A t -structure on the ∞ -category of mixed graded modules

In this work, we shall study in a purely model-independent fashion the ∞ -category of mixed graded modules over a ring of characteristic 0, as deﬁned by D. Calaque, T. Pantev, M. Vaquié, B. Toën and G. Vezzosi. First, we collect some basic results about its main formal properties, clarifying foundational questions in a systematic manner, to serve as a reference for future work. Finally, we shall endow such ∞ -category with a both left and right complete accessible t -structure, showing how this identiﬁes the ∞ -category of mixed graded modules with the left completion of the Beilinson t -structure on the ∞ -category of ﬁltered modules.


Motivations
It is well known that, in characteristic 0, an action of the circle S 1 : = BZ (seen as the classifying derived stack for the constant stack Z) over a derived stack X is linked to differential forms and de Rham theory over X .Namely, in [3] the authors established that p-forms over a derived stack X can be interpreted via functions on the derived loop stack LX , which is canonically endowed with an action of the circle S 1 by rotating loops.The condition of being closed is then encoded in the property that the function is S 1 -equivariant.However, the homotopy theory of S 1 -complexes is equivalent to the homotopy theory of mixed graded complexes, i.e., chain complexes M • endowed with a decomposition where each M p is a sub-complex of M • , and endowed with a mixed differential ε p : M p → M p−1 [−1] satisfying the usual square-to-zero property.The theory of mixed graded complexes in characteristic 0, which was developed in [8,21], has been exploited extensively in the past years, and has also been linked to the usual derived filtered category of Beilinson in [6,19,23,24].Indeed, the de Rham algebra of a differential graded commutative ring A • , with its grading given by [1] , is a mixed graded commutative algebra, where the mixed differential is provided exactly by the de Rham differential.One can then define (shifted) p-forms and closed p-forms on a differential graded commutative algebra A • in terms of elements of dR p (A • ) and homotopy fixed points for the de Rham differentials in dR p (A • ), respectively.Moreover, all these constructions satisfy descent, and make perfectly sense also for more general derived stacks.
This theoretical framework yields new perspectives over derived symplectic geometry and deformation quantization.Using mixed graded complexes and mixed graded cdga's one can define: (1) shifted symplectic forms over a derived stack as shifted closed 2-forms which are non-degenerate in some suitable sense; (2) Lagrangian structures on morphisms of derived stacks; (3) Poisson structures and their deformation quantization; (4) derived algebraic foliations.
In the last years, mixed graded complexes have been employed also in the homotopy theory of Lie algebras and Lie algebroids.Given a differential graded Lie algebra, it is known that its Chevalley-Eilenberg algebra has a richer structure of mixed graded commutative algebra (see also [7,20]).In particular, it is expected that mixed graded complexes can provide a natural setting where to work with formal geometry and deformation theory-this last particular application is what we are most interested in.

Outline of the paper
The content of this paper stems from our work in the context of derived Lie algebras, and in particular from the study of the mixed graded structure of the Chevalley-Eilenberg algebras and coalgebras of Lie algebras, which is studied in greater detail in [22].For this purpose, the main aim of this paper is to collect some of the most important and useful features of the stable ∞-category ε -Mod gr k of mixed graded modules.In Sect. 1 we investigate the computation of limits and colimits of mixed graded modules, their closed monoidal structure, and the relationship of mixed graded modules with purely graded modules and non-graded modules (Sect.1.1).While the statements and the proofs are presented in a model-independent fashion, differently from what one can read in the existing literature, most of the results here gathered are far from being original.In particular, most of them can be found or easily derived from the content of [8,21,24].The main new contribution of this part is, arguably, the characterization of fully dualizable objects of ε -Mod gr k (Proposition 1.1.8).In Sect.2, we study the relationship between the ∞-categories ε -Mod gr k and Mod fil k , where the latter denotes the ∞-category of filtered modules.The main result of this paper is the characterization of ε -Mod gr k as the full sub-∞-category of Mod fil k spanned by filtered modules with complete filtration.While this result has already been proved (see for example [24] and [6]), our work offers a deeper insight on such embedding by taking into account some t-structures on both ∞-categories, namely: Theorem (Theorems 1.3.1 and 2.3.1)There exists a left and right complete t-structure on the stable ∞-category of mixed graded modules whose heart is equivalent to the usual abelian 1-category of chain complexes.Moreover, the embedding of the ∞category of mixed graded modules into the ∞-category of filtered modules admits a left adjoint (−) gr ε : Mod fil k → ε -Mod gr k which identifies the ∞-category of mixed graded k-modules with the left completion Mod fil k of the Beilinson t-structure of [2] on the ∞-category of filtered k-modules (which, in virtue of Proposition 2.2.8, is the full sub-∞-category of modules with complete filtration).

Notations, conventions and main references
• Throughout all this paper, we employ freely the language of derived algebraic geometry, ∞-categories, and homotopical algebra provided by [13,17], from which we borrow the formalism and most notations.Our language is innerly derived: every definition and construction has to be interpreted, without further indication suggesting the contrary, in the context of higher algebra.In particular, by module over a discrete commutative ring k we mean an object of the stable derived ∞-category of k-modules, by limits and colimits we mean homotopy limits and colimits, by tensor product we mean derived tensor product, and so forth.
• Our main references for the homotopy theory of mixed graded complexes are provided by [8,21].• Our main references for the homotopy theory of Lie algebras in characteristic 0, and its relationship with the derived deformation theory, are provided by [10,16].• When dealing with explicit models provided by chain complexes of k-modules, we use a homological notation.• Our standing assumption is that we work in characteristic 0, over a fixed commutative ring k. • Throughout this paper, we shall often work with closed symmetric monoidal ∞-categories C ⊗ enriched over k-modules.In particular, such ∞-categories are endowed with both internal mapping objects, obtained as a right adjoint to ⊗, and mapping k-modules providing the enrichment over Mod k .In order to avoid confusion, we shall denote the former with Map and the latter with Map.

Mixed graded modules
The fundamental objects of study in this article are mixed graded k-modules, which generalize the concept of mixed complexes and provide-at least in characteristic 0-a very useful analogue to complexes endowed with a complete and exhaustive filtration.Mixed graded modules have been studied extensively in the last years in the field of derived differential geometry and theory of Lie algebroids; yet, they are not as well known as filtered k-modules, of which they provide a more well-behaved analogue in characteristic 0. In this first section, we first gather some important definitions and properties of the ∞-category of mixed graded modules, and fix our notations.For this scope, our main sources are [8,21].

Basic definitions and notations
In order to capture the idea behind the notion of a mixed graded k-module, we first recall the concept of mixed k-modules (or mixed complexes, as they are classically called), which-in the words of [12], are objects that are both chain and cochain complexes in a compatible way.Mixed complexes were first introduced in [5], as algebraic S 1 -chain complexes (or chain complexes with an algebraic circle action), in order to study Hochschild and cyclic homology of unital associative algebras in characteristic 0, which naturally come equipped with a mixed structure at the level of chains.
Definition 1.1.1(Mixed complexes, [12]) A mixed complex over a base ring k of characteristic 0 is a chain complex Equivalently, mixed complexes are modules over the free differential graded commutative algebra k[η]: = k[t]/(t 2 ), where η: = t is a generator in homological degree 1 and ∂(η) = 0.Alternatively, they are comodules over the differential graded cocommutative coalgebra k This will be the stepping stone for generalizing the idea of mixed complexes to the derived setting.
with its square-zero extension commutative algebra structure.In this case, we can describe the semi-direct product BG a,k G m,k of the affine group stacks BG a,k and G m,k as the affine group stack whose algebra of functions is equivalent, as a commutative k-algebra, to the formal k-algebra Denoting again the generator in degree −1 with ε, the comultiplication for its cocommutative Hopf structure is given by the assignations t → t ⊗ t and ε → t ⊗ ε.
is the ∞-category of representations of the derived group stack BG a,k G m,k .Equivalently, it is the ∞-category of comodule objects for the Hopf algebra O BG a,k G m,k in Mod k .
Remark 1.1.4 In the setting of commutative differential graded k-algebras and chain complexes, a mixed graded k-module can be thought as a chain complex of k-modules M • , equipped with a decomposition of chain complexes of k-modules (M • ) p p∈Z and with a morphism of chain complexes In the description given in Definition 1.1.3,the action of G m,k yields the weight grading, while the action of BG a,k yields the mixed differential; the fact that we are considering the semi-direct product assures us that the mixed differential decreases the weight grading by −1, i.e. the two actions are intertwined.See also [21,Remark 1.1].
Remark 1.1.5 One could notice that in the explicit models provided by Remark 1.1.4the internal differential and the mixed differential commute, while in the classical notion of mixed complexes of Definition 1.1.1they were required to anti-commute.
But since we are working with bi-graded objects, it is a standard computation to show that the two formalisms are completely equivalent, up to suitably changing the signs of the mixed differential.

