoid objects are preserved under this functor. The following Theorem is an application of Corollary II.3.7 in [1]. See also Corollary II.4.14 in [1].

3 Module↩︎

For a given object \(\mathcal{V}\) in \(\operatorname{Perf}(X)\) and a real number \(\beta\in \mathbb{R}\), we define a derived moduli stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) of \(\beta\)–stable \(\mathcal{V}\)–pairs on \(X\). When \(\beta \in \mathbb{Q}\), the derived category of the stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) will support an action of the algebra defined in the previous Section. Moreover, we compare the stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) to other moduli spaces known in the literature.

3.1 Torsion pairs↩︎

We follow [33] and [25]. Recall the definition of torsion pair.

We list some useful properties of torsion pairs for future reference.

Tilting. Let \(\mathcal{D}=D^b(\mathcal{A)}\) be the derived category of bounded complexes in \(\mathcal{A}\). The tilting of \(\mathcal{A}\) with respect to a torsion pair \((\mathcal{F},\mathcal{T})\) is the smallest extesion–closed subcategory of \(\mathcal{D}\) containing both \(\mathcal{T}\) and \(\mathcal{F}[1]\). We denote it as \(\mathcal{A}^*=\langle\mathcal{F}[1],\mathcal{T}\rangle\), and it can be explicitely described as \[\mathcal{A}^{*} = \left\{G\in \mathcal{D}\Big| \mathcal{H}^i(G) =0 \text{ if } i\neq 0,-1 \;; \;\mathcal{H}^0(G) \in \mathcal{T} ; \;\mathcal{H}^{-1}(G) \in \mathcal{F} \right\} \;,\] where \(\mathcal{H}^i\) denotes the cohomology with respect to the standard t–structure \(\tau_{\operatorname{std}}\). Moreover, we can define a bounded t–structure \(\tau^*\) on \(\mathcal{D}\) as \[\mathcal{D}^{\leq 0} = \{G \in \mathcal{D} \Big| \mathcal{H}^i(G)=0 \text{ for } i>0 \text{ and } \mathcal{H}^0(G)\in \mathcal{T}\}\] \[\mathcal{D}^{\geq 0} = \{G \in \mathcal{D} \Big| \mathcal{H}^i(G)=0 \text{ for } i<-1 \text{ and } \mathcal{H}^{-1}(G)\in \mathcal{F}\}\]

An object \(G\) in the tilted heart \(\mathcal{A}^*\) fits in a canonical short exact sequence \[\label{eqn32:32canonical32ses32tilting} 0 \rightarrow \mathcal{H}^{-1}(G)[1] \rightarrow G \rightarrow \mathcal{H}^0(G) \rightarrow 0\;.\tag{17}\] Then, one can easily deduce that the subcategories \[\mathcal{F}[1]=\{G\in \mathcal{A}^* \;\big|\;\mathcal{H}^0(G) =0 \}\text{ and }\mathcal{T}=\{ G\in \mathcal{A}^* \;\big| \;\mathcal{H}^{-1}(G)=0\}\] form a torsion pair in the tilted heart \(\mathcal{A}^*\). In particular, \(\mathcal{F}[1]\) is the torsion part, and \(\mathcal{T}\) is the torsion free part.

3.2 The tilted heart \(\mathcal{A}^{\beta}\)↩︎

Let \(X\) be a smooth projective curve defined over the field \(\mathbb{C}\) of complex numbers, and let \(\beta\in \mathbb{R}\). Consider the full subcategories of \(\operatorname{Coh}(X)\) given by \[\label{eqn32:32fbetatbeta} \mathcal{F}^{\beta}=\{ G\in \operatorname{Coh}(X) \;|\;\mu(G')\leq \beta \text{ for all non-zero subsheaves } G'\subseteq G \} \;,\tag{18}\] \[\mathcal{T}^{\beta}=\{ G\in \operatorname{Coh}(X) \;|\;\mu(G'') > \beta \text{ for all quotients } G \twoheadrightarrow G''\neq G \}.\] The following is an exercise in [37].

Tilting the standard heart \(\operatorname{Coh}(X)\) with respect to the above torsion pair yields another t-structure which we denote as \(\tau^\beta\), whose heart is given by \[\mathcal{A}^{\beta} = \left\{E\in \operatorname{Perf}(X)\Big| \mathcal{H}^i(E) =0 \text{ if } i\neq 0,-1 \;; \;\mathcal{H}^0(E) \in \mathcal{T}^{\beta} ; \;\mathcal{H}^{-1}(E) \in \mathcal{F}^{\beta} \right\} \;.\] We denote as \[\mathcal{A}^{\beta}_{\operatorname{t.f.}}= \mathcal{F}^{\beta}[1]\mathcal{A}^{\beta}_{\operatorname{tor}}= \mathcal{T}^{\beta}\] the induced torsion pair.

We end the Subsection with a remark which will be useful later on.

