Derived moduli of complexes and derived Grassmannians

In the first part of this paper we construct a model structure for the category of filtered cochain complexes of modules over some commutative ring $R$ and explain how the classical Rees construction relates this to the usual projective model structure over cochain complexes. The second part of the paper is devoted to the study of derived moduli of sheaves: we give a new proof of the representability of the derived stack of perfect complexes over a proper scheme and then use the new model structure for filtered complexes to tackle moduli of filtered derived modules. As an application, we construct derived versions of Grassmannians and flag varieties.


Introduction
Recent developments in Derived Algebraic Geometry have lead many mathematicians to revise their approach to Moduli Theory: in particular one of the most striking results in this area is certainly Lurie Representability Theorem -proven by Lurie in 2004 as the main result of his PhD thesis [21] -which provides us with an explicit criterion to check whether a simplicial presheaf over some ∞-category of derived algebras gives rise to a derived geometric stack. Unfortunately the conditions in Lurie's result are often quite complicated to verify in concrete derived moduli problems involving algebro-geometrical structures, so for several years a rather narrow range of derived algebraic stacks have actually been constructed: in particular the most significant example known was probably the locally geometric derived stack of perfect complexes over a proper scheme X, which was firstly studied by Toën and Vaquié in 2007 (see [41]) without using any representability result à la Lurie. Nonetheless a few years later Pridham developed in [27] several new representability criteria for derived geometric stacks which have revealed to be more suitable to tackle moduli problems arising in Algebraic Geometry, as he himself showed in [28] where he used such criteria to construct a variety of derived moduli stacks for schemes and (complexes of) sheaves. In [12] Halpern-Leistner and Preygel have also recovered Toën and Vaquié's result by using some generalization of Artin Representability Theorem for ordinary algebraic stacks (see [1]), though their approach is not based on Pridham's theory. In this paper we give a third proof of existence and local geometricity of derived moduli for perfect complexes by means of Pridham's representability and then look at derived moduli of filtered perfect complexes: our main result is Theorem 2.33, which essentially shows that filtered perfect complexes of O X -modules -where X is a proper scheme -are parametrized by a locally geometric derived stack. In our strategy a key ingredient to tackle derived moduli of filtrations -in addition to Pridham's representability -is a good Homotopy Theory of filtered modules in complexes: for this reason the first part of this paper is devoted to construct a satisfying model structure on the category FdgMod R , which is probably an interesting matter in itself. Theorem 1.18 shows that FdgMod R is endowed with a natural cofibrantly generated model structure and Theorem 1.29 proves that this is nicely related to the standard projective model structure on dgMod R via the Rees construction. In the end, we conclude this paper by constructing derived versions of Grassmannians and flag varieties, which are obtained as suitable homotopy fibres of the derived stack of filtrations over the derived stack of complexes.

Homotopy Theory of Filtered Structures
This chapter is devoted to the construction of a good homotopy theory for filtered cochain complexes; for this reason we will first recall the standard projective model structure on cochain complexes and then use it to define a suitable one for filtered objects. At last, we will also study the Rees functor from a homotopy-theoretic viewpoint and see that it liaises coherently dg structures with filtered cochain ones.

Background on Model Categories
This section is devoted to review a few complementary definitions and results in Homotopy Theory which will be largely used in this paper; we will assume that the reader is familiar with the notions of model category, simplicial category and differential graded category: references for them include [7], [10], [13], [14], [30] and [39], while [11] provides a very clear and readable overview. Let C be a complete and cocomplete category and I a class of morphisms in C; recall from [14] that: 1. a map is I-injective if it has the right lifting property with respect to every map in I (denote by I-inj the class of I-injective morphisms in C); 2. a map is I-projective if it has the left lifting property with respect to every map in I (denote by I-proj the class of I-projective morphisms in C); 3. a map is an I-cofibration if it has the left lifting property with respect to every I-injective map (denote by I-cof the class of I-cofibrations in C); 4. a map is an I-fibration if it has the right lifting property with respect to every I-projective map (denote by I-fib the class of I-fibrations in C); 5. a map is a relative I-cell complex if it is a transfinite composition of pushouts of elements of I (denote by I-cell the class of I-cell complexes).
Fix some class S of morphisms in C and recall that an object A ∈ C is said to be compact 1 relative to S if for all sequences C 0 −→ C 1 −→ · · · −→ C n −→ C n+1 −→ · · · such that each map C n → C n+1 is in S, the natural map is an isomorphism; moreover A is said to be compact if it is compact relative to C.