Remark 1.1.6
The ∞-category of mixed graded k-modules, being the ∞-category of quasi-coherent sheaves over a derived stack, is naturally stable and it is endowed with a symmetric closed monoidal structure.Viewing O BG a,k G m,k -comodules as graded k-modules endowed with a mixed differential, we can describe the internal tensor product, the internal mapping space and the unit for such monoidal structure as follows.
(1) Given two mixed graded k-modules M • and N • , the tensor product M • ⊗ ε-gr k N • is the mixed graded k-module whose p-th weight component is given by the formula with mixed differential defined on every summand by the formula given by the morphism whose r -th component is the sum of the morphism The enrichment of ε -Mod gr k over Mod k is then given by See also [8,Section 1.1].
For future reference, we provide also a simple result about fully dualizable objects (in the sense of [17,Section 4.6.1]).
given by the adjoint to the identity of M ∨ • .In virtue of [17,Lemma 4.6.1.6],extending such evaluation morphism to a datum of full dualizability is equivalent to showing that tensoring with the evaluation produces an equivalence of k-modules for any mixed graded k-modules N • and P • .This will be a consequence of the following Lemma.and for an arbitrary mixed graded k-module N • , the natural map of mixed graded k-modules is an equivalence.
Proof Let us recall how this map is defined.We have a chain of equivalences of mapping k-modules and so, tensoring the evaluation map To prove it is an equivalence, we check it weight-wise: the left hand side is described in weight p by the formula whereas the right hand side is described in weight p by the formula Map ε -gr With a change of indices i: = −h (hence h + p = p − i = j), since M −i is perfect in Mod k for any integer i, we can rewrite the latter as Finally, since the grading of M • is bounded above and below, we know that Map Mod k (M −i , k) can be non-zero only for finitely many indices, hence the product is actually a direct sum.Therefore, the map is an equivalence.
The splitting morphism By the known equivalence between G m,k -equivariant quasi-coherent sheaves on Spec(k) and graded k-modules, this reduces to the forgetful ∞-functor that sends a graded mixed k-module M • to the underlying graded k-module M • by forgetting the mixed structure.In particular, the forgetful ∞-functor oblv ε : ε -Mod gr k → Mod gr k , being a quasi-coherent pullback ∞-functor, preserves both tensor products and internal mapping spaces.Hence, any dualizable object in ε -Mod gr k , after forgetting the mixed structure, must become dualizable also in Mod gr k .So we just need to prove that any dualizable graded k-module M • is perfect in each weight and endowed with bounded grading.Indeed a map k(0 • corresponds to an element in the weight 0 part of the k-module In particular, a coevaluation morphism for M • in Mod gr k provides, after post-composing with the natural map and then projecting on the p-th direct summand, a coevaluation k → M p ⊗ k Map Mod k M p , k for any integer p.The fact that all the coherences of the definition of a coevaluation morphism are satisfied is a consequence of the fact that • is assumed to be a coevaluation itself: one can easily see it by checking the coherence diagram for the component in weight By the characterization of dualizable objects in Mod k , this shows that M p must be perfect for any integer p.
For the statement about the upper and lower bound of the grading, let us remark that an analogous argument to the one in the proof of Lemma 1.1.9provides always a map where Map gr k is the internal graded mapping k-module ∞-functor for the closed symmetric monoidal ∞-category Mod gr k .If M • is not bounded in both directions, in general it can fail to be an equivalence.For example, if M p 0 for all non-negative integers p, considering N • : = M • , then the weight 0 component of the above map is described by which of course can never be an equivalence without the boundness assumption.
Remark 1.1.12 The ∞-category ε -Mod gr k is equivalent, as a stable symmetric monoidal ∞-category, to the ∞-category of comodules on O BG a,k G m,k .We can alternatively consider the ∞-category of comodules on O 0 G a,k G m,k , that is the ∞category of comodules on the algebra of functions on the affine group stack The two theories are equivalent: in the latter case, the mixed differential ε is a morphism of degree −1 instead of degree 1 (i.e., the mixed structure is the datum of a map ε p : M p → M p−1 [1]).An explicit equivalence between the two comodule theories simply sends a comodule 13 Let k( p) denote the mixed graded k-module consisting of k sitting in pure weight p and homological degree 0. We have an adjunction where the left adjoint simply sends a k-module M to the mixed graded k-module consisting of M concentrated in weight 0, and the right adjoint |−| is the ∞-functor that sends a mixed graded k-module M • to the mapping k-module This right adjoint is called the realization ∞-functor: a strict model of |M • | is provided by the chain complex of k-modules endowed with the total differential, sum of the usual differential of chain complexes and the mixed differential ([8, Proposition 1.5.1]).Notation 1.1.15 In the remainder of this work, the ∞-functor − ⊗ k k(q) : Mod k → ε -Mod gr k which sends a k-module M to the mixed graded k-module M concentrated in weight q with trivial mixed differential shall be denoted simply as (−)(q).Construction 1.1.16 For our purposes, it will be convenient to introduce another realization ∞-functor that can keep track of the k-modules in positive weights of a mixed graded module M • .Let us recall ([8, Section 1.5]) that for all p ∈ Z we have that where the morphism k ] is the unique morphism corresponding to the unit 1 of k under the equivalence 1.1.17.
Definition 1.1.20Again, working with explicit models given by graded chain complexes and mixed differentials, the ∞-functor of Definition 1.1.19sends a mixed graded k-module again endowed with the total differential.There is a natural transformation of ∞functors which is induced by the map of pro-objects k(∞) → k(0) (the latter seen as a constant pro-object).Working with explicit models, we easily see that the natural transformation above is described, on a given mixed graded k-module M • , as the inclusion In particular, the map above is an equivalence whenever M • is trivial in all positive weights.1.1.23By pre-composing (−) 0 with the weight shift by q on the left ∞-endofunctor which informally sends a mixed graded k-module M • = M p p∈Z to the mixed graded k-module M((q)) • : = M p−q p∈Z , and by post-composing Free ε with the shift ∞-functor ((q)) we obtain for all q ∈ Z another adjunction which generalizes the adjunction 1.1.22to all weights.Let us remark that the description of this left adjoint is very explicit: indeed, the proof of [8, Proposition 1.3.8]together with the observations made in [8,Section 1.4.1]shows that such left adjoint simply sends a k-module M to the mixed graded k-module Free ε (M)((q)) consisting of M in weight q, M [1] in weight q − 1, and with mixed structure given by the natural equivalence Let us remark that, for any integer q, one has an adjoint pair which is an ambidextrous adjunction.The ∞-functor (−) q : ε -Mod so the existence of the adjoint Free ε ((q)) yields the following result.The assertion for limits is more subtle, since in general pullback ∞-functors do not preserve them.But in this case, we can use the discussion of 1.1.23:indeed, the composition preserves limits for all p's, since it admits an explicit left adjoint.The product is canonically equivalent to the forgetful ∞-functor oblv ε , and it commutes with limits since it is the product of ∞-functors which commute with limits.
Porism 1.1.26Since both ε -Mod gr k and Mod gr k are presentable ∞-categories, the forgetful ∞-functor oblv ε is both a left and right adjoint in virtue of the Adjoint Functor Theorem ( [13, Corollary 5.5.2.9]).Such adjoints L ε and R ε can be described with explicit models in the following way.
• The left adjoint L ε sends a graded k-module M • to the mixed graded k-module L ε M • defined in weight p by the formula 123 and whose BG a,k -action is described by the morphism given by the canonical equivalence M p M p [1][−1] and the zero map on M p+1 [1].
, and whose BG a,k -action is described by the morphism given by the canonical equivalence It is now clear that the ∞-functor Free ε ((q)) of 1.1.24is canonical equivalent to the composition 27 We shall provide one last, but very useful, construction that highlights how mixed graded k-modules are to be thought as graded k-modules with extra structure.Recall the right split extension of group stacks 1.1.10.The morphism BG a,k G m,k → G m,k induces a pullback ∞-functor at the level of quasi-coherent sheaves By functoriality, one has a natural equivalence id Mod gr In general, triv ε has to be understood as the ∞-functor that endows a graded k-module with the trivial mixed structure, i.e., the zero morphism between each weight.In particular, the unit for the adjunction triv ε NC w cannot be an equivalence.For example, if we consider the graded k-module consisting of k sitting in pure weight 0, then the formula provided in [21] for the weighted negative cyclic ∞-functor yields that for all p 0. However, the ∞-functor triv ε is fully faithful on those graded k-modules which are concentrated in a single weight: indeed, the right adjoint of the ∞-functor is the composition of NC w with the projection on the q-th weight component.In particular, the unit map for the adjunction triv ε • (−)(q) (−) q • NC w is an equivalence, and since (−)(q) is fully faithful for any integer q we have a chain of equivalences of mapping spaces