3.3 Derived stack of \(\beta\)–stable \(\mathcal{V}\)–pairs↩︎

This subsection is devoted to defining and describing the spaces that support the action in Theorem 1.

Stacks of torsion and torsion free objects. We can define a derived stack \(\boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X)\) parametrizing flat families of sheaves in \(\mathcal{T}^{\beta}\). Specifically, \(\boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X)\) is defined by the pullback square \[\label{eqn32:32tstd32open} \begin{tikzcd} \boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X)& {\boldsymbol{\operatorname{Coh}}(X,\tau^\beta)} \\ {\boldsymbol{\operatorname{Coh}}(X)} & \boldsymbol{\operatorname{Perf}}(X) \arrow[from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow[from=1-2, to=2-2] \arrow[""{name=0, anchor=center, inner sep=0}, from=2-1, to=2-2] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=0] \; \end{tikzcd}\tag{20}\] where the bottom horizontal map and the right vertical maps are given by the embedding 12 for the t–structures \(\tau_{\operatorname{std}}\) and \(\tau^\beta\), respectively. Similarly, we define a derived stack \(\boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)\) parametrizing flat families of sheaves in \(\mathcal{F}^{\beta}\). Let \([1] \colon \boldsymbol{\operatorname{Perf}}(X)\rightarrow \boldsymbol{\operatorname{Perf}}(X)\) be the morphism induced by the shift by \(1\) in \(\operatorname{Perf}(X)\). Then we can also define the stack \(\boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)\) via the pullback \[\begin{tikzcd} \boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)&& {\boldsymbol{\operatorname{Coh}}(X,\tau^\beta)} \\ {\boldsymbol{\operatorname{Coh}}(X)} & \boldsymbol{\operatorname{Perf}}(X)& \boldsymbol{\operatorname{Perf}}(X) \arrow[from=1-1, to=1-3] \arrow[from=1-1, to=2-1] \arrow[from=1-3, to=2-3] \arrow[""{name=0, anchor=center, inner sep=0}, from=2-1, to=2-2] \arrow["{[1]}", from=2-2, to=2-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=0] \end{tikzcd}\]

Repeating the construction, we get stacks \[\label{eqn32:32tbeta32open} \boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X)\cong \boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X,\tau^{\beta})\rightarrow \boldsymbol{\operatorname{Coh}}(X,\tau^\beta)\text{and}\boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)[1] \cong \boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X,\tau^{\beta})\rightarrow \boldsymbol{\operatorname{Coh}}(X,\tau^\beta)\tag{25}\] where \(\boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)[1]\) denotes the image of the stack \(\boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)\) under the shift morhphism in \(\boldsymbol{\operatorname{Perf}}(X)\).

Stable pairs. We introduce the notion of stable pair. Compare with Definition III.4.10 in [1].

It follows immediatly from the definition that if \(\mathcal{V}\) is a coherent sheaf, then a \(\beta\)–stable \(\mathcal{V}\)–pair is a short exact sequence in \(\operatorname{Coh}(X)\). We now define a stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) parametrizing stable pairs as in the above Definition, following Definition II.3.2 in [1]. Recall the Waldhausen construction \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{Perf}}(X)\) from 13 . Unravelling the definitions, we see that closed points in the stack \(\mathcal{S}_2\boldsymbol{\operatorname{Perf}}(X)\) are diagrams of the form \[\begin{tikzcd} 0 & \mathcal{V}& F \\ & 0 & T \\ && 0 \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-2, to=2-2] \arrow[from=1-3, to=2-3] \arrow[from=2-2, to=2-3] \arrow[from=2-3, to=3-3] \end{tikzcd}\] where \(\mathcal{V}, F, T\) are objects in \(\operatorname{Perf}(X)\) and the square is a pullback. In other words, closed points in \(\mathcal{S}_2\boldsymbol{\operatorname{Perf}}(X)\) are fiber sequences \([\mathcal{V}\rightarrow F \rightarrow T]\) in \(\operatorname{Perf}(X)\). Under the natural identification \(\mathcal{S}_1\boldsymbol{\operatorname{Perf}}(X)\cong \boldsymbol{\operatorname{Perf}}(X)\), we also have natural maps \[\label{eqn32:32delta32maps} \partial_i \colon \mathcal{S}_2\boldsymbol{\operatorname{Perf}}(X)\longrightarrow \boldsymbol{\operatorname{Perf}}(X)\tag{26}\] for \(i=1,2,3\), given by \[\label{qmtfweok} \partial_0([\mathcal{V}\rightarrow F \rightarrow T]) = T \quad \partial_1([\mathcal{V}\rightarrow F \rightarrow T]) = F \quad \partial_2([\mathcal{V}\rightarrow F \rightarrow T]) = \mathcal{V}\tag{27}\]