Definition 1.1. A model category C is said to be (compactly) cofibrantly generated 2 if there are sets I and J of maps such that: Quillen) Let F : C D : G be an adjoint pair of functors and assume C is a cofibrantly generated model category, with I as set of generating cofibrations, J as set of generating trivial fibrations and W as set of weak equivalences. Suppose further that: 1. G preserves sequential colimits; 2. G maps relative FJ 3 -cell complexes to weak equivalences.
Then the category D is endowed with a cofibrantly generated model structure where FI is the set of generating cofibrations, FJ is the set of generating trivial cofibrations and FW as set of weak equivalences. Moreover (F, G) is a Quillen pair with respect to these model structures.
The end of this section is devoted to review a famous comparison result due to Dold and Kan which establishes an equivalence between the category of non-negatively graded chain complexes of k-vector spaces and that of simplicial k-vector spaces, which has very profound consequences in Homotopy Theory.
Warning 1.5. Be aware that in the end of this section we will deal with (non-negatively graded) differential graded chain structures, while in the rest of the paper we will mostly be interested in cochain objects.
First of all, recall that the normalization of a simplicial k-vector space (V, ∂ i , σ j ) is defined to be the non-negatively graded chain complex of k-vector spaces (NV, δ) where (NV ) n := i ker (∂ i : V n → V n−1 ) and δ n := (−1) n ∂ n . Such a procedure defines a functor N : sVect k −→ Ch ≥0 (Vect k ) .
On the other hand, let V be a chain complex of k-vector spaces and recall that its denormalization is defined to be the simplicial vector space ((KV ) , ∂ i , σ j ) given in level n by the vector space Remark 1.6. Notice that In order to complete the definition of the denormalization of V we need to define face and degeneracy maps: we will describe a combinatorial procedure to determine all of them. For all morphisms α : [m] → [n] in ∆, we want to define a linear map K (α) : (KV ) n → (KV ) m ; this will be done by describing all restrictions K (α, η) : V p [η] → (KV ) m , for any surjective non-decreasing map η ∈ Hom ∆ ([p] , [n]). For all such η, take the composite η • α and consider its epi-monic factorization 4 • η , as in the diagram

Now
• if p = q (in which case is just the identity map), then set K (α, η) to be the natural identification of V p [η] with the summand V p [η ] in (KV ) m ; • if p = q + 1 and is the unique injective non-decreasing map from [p] to [p + 1] whose image misses p, then set K (α, η) to be the differential d p : V p → V p−1 ; • in all other cases set K (α, η) to be the zero map.
The above setting characterizes all the structure of the simplicial vector space ((KV ) , ∂ i , σ j ); again, such a procedure defines a functor The Dold-Kan correspondence described in Theorem 1.7 is known to induce a number of very interesting ∞-equivalences: for more details see for example [5], [10], [37] and [43].

Homotopy Theory of Cochain Complexes
Fix a commutative unital ring R: in this section we will review the standard model structure by which one usually endows the category of (unbounded) cochain complexes R, i.e. the so-called projective model structure; all the section is largely based on [14] Section 2.3, where the homotopy theory of chain complexes over a commutative unital ring is extensively studied: actually all results, constructions and arguments we are about to discuss are essentially dual versions of the ones given there. Recall that the category dgMod R of cochain complex of R-modules (also referred as R-module in complexes) is one of the main examples of abelian category: as a matter of fact it is complete and cocomplete (limits and colimits are taken degreewise), the complex (0, 0) defined by the trivial module in each degree gives the zero object and short exact sequences are defined degreewise; for more details see [43]. Let (M, d) ∈ dgMod R : as usual, we define its R-module of n-cocycles to be Z n (M ) := ker (d n ), its R-module of n-coboundaries to be B n (M ) := Im d n−1 ≤ Z n (M ) and its n th cohomology R-module to be H n (M ) := Z n (M ) /B n (M ); (M, d) is said to be acyclic if H n (M ) = 0 ∀n ∈ Z; cocycles, coboundaries and cohomology define naturally functors from the category dgMod R to the category Mod R of R-modules. Finally, recall that a cochain map f : (M, d) → (N, δ) is said to be a quasi-isomorphism if it is a cohomology isomorphism, i.e. if H n (f ) is an isomorphism ∀n ∈ Z. In the following, we will not explicitly mention the differential of a complex whenever it is clear from the context. Now define the complexes and the only non-trivial connecting map (the one between D R (n) n and D R (n) n+1 ) is the identity.
Remark 1.8. Observe that D R (n) and S R (n) are compact for all n.
Theorem 1.9. Consider the sets The classes (1.1) define a cofibrantly generated model structure on dgMod R , where I dgMod R is the set of generating cofibrations, J dgMod R is the set of generating trivial cofibrations and W dgMod R is the set of weak equivalences.