Naive and clever truncations
For any integer p, we can consider the full sub-∞-category ε -Mod gr, p k of ε -Mod gr k spanned by all those mixed graded k-modules whose q weight component is 0 whenever q > p.
The inclusion commutes with all limits and colimits, since they are computed weight-wise (Lemma 1.1.25).It follows that for all integers p the inclusion ι p , being a limit and colimit preserving ∞-functor between presentable ∞-categories admits both a left and a right adjoint, again in virtue of the Adjoint Functor Theorem.
is the clever truncation in weights p.
The ∞-functor of Definition 1 must be understood as the ∞-functor which sends a mixed graded k-module M • to the mixed graded k-module σ p M • whose q weight component is M p if q p, and 0 otherwise, with obviously induced mixed differential.This is different from the ∞-functor of Definition 2 which in turn is an analogue of the clever truncation of chain complexes in the context of mixed graded k-modules.

Remark 1.2.2
By the adjunction ι p σ p , for any mixed graded k-module M • we have a counit morphism which includes its part concentrated in weights p in itself.
For any q p, we have that We shall commit a slight abuse of notation, and identify σ q M • with its inclusion ι q σ q M • in ε -Mod gr k ; in particular, we shall write the inclusion morphisms above simply as Using the dual properties of the adjunction θ p ι p , we easily obtain dual properties for the clever truncation ∞-functor., which is now the full sub-∞-category consisting of those mixed graded k-modules which are trivial in weights q < p. Now, the inclusion preserves all limits and colimits and hence admits both left and right adjoints.
is the naive truncation in weights p.

The ∞-functor
is the naive truncation in weights p.
Again, the notation and the nomenclature suggest the behavior of the ∞-functors on a mixed graded k-module: the one in Definition 1 simply kills M q for any q < p, while the one in Definition 2 changes the underlying graded k-module even in weight q p, since it involves some sort of totalization.

The t-structure on mixed graded modules
The main result of this section is the following Theorem, which provides a t-structure on mixed graded k-modules describing explicitly both connective and coconnective objects.
Theorem 1.3.1 Let ε -Mod gr k 0 be the full ∞-subcategory of ε -Mod gr k spanned by those mixed graded k-modules M • such that, for any integer q, the k-module M q is (−q)-connective.Dually, let ε -Mod gr k 0 be the full ∞-subcategory spanned by those mixed graded k-modules M • such that, for all q ∈ Z, the k-module M q is (−q)-coconnective.These sub-∞-categories determine an accessible t-structure on the stable ∞-category of mixed graded k-modules, which we call the mixed graded Postnikov t-structure.When k is discrete, the heart of such t-structure is equivalent to the classical abelian 1-category of chain complexes of k-modules dgMod k .

Remark 1.3.2
The name for such t-structure is borrowed from [1], where the standard t-structure on modules over a connective E 1 -ring of [17,Proposition 7.1.1.13]is referred to as the Postnikov t-structure.
Proof of Theorem 1.3.1 Consider the sub-∞-category ε -Mod gr k 0 described in the statement of Theorem 1.3.1.It is a sub-∞-category of ε -Mod gr k which is closed under all colimits and extensions, since they are computed weight-wise (Lemma 1.1.25)and so it suffices to prove the claim in Mod k , where q-connective objects are stable under colimits and extensions for any q (this is [17, Thus, the sub-∞-category ε -Mod where The interesting part of the statement of the Theorem is the one that characterizes the coconnective objects: we have to prove that a mixed graded k-module is coconnective N • precisely if N q is (−q)-coconnective for any integer q. [17, Remark 1. our claim is equivalent to proving that for N • such that N q is (−q − 1)-coconnective for any integer q.We shall need the following two Lemmas.Lemma 1.3.5 Let N • such that N q is (−q − 1)-coconnective for any integer q.Then, N • is a (−1)-coconnective object for the mixed graded Postnikov t-structure.

Proof
We recall that the enrichment in k-modules of mixed graded k-modules is given by the fiber for the mixed structure morphism Map ε -gr The enrichment in spaces is now given by applying the truncation ∞-functor τ 0 to such k-module ( [17, Construction 1.3.1.13]).So, let M • be such that M q is (−q)connective for all q's: we have a fiber sequence Writing explicitly the second and third terms of such fiber sequence, using 1.1.6,we get The second term of such fiber sequence is (at least) (−1)-coconnective, because it is the product of (−1)-coconnective k-modules (being M q (−q)-connective and N q (−q − 1)-coconnective for all q's).Analogously, also the third term is (at least) (−2)-coconnective, hence in particular (−1)-coconnective.Being (−1)-coconnective objects stable under all limits, we get that Map ε -Mod gr k (M • , N • ) is now (−1)coconnective, hence its truncation yields a contractible space.Lemma 1.3.6Suppose that N • is (−1)-coconnective for the mixed graded Postnikov t-structure.Then N q is (−q − 1)-coconnective as a k-module.
Proof Without loss of generality, by shifting weights, we can reduce ourselves to the case that N • is such that, for some integer n 0, there exists a non-trivial homotopy group π n N 0 0. Let k(0) be the mixed graded k-module with trivial mixed structure in pure weight 0. Then by the adjunction 1.1.14,we have equivalences of k-modules If the realization of N • is not (−1)-truncated, then N • cannot be (−1)-coconnective for the mixed graded Postnikov t-structure, since k(0) is obviously connective.So, let us assume that |N • | is (−1)-coconnective as a k-module.Using the naive truncation ∞-functor in degree −1 of Definition 1, by adjunction we have a cofiber sequence Applying the realization ∞-functor, that being a right adjoint between stable ∞categories preserves cofiber sequences, we have another cofiber sequence of the form Since limits and colimits are computed weight-wise in ε -Mod gr k , by the description of the realization ∞-functor provided in 1.1.13,we easily have that |cofib| N 0 , and that moreover In particular, σ −1 N • cannot be trivial, because this would contradict that π n N 0 0 for some n 0. Therefore, considering the k-module k [1](−1), we have is again connective for the mixed graded Postnikov t-structure, and now we have a non-trivial map towards N • , which is given by any non-trivial homology class in π n N 0 .
Lemma 1.3.5 and 1.3.6 together imply that the class of (−1)-coconnective objects for the mixed graded Postnikov t-structure and the class of mixed graded k-modules which are (−q − 1)-coconnective in each weight q coincide, and this concludes the proof of the main statement.The claim about the heart of such t-structure can be proved directly.The idea is that a mixed graded k-module lying in the heart of the t-structure is forced to be a mixed graded k-module whose weight p component is a discrete k-module concentrated in homological degree − p, and the differential for this chain complex is given by the mixed differential.Yet, we shall see this claim as a corollary of Theorem 2.3.1.