We define the stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) of \(\beta\)–stable \(\mathcal{V}\)–pairs defined via the pullback \[\label{eqn32:32def32of32stack32of32stable32pairs}\begin{tikzcd} \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})& \boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\\ {\boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X,\tau^{\beta})\times \boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X,\tau^{\beta})} & {\boldsymbol{\operatorname{Perf}}(X)\times \boldsymbol{\operatorname{Perf}}(X)} \arrow[from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow["{\partial_1[1]\times\partial_0}", from=1-2, to=2-2] \arrow[from=2-1, to=2-2] \end{tikzcd}\tag{28}\] where \(\partial_1[1]\) denotes the composition of the morphism induced by \(\partial_1\) and the shift morphism. Compatibly with the notation 26 , for any object \(\operatorname{P}= [\mathcal{V}\rightarrow F \rightarrow T]\) in \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) we denote \[\partial_1(\operatorname{P}) = F \qquad \partial_0(\operatorname{P}) = T\;.\]

3.4 Quot–spaces in \(\mathcal{A}^{\beta}\)↩︎

In this Subsection, we assume that the framing object \(\mathcal{V}\) belongs to the heart \(\mathcal{A}^{\beta}\).

Unravelling the definition, we see that for any \(S\in \operatorname{dAff}_{\mathbb{C}}\), the Quot space \(\boldsymbol{\operatorname{Quot}}^{\beta}_{S}(X,\mathcal{V})\) parametrizes fiber sequences \[\label{eqn32:32fiber32sequence32in32derived32quot} K \rightarrow \mathcal{V}\rightarrow E\tag{30}\] where \(K\) and \(E\) are flat with respect to the t–structure \(\tau\). In particular, when \(S\) is underived, the objects \(K,\mathcal{V},E\) belong to the heart \(\mathcal{A}^{\beta}\), so 30 becomes a short exact sequence in \(\mathcal{A}^{\beta}\). We deduce that \(\boldsymbol{\operatorname{Quot}}^{\beta}_{S}(X,\mathcal{V})\) is the standard derived enhancement of the Quot space of \(\mathcal{V}\) in the heart \(\mathcal{A}^{\beta}\) defined in [33]. In particular, when \(S\) is an underived scheme, we get an algebraic space \(\operatorname{Quot}^\beta_S(X,\mathcal{V})\) locally of finite presentation over \(S\).
Thanks to Lemma 3 and Remark 2.14 in [25] we also have the following.

We end the section with a piece of notation. We denote as \[\operatorname{Quot}^{\beta}(X,\mathcal{V})= \bigsqcup_{r,d} \operatorname{Quot}_{r,d}^{\beta}(X,\mathcal{V})\] the decomposition into connected components given by the Chern class of \(E\) in the Quot space \(\operatorname{Quot}^{\beta}(X,\mathcal{V})\).

3.5 Stable pairs Vs Quot spaces Vs Bradlow pairs↩︎

In this section we compare the stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) of \(\beta\)–stable \(\mathcal{V}\)–pairs with other moduli spaces occurring in the literature.

Comparing stable pairs and Quot spaces. In some cases, \(\beta\)–stable \(\mathcal{V}\)–pairs can be identified with quotients in the heart \(\mathcal{A}^{\beta}\).

In particular the classical truncation of the moduli stack of \(\beta\)–stable \(\mathcal{L}\)–pairs is the Quot–space of \(\mathcal{L}\) \[\label{eqn32:32QuotPairs} \operatorname{Quot}^{\beta}(X,\mathcal{L}[1])\cong \prescript{cl}{}\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{L})\;.\tag{32}\]

Comparing Stable Pairs and Bradlow pairs. We fix a coherent sheaf \(\mathcal{V}\in \operatorname{Coh}(X)\) whose Chern class is \((r,d)\) and we consider morphisms of the form \(\mathcal{V}\rightarrow F\).

This definition was first given by Bradlow [24] in the case where \(\mathcal{V}= \mathcal{O}_X\). Successively, Lin [40] extended the definition to a more general setting. In particular, it is shown that the stack parametrizing semistable Bradlow pairs in the sense of Definition 11 of rank \(r\) and degree \(d\) admits a projective coarse moduli space \[\label{eqn32:32coarse32bradlow} \mathcal{B}^{\sigma - ss}_{r,d}(X,\mathcal{V})\longrightarrow \operatorname{{B}}^{\sigma - ss}_{r,d}(X,\mathcal{V})\tag{33}\] Moreover, the moduli functor of stable Bradlow pairs is represented by an open subscheme \(\operatorname{B}^{\sigma -s}_{r,d}(X,\mathcal{V}) \subset \operatorname{{B}}^{\sigma - ss}_{r,d}(X,\mathcal{V})\). The relation of the aforementioned moduli space with Quot spaces in \(\mathcal{A}^{\beta}\) has been explored in [26] and [25], following the ideas in [28]. In particular, the following Lemma follows immediatly from Lemma 4.3 in [25] — See also Lemma 2.3 in [28] and Lemma 2.2 in [26]. Fix and a real parameter \(\beta \geq \frac{d}{r}\), and recall the function \(\mu_-\) defined in 24 .