The proof of Theorem 1.9 (which corresponds to [14] Theorem 2.3.11) relies on Theorem 1.3, thus it amounts to explicitly describe fibrations, trivial fibrations, cofibrations and trivial cofibrations determined by the sets (1.1), which we do in the following propositions. Proof. We want to characterize diagrams in dgMod R admitting a lifting. A diagram like (1.2) is equivalent to choosing an element y in N n , while a lifting is equivalent to a pair (x, y) ∈ M n × N n such that p n (x) = y: it follows that p ∈ J dgMod R -inj if and only if p n is surjective for all n ∈ Z.
Proof. First of all, observe that any diagram in dgMod R of the form is uniquely determined by an element in Moreover, there is a bijection between the set of diagrams like (1.3) admitting a lifting and Now suppose that p ∈ I-inj: we want to prove that it is degreewise surjective (because of Proposition 1.10) and a cohomology isomorphism. For any cocycle y ∈ Z n (N ), the pair (y, 0) defines a diagram like (1.3), therefore, as p ∈ I-inj, ∃z ∈ M n such that p n (z) = y and d n (z) = 0, so the induced map Z n (p) : Z n (M ) → Z n (N ) is surjective; in particular the map H n (p) : H n (M ) → H n (N ) is surjective as well. We now show that p n itself is surjective: fix x ∈ N n and consider d n (x) ∈ Z n+1 (N ); as the map Z n+1 (p) is surjective, ∃y ∈ Z n+1 (M ) such that p n+1 (y) = d n (x), thus by the assumption ∃z ∈ M n such that p n (z) = x, hence p is a degreewise surjection. It remains to prove that H n (p) is injective: fix x ∈ N n−1 and consider d n−1 (x) ∈ B n (N ) ≤ Z n (N ); by the surjectivity of Z n (p) ∃y ∈ Z n (M ) such that p n (y) = d n−1 (x), so [y] ∈ ker (H n (p)). The pair (x, y) defines a diagram of the form (1.3), so the assumption on p implies the existence of z ∈ M n−1 such that d n−1 (z) = y and p n−1 (z) = x; in particular, we have that ker (H n (p)) = 0, so H n (p) is injective. Now assume that p is a trivial fibration, i.e. a degreewise surjection with acyclic kernel; consider (x, y) ∈ X: we want to find z ∈ M n such that (x, z, y) ∈ X . The hypotheses on p are equivalent to the existence of a short exact sequence in dgMod R such that H n (K) = 0 ∀n ∈ Z. Take any w ∈ M n such that p n (w) = x; an immediate computation shows that dw−y ∈ Z n+1 (K) and, as K is acyclic, ∃v ∈ K n such that dv = dw−y. Now define z := w − v and the result follows.
The next step is describing cofibrations and trivial cofibrations generated by the sets (1.1), but we need to understand cofibrant objects in order to do that; in the following for any Rmodule P call D R (n, P ) the cochain complex defined by D R (n, P ) :=    P if k = n, n + 1 0 otherwise and in which the only non-trivial connecting map is the identity. Proposition 1.12. If A ∈ dgMod R is cofibrant, then A n is projective for all n. Conversely, any bounded above cochain complex of projective R-modules is cofibrant.
Proof. Suppose A is a cofibrant object in dgMod R and let p : M N be a surjection between two R-modules; the R-linear map p : M N induces a morphismṗ : D R (n, M ) → D R (n, N ) (given by p itself in degree n and n − 1 and by the zero map elsewhere), which is immediately seen to be degreewise surjective with acyclic kernel, hence a trivial fibration by Proposition 1.11. Moreover any R-linear map f : A n → N defines a cochain morphismḟ : A → D R (n, N ) which is given by f in degree n + 1, f d in degree n and 0 elsewhere. By assumption the diagram in dgMod R D R (n, M ) admits a lifting g: now it suffices to look at the above diagram in degree n to see that A n is projective. Now suppose A is a bounded above cochain complex of projective R-modules (i.e. A n = 0 for n 0) and fix a trivial fibration in dgMod R p : M → N and a cochain map g : A → N : we want to prove that g lifts to a morphism h ∈ Hom dgMod R (A, M ), so we construct h n by (reverse) induction. The base of the induction is guaranteed by the fact that A is assumed to be bounded above, so suppose that h k has been defined for all k > n; by Proposition 1.11 p n is surjective and has an acyclic kernel K, so since A n is projective ∃f ∈ Hom Mod R (A n , M n ) lifting g n . Consider the R-linear map F : A n → M n+1 defined as F := d n f − h n+1 d n , which measures how far f is to fit into a cochain map: an easy computation shows that p n+1 F = d n+1 F = 0, so F : A n → Z n+1 (K), but, as K is acyclic, we get that F : A n → B n (K). Of course the map d n+1 gives a surjective R-linear morphism from K n to B n+1 (K), so by the projectiveness of A n F lifts to a map G ∈ Hom dgMod R (A n , K n ). Now define h n := f − G and the result follows.