Porism 1.3.7
The proof of Theorem 1.3.1 does not rely on the assumption that k is a field.We can always define the ∞-category of mixed graded R-modules, for R a Qalgebra in E ∞ -rings, as the ∞-category of quasi-coherent sheaves on the affine group stack BG a,R G m,R .Let Sp gr : = p∈Z Sp be the ∞-category of graded spectra, and let Sp gr 0 be its full sub-∞-category spanned by graded spectra whose q-th weight component is (−q)-connective for all integers q.In virtue of [18,Theorem 4 is again presentable, closed under all colimits and extensions, and so it determines the connective part for an accessible t-structure over ε -Mod gr R .Even for an arbitrary derived scheme X defined over a derived Q-algebra k, one can define ( [24, Section 1.2.1]) the ∞-category of mixed graded modules over X as the ∞-category of sheaves with values in ε -Mod gr k for the Zariski topology on X : Setting ε -Mod gr X 0 to be the ∞-category of those sheaves of mixed graded kmodules over X which are connective as mixed graded S-modules for any S-point Spec(S) → X (the fact that this yields, indeed, a t-structure is essentially [9, Chapter 3, Section 1.5.1]).The only property of a field of characteristic 0 that we use in the proof of Theorem 1.3.1 is the fact that it is connective: indeed, whenever R is connective, then the coconnective part for such t-structure can be characterized as This fact is not true in general if R is not connective (see also [16,Warning 3.5.9]).The analogous statement on the coconnective part for the t-structure on mixed graded modules over a non-affine derived scheme X case can be recovered under the hypothesis that the sheaf of E ∞ -rings is connective (in the sense of [14, Definition 1.20]), using [15, Proposition 2.1.3],or in the non-connective case when X is at least an Artin stack (here, we use [9, Chapter 3, Proposition 1.5.4]).
The characterization of the connective and coconnective objects of the t-structure of Theorem 1.3.1 allows us to describe pretty neatly how the connective cover and the coconnective cover ∞-functors behave on a mixed graded k-module.
Proof Since colimits are computed weight-wise, taking the k-module in weight q is an exact ∞-functor from mixed graded k-modules to k-modules for any integer q.So, considering the canonical fiber sequence and taking the component in weight q we get another fiber sequence By suitably shifting, it follows that this fiber sequence is the essentially unique fiber sequence of k-modules that extends M q with a (−q)-connective part on the left and a (−q − 1)-coconnective part on the right, hence τ ε -gr 0 M • q τ −q M q and τ ε -gr

Relationship with filtered modules
In this section, we shall state the main link between the ∞-categories of filtered kmodules and mixed graded k-modules, and their respective t-structures described in Theorems 2.1.14and 1.3.1.The main result of this section is Theorem 2.3.1, which states that complete (in the homotopical sense) filtrations on k-modules arise uniquely from a mixed structure on the associated graded of the filtration, and moreover that this assignment is t-exact.This can be seen as a slight improvement on [24, Proposition 1.3.1]and [6, Proposition 2.27], which exhibited a fully faithful embedding ε -Mod gr k → Mod fil k and characterized its essential image, without considering any t-structure on those ∞-categories.

Recollection on filtered modules
We start by reviewing some important concepts concerning filtered k-module, following [4,11].where M p is a k-module for any integer p.We shall denote the object associated to such tower with M • , while the k-module M p will be the p weight component of M • .Consistently, a natural transformation between two filtered k-modules M • → N • (that is, a sequence of morphisms f p : M p → N p for any integer p with obvious compatibility with the transition morphisms M p → M p−1 and N p → N p−1 ) shall be denoted by f • .

2.1.3
The selection of the p weight component of a filtered k-module M • is functorial, i.e. there exists an ∞-functor (2.1.7) The complete filtered k-modules are gathered in a full sub-∞-category of Mod fil k , which we shall denote by Mod fil k .Definition 2.1.9With the above notations, the p weight part M p of a filtered k-module M • must be thought as the p-th part of an exhaustive filtration on the k-module M −∞ , while the k-module M ∞ determines whether such filtration is separated Hausdorff in the homotopical setting (which is not always equivalent to classical one-see [11,Remark 2.13] for a classically separated filtration which is not complete in our sense).The p-th graded piece ∞-functor is composition of the previous ∞-functor with the cofiber ∞-functor

2.1.11
The ∞-category Mod fil k is naturally a closed symmetric monoidal ∞-category ( [11, §2.23]); such monoidal structure is inherited in some sense by the closed symmetric monoidal structure on Mod k via the Day convolution.We briefly recall some important properties and useful explicit constructions.
(1) Given two filtered k-modules M • and N • , the tensor product M • ⊗ fil k N • is the filtered k-module whose p-th weight component is given by the formula (2) The unit for ⊗ fil k is the sequence k 0 given by k in non positive weights and by 0 otherwise, with only identities and trivial morphisms, i.e. by the sequence Such completion ∞-functor does not alter the graded pieces of M • .
(3) Both Mod k and Mod fil k have all limits and colimits.On the former, both {(−) p } p∈Z and {Gr p } p∈Z commute with all limits and colimits; on the latter, {Gr p } p∈Z commute with all limits and colimits.( 4 Lemma 2.1.12allows us to say something more about the sub-∞-category of complete filtered k-modules: namely, it is stable.

Since limits and colimits in Mod fil
k are computed weight-wise, we have that Since in a stable ∞-category finite colimits commute with all limits ( [17, Proposition We now state (without proof) this and well known result on the t-structure on the (homotopy category of the) ∞-category of filtered k-modules.Note that this theorem does not need the assumption that k is a classical ring in characteristic 0. Remark 2.1.15 The t-structure of Theorem 2.1.14 is induced by a t-structure on graded k-modules in the following sense.The ∞-category Mod gr k is endowed with a tstructure described as follows: the connective objects are those graded k-modules M • such that M p is (− p)-connective (for the Postnikov t-structure on Mod k ) for all p's, and dually the coconnective objects are those graded k-modules M • such that M p is (− p)-coconnective for all p's.Such t-structure can be lifted via the ∞-functor to a t-structure on Mod fil k that makes Gr • a t-exact ∞-functor between stable ∞categories endowed with a t-structure.This is precisely the Beilinson t-structure.

Left completeness of t-structures
Mixed graded k-modules are not only fully faithfully embedded in filtered k-modules, but such inclusion is universal in some precise sense.Namely, the ∞-category of mixed graded k-modules with the t-structure of Theorem 1.3.1 is the left completion of the ∞-category of filtered k-modules with respect to the Beilinson t-structure of Theorem 2.1.14.In order to precisely formalize and prove this assertion, we need to recall some important concepts concerning t-structures and left completions.