By using this result, it is shown in [26] that if \(\beta\) is chosen in such a way that there is no strictly \(\sigma\)–semistable Bradlow pair, then we have an isomorphism \[\operatorname{B}^{\sigma -s}_{r,d}(X,\mathcal{V}) \cong {\operatorname{Quot}_{r,d}^{\beta}(X,\mathcal{V}[1])} \;\] This morphism sends a \(\sigma\)–semistable Bradlow pair \((F,s)\) to the map \(s[1]\). We compare the coarse moduli space \(\operatorname{{B}}^{\sigma - ss}_{r,d}(X,\mathcal{V})\) with the Quot space \({\operatorname{Quot}_{r,d}^{\beta}(X,\mathcal{V}[1])}\) when \(\mathcal{V}= \mathcal{L}\) is a line bundle of slope smaller then \(\beta\).

Observe that the above Theorem holds for \(\beta \in \mathbb{R}\). Indeed, we know that the Quot space \(\operatorname{Quot}_{r,d}^{\beta}(X,\mathcal{L}[1])\) is representable. However, if \(\beta \notin \mathbb{Q}\), we can’t conclude that it is proper.

4 Left action↩︎

Let \(X\) be a curve as before and let \(\beta \in \mathbb{Q}\). We define a left action of the categorical Hall algebra of \(\operatorname{Coh}^{ss}_{\beta}(X)\) on the pro–category \(\operatorname{Coh^b_{pro}}(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V}))\) defined in Subsection 3.3. Throughout this section, we fix an object \(\mathcal{V}\) in \(\operatorname{Perf}(X)\).

4.1 Construction of the left action↩︎

Left Hecke pattern. We define the stack which induces the convolution diagram 3 from the introduction in the context of our left action. We denote as \(\operatorname{Coh}^{ss}_{\beta}(X)[1]\) the shift by \(1\) of the category \(\operatorname{Coh}^{ss}_{\beta}(X)\).

The following Proposition is a formal consequence of the above definitions and the properties of torsion pairs. It is the key property underlying the associativity of left action in Theorem 1. Compare with III.4.19 in [1].

Summary. We now sketch the construction of the left action. We will define in the following Subsections a stack \(\mathcal{S}^l_1\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) parametrizing extensions of stable pairs in the sense of Definition 12. As before, we denote as \(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1]\) the image of the stack \(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\) under the shift morphism \([1] \colon \boldsymbol{\operatorname{Perf}}(X)\rightarrow \boldsymbol{\operatorname{Perf}}(X)\). Then we have a convolution diagram \[\label{eqn32:32left32action32introduction} \begin{tikzcd} & {\mathcal{S}_1^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})} &&& {\operatorname{E}} \\ {\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1] \times \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})} && \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})& {(G,\operatorname{P})} && {\operatorname{P}'} \arrow["{u^l_1 \times \omega_1}", from=1-2, to=2-1] \arrow["{\omega_0}"', from=1-2, to=2-3] \arrow[maps to, from=1-5, to=2-4] \arrow[maps to, from=1-5, to=2-6] \end{tikzcd}\tag{39}\] We have a natural isomorphism \(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\cong \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1]\) induced by the shift operation. We prove further that the map \(u^l_1 \times \omega_1\) is lci, and the map \(\omega_0\) is proper.

The pull–push operation \[\omega_{0*}(\omega_1 \times u^l_1)^* \colon H^{BM}_*(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X))\otimes H^{BM}_*(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})) \longrightarrow H^{BM}(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V}))\] induces the structure of a left module for the CoHA attached to the category \(\operatorname{Coh}^{ss}_{\beta}(X)\) on the space \(H^{BM}_*(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V}))\). In particular, the associativity property of this action is encoded in the statement of Proposition 17. Even more, we provide a categorification of such action. In order to do this, we upgrade the diagram 39 to a more complicated object, as we explain in the next Subsection.