Remark 1.13. A complex of projective R-modules is not necessarily cofibrant (get a counterexample by adapting [14] Remark 2.3.7); it is possible to give a complete characterization of cofibrant objects in dgMod R in terms of dg-projective complexes (see [2]). Proposition 1.14. i ∈ Hom dgMod R (M, N ) is a cofibration if and only if it is a degreewise injection with cofibrant cokernel.
Proof. Suppose i is a cofibration, i.e. a map having the left lifting property with respect to degreewise surjections with acyclic kernel, by Proposition 1.11; there is an obvious morphism M → D R (n − 1, M n ) given by d n−1 in degree n − 1 and the identity in degree n, while the canonical map D R (n − 1, M n ) → 0 is a trivial fibration, as D R (n − 1, M n ) is clearly acyclic. As a consequence we get a diagram in dgMod R which admits a lifting as i is a cofibration; in particular this implies that i n is an injection. At last recall that the class of cofibration in a model category is closed under pushouts: in particular 0 → coker (i) is a cofibration as it is the pushout of i, thus coker (i) is cofibrant. Now suppose that i is a degreewise injection with cofibrant cokernel C and we are given a diagram of cochain complexes where p is a degreewise surjection with acyclic kernel K (let j : K → X be the kernel morphism): we want to construct a lifting in such a diagram. First of all notice that N n M n ⊕ C n , as C n is projective by Proposition 1.12, so we have where τ n : C n → A n+1 is some R-linear map such that d n τ n = τ n d n , and where σ n : C n → Y n satisfies the relation d n σ n = p n f n τ n + σ n d n . A lifting in the diagram (1.4) then consists of a collection {ν n } n∈Z of R-linear morphisms such that p n ν n = σ n and d n ν n = ν n d n + f n τ n . As C n is projective, fix G n ∈ Hom Mod R (C n , X n ) lifting σ n and consider the map F n : C n → X n+1 defined as F n := d n G−Gd n −f n τ n . It is easily seen that p n+1 F n = 0 and d n+1 F n = −d n+1 G n d n + f n+1 τ n d n−1 , so there is an induced cochain map s : C → ΣK, where ΣK is the suspension complex defined by the relations (ΣK) n = K n+1 and d ΣK = −d K .
As K is acyclic, observe that s is cochain homotopic to 0 (see [14] Lemma 2.3.8 for details), thus there is h n ∈ Hom dgMod R (C n , K n ) such that s = −d n h n + h n+1 d n ; define ν n := G n + j n h n and the result follows.
Proof. Suppose i ∈ J dgMod R -cof, i.e. it has the left lifting property with respect to all fibrations; in particular it is a cofibration so by Proposition 1.14 it is a degreewise injection with cofibrant cokernel C and let c : N → C be the cokernel morphism: we want to show that C is projective as a cochain complex. Fix a fibration p : X → Y and consider the diagram where 0 : M → 0 → X is the zero morphism and f ∈ Hom dgMod R (C, N ) is an arbitrary cochain map. By assumption diagram (1.5) admits a lifting, which is a cochain map h such that hi = 0 and ph = f c; it follows that h factors through a map g ∈ Hom dgMod R (C, M ) lifting f , so C is a projective cochain complex. Now assume i is a degreewise injection with projective cokernel C: again, let c : N → C denote the cokernel morphism and consider a diagram where p is a fibration, i.e. a degreewise surjection because of Proposition 1.10. Since C is projective, there is a retraction r : N → M and it is easily seen that (pf r − g) i = 0, so the map pf r − g lifts to a map s ∈ Hom dgMod R (C, Y ). Again, the projectiveness of C implies that there is a map t ∈ Hom dgMod R (C, X) lifting s; now the map f r − tc gives a lifting in diagram (1.5).
The last claim of the statement follows immediately by the fact that any projective cochain complex is also acyclic (see for example [43] or [14]).
Proposition 1.16. The set W dgMod R of quasi-isomorphisms in dgMod R has the 2-out-of-3 property and is closed under retracts.
Proof. This is a classical result in Homological Algebra: for a detailed proof see [15] Lemma 1.1 (apply it in cohomology).