2.2.1
Let C be a stable ∞-category endowed with a t-structure C 0 , C 0 .For all integers n, we have the n-connective and n-coconnective cover ∞-functors τ n : C → C n and τ n : C → C n .In particular, we have a tower . . .Obviously ε -Mod gr k , being both complete and cocomplete, admits all countable products.Since all limits and colimits are computed weight-wise, a product of countably many mixed graded k-modules M α • α∈N , all of which are connective for the mixed graded Postnikov t-structure, is again connective.Indeed, in each weight we have an equivalence of k-modules where all the M α q 's are now (−q)-connective, hence the statement follows from the fact that Mod k is left complete by [17,Proposition 7.1.1.13].

On the converse, recall that the ∞-category Mod fil
k endowed with the Beilinson t-structure of Theorem 2.1.14 is not left complete ( [4, Section 5]).In fact, the objects lying in Mod fil k ∞ are those filtered k-modules M • such that Gr n M • vanishes for all integers n, i.e. filtered k-modules corresponding to essentially constant diagrams Z −→ Mod k .In particular, for any non trivial k-module M the constant diagram on it is ∞-connective without being 0. Yet, we can characterize what the completion of the Beilinson t-structure on filtered k-modules looks like: this is precisely Mod fil k (which justifies, a posteriori, the choice of notation for such sub-∞-category).While this seems to be known to experts, to our knowledge there is no proof of this fact available in the existing literature.k is endowed with a tstructure which is provided by the restriction of the Beilinson t-structure to complete filtered k-modules.Moreover, the completion ∞-functor is a t-exact ∞-functor that naturally identifies Mod fil k with the left completion of the Beilinson t-structure on all filtered k-modules.
For the sake of clarity, we shall split the proof of Proposition 2.2.8 in several lemmas.Lemma 2.2.9The Beilinson t-structure of Theorem 2.1.14on filtered k-modules restricts to a t-structure on the full sub-∞-category of complete filtered k-modules.
In order to prove Lemma 2.2.9, we shall need the following important technical result.

Lemma 2.2.10 If M
• is a filtered k-module which is eventually coconnective (that is, n-coconnective for some n) for the Beilinson t-structure of Theorem 2.1.14,then M • is complete.
Proof Since for any integer p, the k-module M p is (n − p)-coconnective for the usual t-structure on k-modules by assumption, and coconnective objects are stable under all limits, we just need to observe that M ∞ : = lim q∈Z M q must be n − q-coconnective for each integer q, i.e., it must belong to the ∞-category But the Postnikov t-structure on k-modules is left complete, hence M ∞ must be trivial because of Proposition 2.2.6.

Proof of Lemma 2.2.9 Since Mod fil
k is a stable full sub-∞-category of Mod fil k and its inclusion admits a right adjoint, it is closed under all limits and finite colimits, and in particular under all loops and suspensions.Moreover, if M • is a complete filtered k-module which is connective for the Beilinson t-structure, it is obviously left orthogonal to any complete filtered k-module which is (−1)-connective for the Beilinson t-structure, since Mod fil k is a full sub-∞-category of Mod fil k .So, we only need to check that if M is endowed with a complete filtration, then both τ fil n M • and τ fil n M • lie in Mod fil k , where τ fil n and τ fil n denote the n-connective and n-coconnective cover ∞functor for the Beilinson t-structure on Mod fil k , respectively.This is clear for τ fil n , since n-coconnective objects are always complete in virtue of Lemma 2.2.10, and this implies that the same holds also for τ n : indeed, consider the left adjoint to the inclusion Mod fil k ⊆ Mod fil k of Part (2) of Lemma 2.1.12.For any filtered module M • the unit map of the adjunction yields a map of fiber sequences (given by the unit map of the adjunction) If M • is complete, then both the second and third vertical arrows are equivalences, and so the first must be an equivalence as well.It follows that both the connective and coconnective cover ∞-functors restrict naturally to the stable full sub-∞-category Mod fil k , which implies our assertion.
for any integer n.
• For any m n, the map Employing this canonical model, the canonical equivalence Mod k lies in Fun Z, Mod fil k if and only if F n is (−n)-coconnective for all n.In particular, using again Lemma 2.2.10, it follows that at least as mere stable ∞-categories, Mod fil k Fun Z , Mod fil k .The last thing we need to prove is that this equivalence is t-exact.Recall ( [17, Proposition 1.2.1.17.(2)]) that the natural t-structure on Fun Z , Mod fil k is defined as follows: the connective objects are those ∞-functors F : Z → Mod fil k that factor through Mod fil k 0 , while coconnective objects are those ∞-functors that factor through Mod fil k 0 ; an analogous statement holds for Fun Z , Mod fil k .But, being complete with respect to its t-structure, the canonical ∞-functor

Main theorem
We can now state the main result of this section, and of this paper as well.Even if the ∞-functors that exhibit ε -Mod gr k as the left of Mod fil k are precisely the same as the ones considered in [24], for the sake of clarity we recall all the details of their construction in this subsection.

2.3.2
Consider the naive truncation ∞-functor of mixed graded k-modules of Definition 1. Thanks to Remark 1.2.2, for any mixed graded k-module M • and for any triple of integers r q p we have natural morphisms in ε -Mod This means that gathering all the naive truncations ∞-functors {σ p } p∈Z we get an ∞-functor Using the isomorphism Z ∼ = Z given by changing signs, we can write the ∞functor above as σ − p M • is bounded above (i.e., its weight components are 0 for all q > − p), so its Tate realization is equivalent to Either way, in terms of explicit models of chain complexes, this means that where the right hand side is endowed with the total differential.transition maps, in terms of the above equivalence, are simply given by the obvious inclusions In the following, we shall often denote the object (M • ) fil p with the suggestive notation q p M −q [−2q], leaving implicit the fact that this is not actually a product of k-modules.
We prove some important properties of the associated filtered ∞-functor.Proposition 2.3.6 The associated filtered ∞-functor of Definition 2.3.3 is a t-exact ∞-functor between stable ∞-categories with t-structure.Proof Remark 2.3.4 allows us to describe neatly the graded pieces of the filtered k-module M fil • for any mixed graded k-module M • .In fact, the cofiber sequence • can be now represented by explicit models as where again the product has to be thought as endowed with the total differential, and by direct inspection of the long exact sequence of homotopy groups we get that Gr p M fil On the converse, let M • be an n-coconnective mixed graded k-module.Then M fil • p is equivalent to a product of k-modules sitting in (Mod k ) n− p , and since n-coconnective objects are stable under limits this yields that M fil • p is (n − p)coconnective for any integer n.So (−) fil preserves both connective and coconnective objects, hence it is t-exact.
To prove Theorem 2. Proof Being ε -Mod gr k and Mod fil k two ∞-categories which are both presentable and accessible, it is sufficient to show that (−) fil commutes with all limits and with κfiltered colimits for κ some regular cardinal.Since Mod fil k is an ∞category of functors, hence limits and colimits are computed weight-wise, it suffices to show that for any p the ∞-functor σ − p (−) t commutes with limits and is accessible, i.e., it commutes with κ-filtered colimits for some regular cardinal κ.Since such ∞-functor is the composition of two ∞-functors, one of which (namely, σ − p ) commutes with limits and is accessible, being a left adjoint, we are left to prove that |−| t commutes with limits and is accessible as well.
• The ∞-functor |−| t commutes with all limits: this can be seen via direct computations from the description of |−| t and by recalling that limits distribute over limits.
• The Tate realization ∞-functor is given explicitly by a countable product, so it commutes with κ 2 -filtered colimits for some regular cardinal κ 2 bigger than ℵ 0 .
The claim now follows, once again, from the Adjoint Functor Theorem.We want to study some properties of the ∞-functor of Definition 2.3.8 as well.One of its key features is described in the following Proposition.
On the other hand, by the adjunction oblv ε R ε one has By the fully faithfulness of the Yoneda embedding, it is enough to check that for any graded k-module N • .This is seen by direct inspection: in fact, (R ε N • ) fil is the filtered k-module whose p-th component is obtained by the Tate realization of the naive truncation σ induces the zero map on the Tate realizations, and so we get exactly the filtered k-module R fil N • we described above, which completes the proof.