4.2 Stack of left extensions↩︎

Recall the stack of \(\mathcal{V}\)–pairs \(\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\) of Definition 9. Construction I.3.6 in [1] provides us with a simplicial derived stack \[\label{eqn32:32flags32relative} \mathcal{S}_{\bullet}^l\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\tag{40}\] Unravelling the definition, the stack \(\mathcal{S}^l_1\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\) fits into the fiber product \[\label{eqn32:32fiber32product32defining32S1flags} \begin{tikzcd} {\mathcal{S}^l_1\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})} & {\mathcal{S}_3\boldsymbol{\operatorname{Perf}}(X)} \\ {Spec(\mathbb{C})} & {\boldsymbol{\operatorname{Perf}}(X)} \arrow[from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow["{\partial_{23}}", from=1-2, to=2-2] \arrow["\mathcal{V}", from=2-1, to=2-2] \end{tikzcd}\tag{41}\] where the bottom horizontal map is the morphism induced by the object \(\mathcal{V}\), and the map \(\partial_{23} \colon \mathcal{S}_3\boldsymbol{\operatorname{Perf}}(X)\rightarrow \mathcal{S}_1\boldsymbol{\operatorname{Perf}}(X)\cong \boldsymbol{\operatorname{Perf}}(X)\) is induced by the face map \([3] \rightarrow[1]\) in \(\Delta^{op}\) which avoids \(2\) and \(3\). In particular, the stack \(\mathcal{S}^l_1\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\) parametrizes extensions of the form ?? , where no condition is imposed on the perfect complexes occurring in the diagram. Moreover, the fiber diagram 41 induces natural maps \[\label{eqn32:32convolution32left} \begin{tikzcd} & {\mathcal{S}_1^l\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})} &&& {\operatorname{E}} \\ {\boldsymbol{\operatorname{Perf}}(X)\times \boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})} && \boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})& {(G,\operatorname{P})} && {\operatorname{P}'} \arrow["{u^l_1 \times \omega_1}", from=1-2, to=2-1] \arrow["{\omega_0}"', from=1-2, to=2-3] \arrow[maps to, from=1-5, to=2-4] \arrow[maps to, from=1-5, to=2-6] \end{tikzcd}\tag{42}\] We now discuss the higher simplicial levels. The morphism \(u_1^l\) can be extended to a morphism of simplicial stacks \[\label{eqn32:32relative32simp32flags} \mathcal{S}_{\bullet}^l\boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\longrightarrow \mathcal{S}_{\bullet}\boldsymbol{\operatorname{Perf}}(X)\tag{43}\] which satisfies the relative 2–Segal property. As the 2–Segal property encodes the Hall algebra structure, the relative 2–Segal property encodes the Hall algebra action — See Corollary 5.4 in [35] for a precise formulation.

Proceeding as in II.3.2 in [1] — see Construction I.5.5 — we can define a simplicial stack \(\mathcal{S}_{\bullet}^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\), which parametrizes extensions as in Definition 12. Moreover we have a natural morphism \[\label{eqn32:32relative32simp32pairs} \mathcal{S}_{\bullet}^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\longrightarrow \mathcal{S}_{\bullet}\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1]\tag{44}\] In particular, the maps 39 are well–defined. The crucial step in the proof of Theorem 1 is that the morphism 44 also satisfies the relative 2–Segal property.

The 2–Segal property of the morphism 44 is induced by 43 in two steps. We follow the procedure in the proof of Proposition III.4.21 in [1].

Step 1. Consider the auxiliary stack \(\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})\) defined via the pullback \[\label{eqn:32aux32stack} \begin{tikzcd} {\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})} & \boldsymbol{\operatorname{Perf}}^{\dagger}(X,\mathcal{V})\\ {\boldsymbol{\operatorname{Coh}}(X,\tau^\beta)} & \boldsymbol{\operatorname{Perf}}(X) \arrow[from=1-1, to=1-2] \arrow[ from=1-1, to=2-1] \arrow["{\partial_0}", from=1-2, to=2-2] \arrow[""{name=0, anchor=center, inner sep=0}, from=2-1, to=2-2] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=0] \end{tikzcd}\tag{45}\] where the t–structure \(\tau^\beta\) was defined in Subsection 3.2. Such stack parametrizes fiber sequences \([\mathcal{V}\rightarrow F \rightarrow T]\) in \(\operatorname{Perf}(X)\) with the only condition of \(T\) being in the heart \(\mathcal{A}^{\beta}\). Applying construction II.3.2 in [1] as before, we define the relative simplicial stack \[\label{eqn32:32aux32relative32stack} u_{\bullet}^l \;\colon \mathcal{S}_{\bullet}^l\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V}) \longrightarrow \mathcal{S}_{\bullet}\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1] \;.\tag{46}\]

Step 2. The stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) maps naturally to the stack \(\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})\). Moreover, we have induced maps at the higher simplicial levels. Thus, we can set up a square \[\label{eqn32:32non32aux32square} \begin{tikzcd} {\mathcal{S}_1^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})} & {\mathcal{S}_1^l\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})} \\ {\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})} & {\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})} \arrow[from=1-1, to=1-2] \arrow["{\omega_0}"', from=1-1, to=2-1] \arrow["{\omega_0}", from=1-2, to=2-2] \arrow[from=2-1, to=2-2] \end{tikzcd}\tag{47}\]

It follows from Lemma 7 that the square \[\begin{tikzcd} {\mathcal{S}_1^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})} & {\mathcal{S}_1^l\boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})} \\ {\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1] \times \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})} & {\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1] \times \boldsymbol{\operatorname{Perf}}^{\dagger}_{0-\mathcal{A}^{\beta}}(X,\mathcal{V})} \arrow[from=1-1, to=1-2] \arrow["{u_1^l \times \omega_1}"', from=1-1, to=2-1] \arrow["{u_1^l \times \omega_1}", from=1-2, to=2-2] \arrow[from=2-1, to=2-2] \end{tikzcd}\] is a pullback. Then, Lemma 6 allows us to apply Corollary I.5.7 in [1] to conclude the proof of Proposition 19.