The above results (especially Proposition 1.11, Proposition 1.15 and Proposition 1.16) say that the category dgMod R endowed with the structure (1.1) fits into the hypotheses of Theorem 1.3, so Theorem 1.9 has been proven. Now assume R is a (possibly differential graded) commutative k-algebra, where k is a field of characteristic 0: under such hypothesis there is also a canonical simplicial enrichment on dgMod R (all the rest of the section is adapted from [28]). Formulae (1.6) make dgMod R into a differential graded category over k, thus the simplicial structure on dgMod R will be given by setting where K is the simplicial denormalization functor giving the Dold-Kan correspondence (see Section 1.1) and τ ≥0 is good truncation.

Homotopy Theory of Filtered Cochain Complexes
Let R be any commutative unital ring: in this section we will endow the category of filtered cochain complexes with a model structure which turns to be compatible (in a sense which will be clarified in Section 1.6) with the projective model structure on dgMod R . Recall that a filtered cochain complex of R-modules (also referred as filtered R-module in complexes) consists of a pair (M, F ), where M ∈ dgMod R and F is a decreasing filtration on it, i.e. a collection F k M k∈N of subcomplexes of M such that F k+1 M ⊆ F k M and F 0 M = M ; as a consequence an object (M, F ) ∈ FdgMod R looks like a diagram of the form . .

?
O O A morphism of filtered complexes is a cochain map preserving filtrations 6 , so denote by FdgMod R the category made of filtered R-modules in complexes and their morphisms. The category FdgMod R is both complete and cocomplete: as a matter of fact let (M α , F α ) α∈I and (N β , F β ) be respectively an inverse system and a direct system in FdgMod R : we have that In particular the filtered complex (0, T ), where 0 is the zero cochain complex and T is the trivial filtration over it, is the zero object of the category FdgMod R . Define the filtered complexes Remark 1.17. Observe that (D R (n, p) , F ) and (S R (n, p) , F ) are compact for all n and all p.
In the following we will sometimes drop explicit references to filtrations if the context makes them clear.
The classes (1.8) define a cofibrantly generated model structure on FdgMod R , where I FdgMod R is the set of generating cofibrations, J FdgMod R is the set of generating trivial cofibrations and W FdgMod R is the set of weak equivalences.
As done in Section 1.1, proving Theorem 1.18 amounts to provide a precise description of fibrations, trivial fibrations, cofibrations and trivial cofibrations determined by the sets (1.8), which we do in the following propositions.
Proof. We want to characterize diagrams in FdgMod R admitting a lifting. A diagram like (1.9) corresponds to the sequence of diagrams and -as we did in the proof of Proposition 1.10 -we see that the sequence (1.10) corresponds bijectively to an element Proof. We want to characterize diagrams in FdgMod R admitting a lifting. A diagram like (1.9) corresponds to the sequence of diagrams in dgMod R and -as we did in the proof of Proposition 1.11 -we see that the sequence (1.12) corresponds bijectively to an element ((x 0 , y 0 ) , (x 1 , y 1 ) , . . . , (x p , y p )) ∈ X 0 × X 1 × · · · × X p , where and moreover the pair (x p , y p ) determines all the previous ones through the inclusion maps defining the filtration F . Now by Proposition 1.11 a diagram like (1.11) admits a lifting if and only if F p p is degreewise surjective and induces an isomorphism in cohomology, thus the result follows letting n and p vary.
As done in Section 1.2, we study cofibrant objects defined by the structure (1.8).
and any morphism g ∈ Hom FdgMod R ((A, F ) , (N, F )). By assumption, there exists a morphism h lifting g, so the diagram commutes. In particular this means that the big diagram now it suffices to apply Proposition 1.12 to show that F k A n is a projective R-module ∀n ∈ Z, ∀k ∈ N. Now assume that F k A is a cofibrant cochain complex (which in particular implies that F k A n is a projective R-module ∀n ∈ Z by Proposition 1.12) and F is bounded above: we want to prove that (A, F ). Let p ∈ Hom FdgMod R ((M, F ) , (N, F )) be a trivial fibration and pick a morphism g ∈ Hom FdgMod R (A, M ): we want to show that there is a morphism h lifting g. By reverse induction, assume that F p h : F p A → F p M has been defined for all p ≥ k (the boundedness of F ensures that we can get started): we want to construct a lifting in level k − 1. Consider the diagram The assumption on the filtration in Proposition 1.21 is probably too strong: it can be substituted with any hypothesis giving the base of the above inductive argument.
it has the left lifting property with respect to fibrations; in particular it is in I FdgMod R -cof, so we only need to prove that H n F k i is an isomorphism ∀n ∈ Z, ∀k ∈ N. Let p ∈ Hom FdgMod R ((X, F ) , (Y, F )) be any fibration, so by Proposition 1.19 F k p n is surjective ∀n ∈ Z, ∀k ∈ N: by assumption the diagram admits a lifting and, unfolding it, we get that the diagram in dgMod R lifts as well. Letting p vary among all fibrations in FdgMod R we see that F k i has the right lifting property with respect to all degreewise surjections in dgMod R , so by Proposition 1.10 and Proposition 1.15 it is a trivial cofibration in dgMod R ; in particular this means that H n F k i is an isomorphism ∀n ∈ Z, ∀k ∈ N, so the result follows. Proof. The result follows immediately by applying Proposition 1.16 levelwise in the filtration.