Remark 2.3.11
Working with explicit models, the mixed differential in the mixed graded k-module (M • ) ε -gr is given by the boundary morphism in the cofiber sequence after a suitable (i.e., correctly functorial) choice of the cofibers in each weight.
The above characterization of the weight components of the associated mixed graded ∞-functor leads immediately to two important consequences.Proof Let us recall that the ∞-functor (−) fil is lax monoidal: it can be checked by straightforward computation, by using the fact that in each piece it is given by the mapping space out of a cocommutative coalgebra object (namely, k(∞)).Being the left of a lax monoidal ∞-functor, (−) ε -gr naturally possesses an oplax monoidal structure, and so given two filtered k-modules M • and N • we have a natural map Since equivalences of mixed graded k-modules are detected by the underlying graded k-modules and forgetting the mixed structure is a strongly monoidal ∞-functor, it suffices to show that the underlying graded map is an equivalence.Now, Proposition 2.3.9 tells us that the graded k-module on the left is described in its p weight component by the formula But now Lemma 2.1.12.5 tells that the above object is equivalent to which is exactly the p weight component of the tensor product of (M • ) gr ⊗ gr k (N • ) gr .So, the natural map above is indeed an equivalence.

Corollary 2.3.13 The
Proof Let M • be an n-connective filtered k-module.Then Gr p M • is an (n − p)connective k-module for any integer p, therefore On the converse, if M • is an n-coconnective filtered k-module, then M p is an (n− p)coconnective k-module for any p.This implies that Gr p M • is an (n − p)-coconnective k-module for any p as well: in fact, Gr p M • is the cofiber of M p+1 −→ M p .Since M • is n-coconnective, M p+1 and M p are respectively (n − p − 1)-coconnective and (n − p)-coconnective k-modules.By inspecting the long exact sequence induced on the homotopy groups by the cofiber sequence is (n − p)-coconnective for all p's.
The following two claims are the last ingredients we need to prove Theorem 2.  Proof Let M • be a filtered k-module which is n-coconnective for the Beilinson tstructure for some integer n.We briefly study what (N • ) ε -gr fil is: its p-th part is the k-module given by the Tate realization But we have that and since the shift ∞-functors commute with arbitrary products we have that Gr q N • 123 The transition maps are just given by inclusions.Let Gr q N • ⎞ ⎠ denote the fiber of the p-th component of η.Then we have the following diagram, where every row and every column is a fiber sequence.
The map on the bottom right is an equivalence, and this forces cofib F p+1 → F p to be zero.This implies that F p+1 → F p is an equivalence; by induction, we can conclude that F p is the fiber of Gr q N • for any integer m.So, the fact that η m is an equivalence for any m is equivalent to proving that F p is 0, but this is a consequence of Lemma 2.2.10.Indeed, if F p was not 0, we would have a non-trivial k-module with a non-trivial map F p → N m for all m's.Such map would be forced to factor through N ∞ ; in particular, it would yield that N ∞ is non-trivial, hence that N • is not complete.But this contradicts the (eventually) coconnectivity assumption on N • .Remark 2.3.16 In the proof of Proposition 2.3.15,we did not need to restrict ourselves to eventually coconnective objects.In fact, the key property of eventually coconnective filtered k-modules that we used was their completeness (Lemma 2.2.10): the unit morphism of the adjunction (−) gr ε (−) fil is an equivalence on all complete filtered k-modules, which are not necessarily eventually coconnective (take for example the filtered k-module given by the 0 sequence everywhere, except for k [1] in weight 0).The fiber F p : = fib N p → (N • ) ε -gr fil p of the proof of Proposition 2.3.15is, in fact, N ∞ itself: the unit η fails to be an equivalence precisely on non-complete filtered k-modules.gr ε are both t-exact ∞-functors which induce equivalences on coconnective objects, it follows that the hearts of the two t-structures are naturally equivalent and both yield, when k is a discrete ring, the usual 1-category of chain complexes of k-modules: this is the last statement that we left unproven in Theorem 1.3.1.

Porism 2.3.19
The strongly monoidal structure of the ∞-functor (−) ε -gr proved in Corollary 2.3.12,together with the description of the monoidal structure on complete filtered k-modules provided by Lemma 2.1.12.4,implies that the equivalence of Theorem 2. If both M • and N • are endowed with a filtration bounded in non-positive weights, the above formula is 0 for any positive integer n.Hence the limit on the tower of k-modules corresponding to M • ⊗ fil k N • is zero, hence the filtered tensor product in this case preserves completeness.
The unit for ⊗ ε-gr k is the mixed graded k-module k(0), consisting of k sitting in pure weight 0 with trivial mixed structure.(3) Given two mixed graded k-modules M • and N • , the internal mapping space Map ε -gr k (M • , N • ) is the mixed graded k-module whose p-th weight component is given by the formula Map ε -gr

Notation 1 . 1 . 7 Proposition 1 . 1 . 8 .
In the following, we shall denote by ε -Perf gr,− k the full sub-∞category of ε -Mod gr k spanned by those mixed graded k-modules M • which are perfect in each weight and such that M p 0 for all p 0. Dually, we shall denote by ε -Perf gr,+ k the full sub-∞-category of ε -Mod gr k spanned by those mixed graded kmodules M • which are perfect in each weight and such that M p 0 for all p 0. In a similar fashion, we shall denote by ε-Perf gr, p k (respectively, ε-Perf gr, q k) the full sub-∞-category of ε -Mod gr k spanned by those mixed graded k-modules M • which are perfect in each weight and such that M n 0 for all n < p (respectively, for all n > q).Let us remark that we have inclusions of ∞-categories p and q.The full sub-∞-category of fully dualizable objects of mixed graded k-modules coincides with the ∞-category Proof Let us denote by M ∨• the mixed graded k-linear dual Map ε -gr

Construction 1 . 1 . 21
For all p ∈ Z we have a p weight part ∞-functor (−) p : ε -Mod gr k −→ Mod k described (model independently) in the following way.Let us recall the forgetful ∞functor oblv ε : ε -Mod gr k → Mod gr k described in 1.1.11as the quasi-coherent pullback ∞-functor induced by atlas BG m,k → B BG a,k G m,k .Since we have an equivalence of ∞-categories Mod gr k p Mod k , one can project onto the p-th coordinate: this composition yields the desired ∞functor.The ∞-functor (−) 0 has a left adjoint Free ε : Mod k −→ ε -Mod gr k (1.1.22)(the free mixed graded k-module construction, see [8, 1.4.1]).

Lemma 1 . 1 .
25 Limits and colimits in ε -Mod gr k are computed weight-wise.Proof Let us recall that the forgetful functor oblv ε : ε -Mod gr k −→ Mod gr k corresponds geometrically to the pullback functor 1.1.11.Every pullback ∞-functor commutes with colimits ( [15, Proposition 2.7.17]) and thus colimits in ε -Mod gr k are computed as in Mod gr k .But Mod gr k q∈Z Mod k , and colimits on the right hand side are computed weight-wise.