4.3 Proof of the left action↩︎

In this subsection we complete the proof of the left action part in Theorem 1 by checking the geometric properties.

Proper push–forward. Proceeding as in Notation III.4.22 in [1], we can define a derived stack \[\mathcal{S}_1^l\boldsymbol{\operatorname{FlagCoh}}^{(1)}_{\operatorname{t.f.,\beta-ss[1]}}(X,\tau^\beta)\] whose closed points are short exact sequences \(0 \rightarrow T\rightarrow T'\rightarrow G \rightarrow 0\) in \(\mathcal{A}^{\beta}\) where \(T, T'\in \mathcal{A}^{\beta}_{\operatorname{t.f.}}\) and \(G\in \operatorname{Coh}^{ss}_{\beta}(X)[1]\). Consistently with the above notations, we set \[\partial_0(T\rightarrow T'\rightarrow G)=G,\;\; \partial_1(T\rightarrow T'\rightarrow G)=T', \;\;\partial_2(T\rightarrow T'\rightarrow G)=T \;.\]

Consider the map \[\label{eqn32:32pitfzero} \pi^{\operatorname{t.f.}}_{0}\colon= \partial_0 \;\colon \;\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\longrightarrow \boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X,\tau^{\beta})\tag{49}\] sending a stable pair \([\mathcal{V}\rightarrow F \rightarrow T]\) to the \(\mathcal{A}^{\beta}\)–torsion free component \(T\). We also have a map \[\pi_1^{\operatorname{t.f.}} \;\colon \mathcal{S}_1^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\longrightarrow \mathcal{S}_1^l{\boldsymbol{\operatorname{FlagCoh}}}^{(1)}_{\operatorname{t.f.,\beta-ss[1]}}(X,\tau^\beta)\;\] which sends an extension ?? to the sub–extension \(0 \rightarrow T\rightarrow T'\rightarrow G \rightarrow 0\).

Flat pull–back. We consider the map \[\label{eqn32:32left32pullback32map} u_1^l \times \omega_1 \colon \mathcal{S}^l_1 \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\longrightarrow \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1] \times \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\tag{50}\]

Left action. In order to extract a categorical action from the above data, we would like to apply the functor 15 to the correspondence encoded by the simplicial derived stacl \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\). However, according to the defining diagram 28 , the stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) is cut out by open conditions in \(\mathcal{S}_2\boldsymbol{\operatorname{Perf}}(X)\). In particular we can’t conclude that the connected component of the stack \(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\) are quasi–compact. In order to state Theorem 23 and Theorem 28 in their full generality, we use the refinement \[\operatorname{Coh^b_{pro}}\colon \operatorname{Corr}^\times(\operatorname{Ind(dGeom^{qc}))_{rpas,lci}} \longrightarrow \operatorname{Pro(Cat^{st}_{\infty})}\] where the notation \(\operatorname{Ind(dGeom^{qc})}\) stands for Ind–geometric derived stacks, and \(\operatorname{Pro(Cat^{st}_{\infty})}\) is the \(\infty\)–category of Pro stable categories. See the appendix of [19] for the precise definition of the above functor. The shift morphism \[[1] \;\colon \;\boldsymbol{\operatorname{Perf}}(X)\longrightarrow \boldsymbol{\operatorname{Perf}}(X)\] induces a natural isomorphism between the stack \(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\) and its image \(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)[1]\). Thus we can apply Corollary II.3.12 in [1] to deduce the following.

5 Right Action↩︎

Throughout this section \(\mathcal{V}\) is a coherent sheaf on \(X\).

5.1 Construction of the right action↩︎

\(\beta\)–stable \(\mathcal{V}[1]\)–copairs In order to construct the right action, we need to tweak the definition of stable pairs.

The following is tautological.

Right Hecke pattern We introduce the stack of extensions that encodes the right action.

In analogy with the discussion in Subsection 4.1, we remark that the above Proposition is the essential ingredient in the proof of the associativity of the right action in Theorem 1.