The above results (especially Proposition 1.20, Proposition 1.23 and Proposition 1.24) say that the category FdgMod R endowed with the structure (1.8) fits into the hypotheses of Theorem 1.3, so Theorem 1.18 has been proven.
Remark 1.25. We have not provided a complete description of cofibrations as this is not really needed in order to establish that data (1.8) endow FdgMod R with a model structure; clearly all are cofibrations for such model structure, but it is not clear (nor expected) that these are all of them. Actually we believe that a careful characterization of cofibrations should be quite complicated. Now assume R is a k-algebra, where k is a field of characteristic 0: we now endow FdgMod R with the structure of a simplicially enriched category.
where, by a slight abuse of notation, we mean that Formulae (1.13) make FdgMod R into a differential graded category over k, so we can naturally endow it with a simplicial structure by taking denormalization, i.e. by setting (1.14)

The Rees Functor
Let R be a commutative unital ring; the model structure over FdgMod R given by Theorem 1.18 is really modelled on the unfiltered situation: unsurprisingly, the homotopy theories of filtered modules in complexes and unfiltered ones are closely related, and the functor connecting them is given by the classical Rees construction. Recall that the Rees module associated to a filtered R-module (M, F ) is defined to be the graded R [t]-module given by so the Rees construction transforms filtrations into grading with respect to the polynomial algebra R [t]. Also, it is quite evident from formula (1.15) that the Rees construction is functorial, so there is a functor Rees : at our disposal, which in turn induces a functor to the category of graded dg-modules over R [t]; in particular we like to view the latter as the category G m -dgMod R[t] of R [t]-modules in complexes equipped with an extra action of the multiplicative group compatible with the canonical action The projective model structure on dgMod R[t] admits a natural G m -equivariant version.
Theorem 1.26. Consider the sets The classes (1.18) determine a cofibrantly generated model structure over is the set of generating trivial cofibrations and W Gm-dgMod R[t] is the set of weak equivalences.
Arguments and lemmas discussed in Section 1.2 to prove Theorem 1.1 carry over to this context once we restrict to G m -equivariant objects and maps.
Fibrations in the model structure determined by Theorem 1.18 are very nicely described: this is the content of the next propositions.
Proof. The proof of Proposition 1.10 adapts to the G m -equivariant context.
The following result collects various properties of functor (1.16): all claims are well-known, we only state them in homotopy-theoretical terms.
Theorem 1.29. The Rees functor has the following properties: 1. it has a left adjoint functor, given by which is natural in all variables; 3. its essential image consists of the full subcategory of t-torsion-free R [t]-modules in complexes; 4. it induces an equivalence on the homotopy categories; 5. it preserves compact objects; 10 9 Here Hom R[t] (Rees ((M, F )) , Rees ((N, F ))) Gm stands for the set of Gm-equivariant morphisms of R [t]-modules in complexes between Rees ((M, F )) and Rees ((N, F )). 10 In particular this means that the Rees functor maps filtered perfect complexes to perfect complexes: we will be more precise about this in Section 2.2 and Section 2.3.

it maps fibrations to fibrations.
In particular the Rees construction provides a Quillen equivalence between the categories FdgMod R and G m -dgMod R[t] , both endowed with the projective model structure.
Proof. We give references for most of the claims enunciated: the language we are using might be somehow different from the one therein, but the results and arguments we quote definitely apply to our statements.
this in turn implies that is degreewise surjective as a map of G m -equivariant R [t]-modules in complexes, thus the statement follows because of Proposition 1.28.
In particular Claim (1), Claim (4) and Claim (6) can be rephrased by saying that is a right Quillen equivalence.
Remark 1.30. We can say that the model structure on FdgMod R defined by Theorem 1.18 is precisely the one making the Rees functor into a right Quillen functor; more formally consider the pair given by the Rees functor and its left adjoint described in Theorem 1.29.1: than such a pair satisfies the assumption of Theorem 1.4 and moreover the model structure induced on FdgMod R through the latter criterion is the one determined by Theorem 1.18.
where the object on the left-hand side is the Ext group in the category FdgMod R , i.e.