Warning 1 . 1 .
29 The ∞-functor triv ε : Mod gr k → ε -Mod gr k commutes with all limits and colimits since the forgetful ∞-functor oblv ε creates them, and oblv ε • triv ε id Mod gr k .Moreover, it is a straight-forward consequence of the discussion in 1.1.6that triv ε is also strongly monoidal.It is, however, not fully faithful: such pullback ∞functor is in fact the left adjoint to the push-forward ∞-functor ε -Mod gr k → Mod gr k which agrees with the weighted negative cyclic ∞-functor NC w : ε -Mod gr k → Mod gr k of [21, Remark 1.5].Unraveling all definitions, this ∞-functor sends any mixed graded k-module M • to the graded k-module NC w (M • ) described in each weight by the formula NC w (M • ) p : = |M • (( p))| .

gr k 0 0 ( 1 . 3 . 3 )
defines a t-structure such that ε -Mod gr k 0 is precisely the ∞-category of connective objects, thanks to [17, Proposition 1.4.4.11].In detail: under these hypotheses, the ∞-category ε -Mod gr k 0 is a colocalization of ε -Mod gr k , hence the Adjoint Functor Theorem provides an ∞-functor τ ε -gr 0 : ε -Mod gr k −→ ε -Mod gr k right adjoint to the obvious inclusion.Thus, the (−1)-coconnective objects are defined to be those mixed graded k-modules N • such that τ ε -gr 0 N • 0 in ε -Mod gr k 0 , and so each mixed graded k-module is sandwiched in a (∞-functorial) cofiber sequence τ ε -gr 0 2.1.3]assures us that (−1)coconnective objects are uniquely determined by the property of being right orthogonal to any connective object.Since the enrichment in spaces of a k-linear ∞-category C is given by the truncation in degrees 0 of the mapping k-module, this means that the k-module Map ε -Mod gr k (M • , N • ) has to be (−1)-coconnective for any M • connective.Since in ε -Mod gr k colimits are computed weight-wise, and by definition .1], we have an equivalence of ∞-categories Sp gr QCoh BG m,S , where G m,S : = Spec(S[Z]) is the flat multiplicative group (spectral) scheme.Hence, by the abstract nonsense recollected at the beginning of the proof of Theorem 1.3.1, the ∞-category ε -Mod gr R 0 : = ε -Mod gr R × Sp gr Sp gr 0

Definition 2 . 1 . 10
Given a filtered k-module M • , consider the inclusion of 1 { p + 1 → p} in Z .This provides an ∞-functor Mod fil k : = Fun Z , Mod k → Fun 1 , Mod k which selects the morphism M p+1 → M p in a filtered k-module M • .

( 1 )
where the first copy of k sits in weight 0. (3) Given two filtered k-modules M • and N • , the internal mapping space Map fil k (M • , N • ) is the filtered k-module whose p-th weight component is given by the end formula Map fil k (M • , N • ) p : = q∈Z Map k M q , N p+q .Lemma 2.1.12([4, Lemma 5.2]) The collection of ∞-functors {Gr p } p∈Z and (−) ∞ is jointly conservative on Mod fil k .On the sub-∞-category Mod fil k , the ∞-functors {Gr p } p∈Z are already jointly conservative.(2) The inclusion Mod fil k ⊆ Mod fil k admits a left adjoint (−) : Mod fil k −→ Mod fil k , which sends M • to its completion M • described in its p weight component by ) There exists a (unique) closed symmetric monoidal structure on Mod fil k compatible with the one on Mod fil k via the completion ∞-functor.(5) For any M • and N • in Mod fil k or Mod fil k , we have an equivalence

Proposition 2 . 1 . 13
The full sub-∞-category Mod fil k of Mod fil k is stable.Proof The ∞-category Mod fil k has a zero object, since the trivial filtration is clearly complete.Moreover, since their inclusion in all filtered k-modules admits a left adjoint, it follows that all limits in Mod fil k are computed as in Mod fil k .So, we only need to prove that given a diagram of complete filtered k-modules of the form

Theorem 2 . 1 . 14 (
Beilinson t-structure, [2, Appendix A] and [4, Theorem 5.4]) Let Mod fil k 0 be the full ∞-subcategory of Mod fil k spanned by those filtered modules such that Gr p M • is (− p)-connective for all integers p. Dually, let Mod fil k 0 be the full ∞subcategory spanned by those filtered modules such that M p is (− p)-coconnective for all integers p. Then these two ∞-subcategories define a t-structure on Mod fil k whose heart Mod fil k ♥ is equivalent, as an abelian 1-category, to the usual 1-category of chain complexes of k-modules dgMod k when k is a discrete ring.

Lemma 2 . 2 . 11
The restriction of the Beilinson t-structure to the ∞-category of complete filtered k-modules is a left complete t-structure.Proof Since Modfil k is closed under all limits existing in Mod fil k , the hypotheses of the left completeness criterion of Proposition 2.2.6 are satisfied, so we can simply check what the ∞-connective objects of Mod fil k are: they are ∞-connective objects of Mod fil k (hence, constant filtrations) which are also complete as filtered k-modules.But it is immediate to see that the only constant filtration which is complete is the constant filtration on the trivial k-module 0. Thus, Mod fil k is left complete.Lemma 2.2.12The left completion ∞-functor (−) : Mod fil k −→ Mod fil k , identifies Mod fil k , endowed with the restriction of the Beilinson t-structure, with the left completion of the Beilinson t-structure on Mod fil k .Proof In virtue of Lemma 2.2.11, the restriction of the Beilinson t-structure on complete filtered k-modules is a left complete t-structure.This means, in virtue of [17, Proposition 1.2.1.17],that the natural ∞-functor from the ∞-category Mod fil k towards its left completion is an equivalence.As already described in Proposition 2.2.4,we can identify the left completion of the Beilinson t-structure on Mod fil k with the full sub-∞-category Fun Z , Mod fil k ⊆ Fun Z , Mod fil k spanned by those ∞-functors

k
simply sends an object M • to the tower of its truncations τ fil −n M • n∈Z , while its inverse takes the limit of a functor over Z .So, we have a Fun Z , Mod fil k where the vertical arrow on the right is simply the post-composition with the natural inclusion.Since the coconnective truncation ∞-functors for filtered modules and complete filtered modules are the same, this diagram commutes; moreover, the functor Fun Z , Mod fil k −→ Fun Z , Mod fil k is an equivalence as well, because a functor F : Z → Mod fil

→Fun
fil k is a t-exact equivalence, and Fun Z , Mod fil k −→ Fun Z , Mod fil k is again t-exact, because by definition Mod fil Fun Z , Mod fil k sends connective objects to connective objects, and coconnective objects to coconnective objects.Now, since every complete filtered k-module is canonically equivalent to the limit over the tower of its coconnective truncations, it is clear that the composition Mod fil k −→ Fun Z , Mod fil k Fun Z , Mod fil k −→ Mod fil k which is canonically equivalent to the inclusion Mod fil k ⊆ Mod fil k .Hence (−) : Mod fil k → Mod fil k does exhibit the ∞-category of complete filtered k-modules as the left completion of the Beilinson t-structure on all filtered k-modules.

Theorem 2 . 3 . 1
Let Mod fil k denote the full sub-∞-category of Mod fil k spanned by kmodules with complete filtration.There exists a fully faithful embedding (−) fil : ε -Mod gr k −→ Mod fil k which is moreover t-exact with respect to both t-structures, such that the following diagram of ∞-functors commutes.fil Moreover, such embedding admits a t-exact left adjoint (−) ε -gr : Mod fil k −→ ε -Mod gr k which exhibits the ∞-category ε -Mod gr k , endowed with the mixed graded Postnikov tstructure of Theorem 1.3.1, as the left completion of Mod fil k with respect to the Beilinson t-structure of Theorem 2.1.14.