Introduction to the section. We now explain our plan to complete the proof of Theorem 1. With a similar procedure as in the previous Sections, we define a stack \(\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\) of stable copairs as in Definition 13, and a stack \(\mathcal{S}^r_1\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\) parametrizing extensions of stable copairs in the sense of Definition 14. We get in this way a convolution diagram

\[\label{eqn32:32convolution32right} \begin{tikzcd} & {\mathcal{S}^r_1\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})} &&& {\reflectbox{\operatorname{E}}} & \\ {\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\times \boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)} && \boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})& {(\reflectbox{\operatorname{P}},G)} && {\reflectbox{\operatorname{P}}'} \arrow["{\omega_0\times u^r_1}", from=1-2, to=2-1] \arrow["{\omega_1}"', from=1-2, to=2-3] \arrow[from=1-5, to=2-4] \arrow[from=1-5, to=2-6] \end{tikzcd}\tag{52}\] We prove further that the map \(\omega_0 \times u_1^r\) is lci, and the map \(\omega_1\) is proper. Then, the pull–push operation \[\omega_{1*} (\omega_0\times u^r_1)^* \colon H^{BM}_*(\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})) \otimes H^{BM}_*(\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)) \longrightarrow H^{BM}_*(\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V}))\] induces the structure of a right module for the CoHA attached to the category \(\operatorname{Coh}^{ss}_{\beta}(X)\) on the space \(H^{BM}_*(\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V}))\). In particular, the associativity property of this action is encoded in the statement of Proposition 24. Finally, thanks to Lemma 9 we get a natural isomorphism of derived stacks \[\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\cong \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\] which allows us to get a right action on the space \(H^{BM}_*(\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V}))\). We make two remarks explaining the role of the tilting procedure in our construction.

1. From a representation theoretical point of view, we get a left and a right action \[\operatorname{HA}_{\operatorname{Coh}^{ss}_{\beta}(X)} \curvearrowright \bigoplus_{r,d}H^{BM}_*(\boldsymbol{\operatorname{P}}^\beta_{r,d}(X,\mathcal{V})) \curvearrowleft \operatorname{HA}_{\operatorname{Coh}^{ss}_{\beta}(X)}\] where the direct sum in the central term comes from the decomposition in Remark 13. We unravel the actions by keeping track of the topological type. The convolution diagram 42 induces a fiber sequence \(F\rightarrow F'\rightarrow G\). Rotating, we get the short exact sequence in \(\operatorname{Coh}(X)\) \[\label{eqn32:32naive321} 0 \rightarrow G[-1] \rightarrow F \rightarrow F' \rightarrow 0\tag{53}\] where the topological type of \(F'\) is smaller than that of \(F\). Viceversa, the diagram 52 induces a short exact sequence in \(\operatorname{Coh}(X)\) \[\label{eqn32:32naive322} 0 \rightarrow G \rightarrow F' \rightarrow F \rightarrow 0\tag{54}\] where the topological type of \(F'\) is bigger than that of \(F\). Comparing 53 and 54 we deduce that operators coming from the left action decrease the topological type, and operators coming from the right action increase the topological type. Compare with Example 1.

2. From a geometric point of view, working in different t–structures allows us to construct morphisms with the desired properties. For example, the morphism \[\omega_0 \colon \mathcal{S}_1^l\boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V})\longrightarrow \boldsymbol{\operatorname{P}}^{\beta}(X,\mathcal{V}), \quad \operatorname{E} \longmapsto \operatorname{P}\] is not proper, so it cannot be used to induce an operator of right action. This problem is solved by working in a different heart.

5.2 Stack of right flags↩︎

We define a derived stack of copairs \(\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\) parametrizing stable copairs. Following [1] we apply the left version of Construction I.3.6 to the 2–Segal stack \(\mathcal{S}_{\bullet}\boldsymbol{\operatorname{Perf}}(X)\), and we get a relative 2–Segal space \[u^r_{\bullet}\;\colon \;\mathcal{S}_{\bullet}^r\boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])\longrightarrow \mathcal{S}_{\bullet}\boldsymbol{\operatorname{Perf}}(X)\] As before, closed points in tthe stack \(\mathcal{S}_0\boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])\cong \boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])\) correspond to fiber sequences \[[F \rightarrow T \rightarrow \mathcal{V}[1]]\] in \(\operatorname{Perf}(X)\) with no further conditions. Moreover, the stack \(\mathcal{S}_1\boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])\) parametrizes diagrams of the form ?? without any stability condition. Moreover, we have natural maps \[\begin{tikzcd} & {\mathcal{S}_1^r\boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])} & \\ {\boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])\times \boldsymbol{\operatorname{Perf}}(X)} && \boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1]) \arrow["{\omega_0 \times u^r_1}", from=1-2, to=2-1] \arrow["{\omega_1}"', from=1-2, to=2-3] \end{tikzcd}\] which agree with the notations in the previous Subsection.

We define the derived stack of copairs via the pullback \[\begin{tikzcd} \boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})& \boldsymbol{\operatorname{Perf}}^{\ddagger}(X,\mathcal{V}[1])\\ {\boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)\times \boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X)} & {\boldsymbol{\operatorname{Perf}}(X)\times \boldsymbol{\operatorname{Perf}}(X)} \arrow[from=1-1, to=1-2] \arrow[from=1-1, to=2-1] \arrow["{\partial_1\times\partial_0}", from=1-2, to=2-2] \arrow[from=2-1, to=2-2] \end{tikzcd}\] As an immediate consequence of Lemma 9 we have the following.