Derived Moduli of Filtered Complexes
From now on k will always denote a field of characteristic 0 and R a (possibly differential graded) commutative algebra over k; let X be a smooth proper scheme over k: the main goal of this chapter is to study derived moduli of filtered perfect complexes of O X -modules. In order to do this we will first recall some generalities about representability of derived stacks -following the work of Lurie and Pridham -and then we will use these tools to construct derived geometric stacks classifying perfect complexes and filtered perfect complexes. Such stacks are related by a canonical forgetful map: as we will see in the last section of the chapter, the homotopy fibre of this map will provide us with a coherent derived version of the Grassmannian.

Background on Derived Stacks and Representability
This section is devoted to collect some miscellaneous background material on derived geometric stacks which will be largely used in the other sections of this chapter: in particular we will review a few representability results -due to Lurie and Pridham -giving conditions for a simplicial presheaf on dgAlg ≤0 R to give rise to a (truncated) derived geometric stack. We will assume that the reader is familiar with the notion of derived geometric n-stack and the basic tools of Derived Algebraic Geometry as they appear in the work of Lurie, Toën and Vezzosi: foundational references on this subject include [21], [22], [40] and [42]; in any case along most of the paper it will be enough to think of a derived geometric stack as a functor F : dgAlg ≤0 R → sSet satisfying hyperdescent and some technical geometricity assumption -i.e. the existence of some sort of higher atlas -with respect to affine hypercovers. These two conditions are precisely those turning a completely abstract functor to some kind of "geometric space", where the usual tools of Algebraic Geometry -such as quasi-coherent modules, formalism of the six operations, Intersection Theory -make sense. Also note that the case of derived schemes is much easier to figure out: as a matter of fact by [26] Theorem 6.42 a derived scheme X over k can be seen as a pair π 0 X, O X, * , where π 0 X is an honest k-scheme and O X, * is a presheaf of differential graded commutative algebras in non-positive degrees on the site of affine opens of π 0 X such that: Warning 2.1. Be aware that there are some small differences between the definition of derived geometric stack given in [21] -which is the one we refer to in this paper -and the one given in [42]: for a comparison see the explanation provided in [26] and [40]. Now we are to recall representability for derived geometric stacks: all contents herein are adapted from [27] and [28]. Recall that a functor F : dgAlg ≤0 R → sSet is said to be homotopic or homotopy-preserving if it maps quasi-isomorphisms in dgAlg ≤0 R to weak equivalences in sSet, while it is called homotopyhomogeneous if for any morphism C → B and any square-zero extension is a weak equivalence. Let F : dgAlg ≤0 R → sSet be a homotopy-preserving homotopy-homogeneous functor and take a point R ; recall from [27] that the tangent space to F at x is defined to be the functor and define for any differential graded A-module M and for all i > 0 the groups .
In the notations of formula (2.1) we have that: is an abelian group and the abelian structure is natural in M and F; be a square-zero extension in dgAlg ≤0 R and set y := f * x: there is a long exact sequence of groups and sets Proof. Claim 1 and Claim 2 correspond to [27] ) should be thought morally as some sort of pointwise cohomology theory for the functor F; such a statement is actually true -in a rigorous mathematical sense -whenever F is a derived geometric n-stack over R and x : RSpec (A) → F is a point on it: as a matter of fact in this case At last, recall that a simplicial presheaf on dgAlg ≤0 R is said to be nilcomplete if for all 11 The symbol − × h − − denotes the homotopy fibre product in sSet.
is a weak equivalence, where {P r A} r>0 stands for the Moore-Postnikov tower of A (see [10] for a definition). Now we are ready to state Lurie-Pridham Representability Theorem for derived geometric stacks.
R → sSet is a derived geometric n-stack almost of finite presentation if and only if the following conditions hold: 3. F is homotopy-homogeneous; 4. F is nilcomplete; 5. π 0 F is a hypersheaf (for the étale topology); 6. π 0 F preserves filtered colimits; 7. for finitely generated integral domains A ∈ H 0 (R) and all x ∈ F (A), the groups D j x (F, A) are finitely generated A-modules;

for finitely generated integral domains
9. for all finitely generated integral domains A ∈ Alg H 0 (R) and all x ∈ F (A) the functors D j (F, −) preserve filtered colimits for all j > 0; 10. for all complete discrete local Noetherian H 0 (R)-algebras A the map is a weak equivalence.