Definition 2 . 3 . 3 Remark 2 . 3 . 4
σ with the Tate realization ∞-functor of Definition 1.1.19,we land in the ∞-category Fun Z , Mod k = :Mod fil k .The ∞-functor (−) fil : ε -Mod gr k σ −→ Fun Z , ε -Mod gr k |−| t •− − −−− → Mod fil k is the associated filtered ∞-functor.Given a mixed graded k-module M • , its associated filtered k-module is given in weight p by the Tate realization of its truncation in weights − p, i.e. σ − p M • t and all the morphisms are the inclusions exhibited in 2.3.2.Let us study in greater detail the k-module σ − p M • t : for any p, this mixed graded k-module

•M
− p [−2 p].So, let us assume that M • was an n-connective mixed graded k-module for the t-structure of Theorem 1.3.1:then, for any integer p, M p was an (n − p)-connective k-module.This implies that Gr p M fil • M − p [−2 p] is an (n− p)-connective k-module as well, so (−) fil preserves connective objects.

Proposition 2 . 3 . 7
3.1, we have to show that ε -Mod gr k is a localization of Mod fil k , i.e. the ∞-functor (−) fil is a fully faithful right adjoint whose essential image consists of complete objects.The ∞-functor (−) fil : ε -Mod gr k → Mod fil k admits a left adjoint.

Proposition 2 . 3 . 9
Given a filtered k-module M • , the p weight component of its associated mixed graded k-module (M • ) ε -gr is naturally equivalent to Gr − p M • [−2 p].Proof Let (−) gr : Mod fil k −→ Mod gr k denote the ∞-functor obtained by the composition of (−) ε -gr with the forgetful ∞functor oblv ε : ε -Mod gr the ∞-functor obtained by patching the ∞-functors Gr − p (−)[−2 p] : Mod fil k −→ Mod k .In particular, the p-th component of this ∞-functor is Gr − p (−) [−2 p].We will prove that for any filtered k-module M • , there is a canonical equivalence of k-modules Gr • M • (M • ) gr .We first observe that Gr • : Mod fil k → Mod gr k is the left adjoint to the ∞-functor R fil : Mod gr k −→ Mod fil k which sends a graded k-module N • to the filtered k-module p-th component is N − p [−2 p] and whose transition maps are all zero maps (this is essentially [11, Lemma 3.30], after an homological shift by −2 p and the equivalence Mod gr k Mod gr k which swaps the graded parts in positive and negative degrees).In the same way, in virtue of Porism 1.1.26the ∞-functor which forgets the mixed structure is the left adjoint to the ∞-functor R ε : Mod gr k −→ ε -Mod gr k .Now let M • denote a filtered k-module, and N • denote a graded k-module.By the adjunction Gr • R fil , we have the following equivalences of mapping spaces Map Mod gr k

2. 3 . 17
All previous propositions, lemmas and remarks imply together Theorem 2.3.1.In fact, Proposition 2.2.8 allows us to identify the left completion Mod fil k with the ∞category of complete filtered k-modules in the sense of Definition 2.1.8,endowed with the restriction of the Beilinson t-structure.The equivalence of Theorem 2.3.1 is then provided by the composition of the natural inclusion Mod fil k ⊆ Mod fil k with the associated mixed graded ∞-functor of Definition 2.3.8.Its inverse is described by the composition of the associated filtered ∞-functor of Definition 2.3.3 with the completion ∞-functor of Lemma 2.1.12.2..The fact that these ∞-functors are indeed equivalences follows from Propositions 2.3.14 and 2.3.15.The t-exactness of the equivalence Mod fil k ε -Mod gr k follows from the t-exactness of the ∞-functors (−) fil and (−) gr ε .2.3.18Finally, since (−) fil and (−)

Porism 2 . 3 . 20
3.1 is strongly monoidal.Explicitly, given two complete filtered k-modules M • and N • , their tensor product in Mod fil k , given by the completion of their tensor product as mere filtered k-modules ( [11, Theorem 2.2.5]), is equivalently described asM • ⊗ fil k N • (M • ) ε -gr ⊗ ε-gr k (N • ) ε -gr fil .In Sect.1.1, we mentioned how the Tate realization ∞-functor was strongly monoidal when restricted to non-negatively mixed graded k-modules.This claim follows immediately from Theorem 2.3.1:indeed, we have an equivalence in Fun ε -Modgr k , Mod k |−| t (−) −∞ • (−) filand since (−) ε -gr is a localization ∞-functor, we have another equivalence|−| t • (−) ε -gr (−) −∞ if one restricts (−) ε -gr to Mod fil k .Since the ∞-functor (−) −∞ is strongly monoidal ( [11, Section 2.23]), one simply has to show that the tensor product of two complete filtered k-modules M • and N • such that M p and N p are zero for all positive integers p is again complete.The explicit formula for the filtered tensor product provided in 2.1.11yields thatM • ⊗ fil k N • n colim p+q n M p ⊗ k N q .

Porism 2 . 3 . 21
Arguing analogously to Porism 2.3.20,we can prove that the Tate realization ∞-functor preserves colimits when it is restricted to mixed non-negatively graded k-modules.Indeed, we have a commutative square the ∞-functor ∞-category Fun N , Mod k of k-modules with filtration bounded in non-positive weights, and the inclusion is induced by the map of posets N ⊆ Z .The ∞-functor (−) −∞ commutes with colimits because it is a colimit ∞-functor itself, so it suffices to show that (−) fil : ε -Mod gr, 0 k → Mod fil, 0 k commutes with colimits.This is true because colimits of non-positively filtered kmodules are again non-positively filtered, since colimits are computed weight-wise.Hence they are again complete, and so the colimit in mixed graded k-modules agrees with the colimit in filtered k-modules.In particular, the ∞-functor|−| t : ε -Mod gr, 0 k −→ Mod k (2.3.22)admits a right adjoint, which we can compute explicitly using the fact that the ∞functor 2.3.22 is the composition of Mod k .In fact, each ∞-functor of this composition is a left adjoint, so the right adjoint to 2.const is the diagonal ∞-functor sending a k-module to the constant sequence, σ 0 is the naive truncation ∞-functor for filtered k-modules (built similarly as the ∞-functor of Part 1 of Definition 1.2.1), and the last ∞-functor is the mixed graded k-module construction (which is an equivalence because every filtered k-module is complete if it is 0 for all positive integers).In particular, this right adjoint agrees with (−)(0) : Mod k −→ ε -Mod gr, 0 k ⊆ ε -Mod gr k .
Let BG a,k the classifying stack for the affine smooth group scheme A 1 : = Spec(k[t]): it is an affine group stack which, in any characteristic, is equivalent to the spectrum of the derived commutative ring Sym Definition 1.1.2k (k[−1]).However, when k is a base ring which contains Q, it is well-known that Sym k • , N • ) for any mixed graded k-module N • and for any mixed graded k-module M • which is bounded and perfect in each weight, and this equivalence is provided exactly by tensoring the adjoint map to the evaluation M ∨ • ⊗ Proposition 7.1.1.13]).Moreover it is presentable: indeed, it fits in a (∞, 2)-pullback diagram of presentable ∞-categories (2) us recall this important result concerning left completions.([17,1.2.1.17]).Let C be a stable ∞-category endowed with a t-structure.Then:(1)The left completion C is also stable.(2)TheleftcompletionC is naturally endowed with a t-structure which can be described as follows.Given an identification of C with the full ∞-subcategory of Fun Z op , C spanned by those functors in which F n factors through C n and such that F m → F n induces an equivalence τ n F m → F n for all n m in Z, then the connective (resp.coconnective)objects of C are those ∞-functors which factor through C 0 (resp.C 0 ).(3)The canonical ∞-functor C → C is exact and induces an equivalence C 0 → C 0 .Proof In order to prove our claim, we use the following criterion.( [17, Proposition 1.2.1.19])Let C a stable ∞-category equipped with a t-structure.Suppose that C has all countable products, and C 0 is closed under countable products.Then C is left exact with respect to its t-structure if and only if C We will say that C is left complete if the canonical morphism C → C is an equivalence of stable ∞-categories.∞ : = n∈Z C n consists only of zero objects of C.