We apply again Construction I.5.5 in [1] to get a relative simplicial stack \[\label{eqn32:32relative32simplicial32copairs} u_{\bullet}^r \; \colon \;\mathcal{S}_{\bullet}^r\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\longrightarrow \mathcal{S}_{\bullet}\boldsymbol{\operatorname{Coh}}^{ss}_{\beta}(X)\tag{55}\] As before, the stack \(\mathcal{S}_1^r\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\) parametrizes stable extensions in the sense of Definition 14. In particular, the maps 52 are well defined. In light to Proposition 24, we can use an argument analogous to that of Proposition 19 to prove the following.

5.3 Proof of the right action↩︎

Proper push–forward. Proceeding as in Notation III.4.22 in [1], we can define a derived stack \[\mathcal{S}_1^r\boldsymbol{\operatorname{FlagCoh}}^{(1)}_{\operatorname{\beta--ss,t..f.}}(X)\] whose closed points are exact sequences \(0 \rightarrow G\rightarrow F' \rightarrow F \rightarrow 0\) in \(\operatorname{Coh}(X)\) where \(F, F'\in \mathcal{F}^{\beta}\) and \(G\in \operatorname{Coh}^{ss}_{\beta}(X)\). Let us denote as \[\partial_0(G\rightarrow F' \rightarrow F) = F \quad \partial_1(G\rightarrow F' \rightarrow F) = F'\quad \partial_2(G\rightarrow F' \rightarrow F) = G \;.\] the forgetful maps.

Consider the map \[\label{eqn32:32copair32to32tf} \nu_0^{\operatorname{t.f.}} :=\partial_2 \;\colon \;\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\longrightarrow \boldsymbol{\operatorname{Coh}}_{\operatorname{t.f.}}(X)\tag{57}\] which sends a stable copair \([F\rightarrow T \rightarrow \mathcal{V}[1]]\) to the torsion free component \(F\). Moreover, we have a morphism \[\nu_1^{\operatorname{t.f.}} \;\colon \;\mathcal{S}_1^r\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\longrightarrow \mathcal{S}_1^r\boldsymbol{\operatorname{FlagCoh}}^{(1)}_{\operatorname{\beta--ss,t..f.}}(X)\] sending a stable extension ?? to the sub–extension \(G\rightarrow F'\rightarrow F\).

Flat pull–back. As above, we consider a stack \[\mathcal{S}_1^r\boldsymbol{\operatorname{FlagCoh}}^{(1)}_{\operatorname{\beta-ss,tor}}(X)\] whose closed points are short exact sequences in \(\operatorname{Coh}(X)\) of the form \(0 \rightarrow G\rightarrow T' \rightarrow T \rightarrow 0\), where \(T , T'\in \mathcal{T}^{\beta}\) and \(G\in \operatorname{Coh}^{ss}_{\beta}(X)\). Let us denote as \[\partial_0(G\rightarrow T' \rightarrow T) = T \quad \partial_1(G\rightarrow T' \rightarrow T) = T'\quad \partial_2(G\rightarrow T' \rightarrow T) = G \;.\] the forgetful maps.

Consider the map \[\label{eqn32:32copair32to32tor} \nu^{\operatorname{tor}}_0 := \partial_1 \;\colon \;\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\longrightarrow \boldsymbol{\operatorname{Coh}}_{\operatorname{tor}}(X)\tag{58}\] sending a stable copair \([F\rightarrow T \rightarrow \mathcal{V}[1]]\) to the torsion component \(T\). We also have a natural map \[\nu^{\operatorname{tor}}_{1} \; \colon \;\mathcal{S}_1^{r}\boldsymbol{\reflectbox{\operatorname{P}}}^{\beta}(X,\mathcal{V})\longrightarrow \mathcal{S}_1^r\boldsymbol{\operatorname{FlagCoh}}^{(1)}_{\operatorname{\beta-ss,tor}}(X)\] which associates to every extension of copairs \(\eqref{eqn32:32co-extension}\) the sub–extension \(G\rightarrow T' \rightarrow T\).

Right action. In view of what we have already shown, we can apply Corollary II.3.14 in [1] to get the following.

Under the natural isomorphism of Lemma 9, we finally get both left and right operators on the same space.

Finally, if we specialize to the case where \(\mathcal{V}= \mathcal{L}[1]\) is a line bundle in \(\mathcal{F}^{\beta}\). Since the quot space \(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1])\) has proper connected components, we can consider the \(\infty\)–category \(\operatorname{Coh}^{\operatorname{b}}(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1]))\). Thanks to the isomorphism 32 we get a two–sided categorical action on \(\operatorname{Coh}^{\operatorname{b}}(\operatorname{Quot}^{\beta}(X,\mathcal{L}[1]))\).

References↩︎

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1. More precisely, the Hilbert stack \(\mathfrak{Hilb}^n(S) \cong \operatorname{Hilb}^n(S) \times B\mathbb{G}_m\).↩︎

2. This is analogous to Construction 2.5 in [19].↩︎