Remark 2.5. As we have already mentioned, a derived geometric n-stack roughly corresponds to a n-truncated homotopy-preserving simplicial presheaf on dgAlg ≤0 R which is a hypersheaf for the (homotopy) étale topology and which is obtained from an affine hypercover by taking successive smooth quotients. Theorem 2.4 says that in order to ensure that some given homotopy-homogeneous functor F : dgAlg ≤0 R → sSet is a derived geometric stack it suffices to verify that its underived truncation π 0 F : Alg H 0 (R) → sSet is a n-truncated stack (in the sense of [17] and [34]) and that for all x ∈ F (A) the cohomology theories D * x (F, −) satisfy some mild finiteness conditions. The most technical assumption in Theorem 2.4 is probably Condition (4), i.e. nilcompleteness: this is actually avoided when working with nilpotent algebras. Consider the full subcategory dg b Nil ≤0 R of dgAlg ≤0 R made of bounded below differential graded commutative R-algebras in non-positive degrees such that the canonical map A → H 0 (A) is nilpotent: the following result is Pridham Nilpotent Representability Criterion.
R → sSet is the restriction of an almost finitely presented derived geometric n-stack F : dgAlg ≤0 R → sSet if and only if the following conditions hold: 1. F is n-truncated; 2. F is homotopy-preserving 12 ; 3. F is homotopy-homogeneous; 4. π 0 F is a hypersheaf (for the étale topology); 5. π 0 F preserves filtered colimits; 6. for finitely generated integral domains A ∈ H 0 (R) and all x ∈ F (A), the groups D j x (F, A) are finitely generated A-modules; 7. for finitely generated integral domains A ∈ H 0 (R), all x ∈ F (A) and all étale morphisms are isomorphisms; 8. for all finitely generated integral domains A ∈ Alg H 0 (R) and all x ∈ F (A) the functors D j (F, −) preserve filtered colimits for all j > 0; 9. for all complete discrete local Noetherian H 0 (R)-algebras A the map is a weak equivalence.
Moreover F is uniquely determined by F up to weak equivalence.
In the last part of this section we will recall from [28] a few criteria ensuring homotopicity, homogeneity and underived hyperdescent of a functor F : dgAlg ≤0 R → sSet, which from now on will always be thought of as an abstract derived moduli problem. Most definitions and results below will involve sCat-valued derived moduli functors rather than honest simplicial presheaves on dgAlg ≤0 R : the reason for this lies in the fact that it is often easier to tackle a derived moduli problem by considering a suitable sCat-valued functor F : dg b Nil ≤0 R → sCat and then use Theorem 2.6 to prove that the diagonal of its simplicial nerve diag (BF) : dg b Nil ≤0 R → sSet gives rise to a honest truncated derived geometric stack; we will see instances of such a procedure in Section 2.2 and Section 2.3, for more examples see [28] Section 3 and Section 4. Moreover Cegarra and Remedios showed in [3] that the diagonal of the simplicial nerve is weakly equivalent to the functorW obtained as the right adjoint of Illusie's total décalage functor (see [10] or [19] for a definition), so we can substitute diag (BF) withW F in the above considerations: for more 12 When dealing with functors defined on dg b Nil ≤0 R actually it suffices to check that tiny acyclic extensions are mapped to weak equivalences. details see [28].
be a diagram of simplicial categories; recall that the 2-fibre product C × B D is defined to be the simplicial category for which simplicial categories is said to be a 2-fibration if the following conditions hold: ) is a fibration in sSet; 2. for any c 1 ∈ C, d ∈ D and homotopy equivalence h : a homotopy equivalence k : c 1 → c 2 in C and an isomorphism θ : Definition 2.8. A morphism F : C → D of simplicial categories is said to be a trivial 2-fibration if the following conditions hold: G(B) G (A) is a 2-fibration. If η is also 2-homotopic, it is said to be formally 2-quasi-smooth. The functor F is said to be formally 2-quasi-(pre)smooth if so is the morphism F → •.
R → sCat is said to be 2-homogeneous if for all squarezero extensions A → B and all morphisms C → B the natural map is essentially surjective on objects and an isomorphism on Hom spaces. Now given a simplicial category C, denote by W (C) the full simplicial subcategory of C in which morphisms are maps whose image in π 0 C is invertible (in particular this means that π 0 W (C) is the core of π 0 C). Also denote by c (π 0 C) the set of isomorphism classes of the (honest) category π 0 C. The following result relates quasi-smoothness to homogeneity and will be very useful in the rest of the paper. 3. the map W (F) → F is formally étale, meaning that for any square-zero extension A → B the induced map is an isomorphism; 4. W (F) is 2-homogeneous and formally 2-quasi smooth, as well.
At last, let us recall for future reference the notion of openness for a simplicial category.
is a weak equivalence.
is surjective.

Derived Moduli of Perfect Complexes
Let X be a smooth proper scheme over k and recall that a complex E of O X -modules is said to be perfect if it is compact as an object in the derived category D (X); in simpler terms E is perfect if it is locally quasi-isomorphic to a bounded complex of vector bundles. A key example of perfect complex is given by the total right derived functor of the push-forward of the relative De Rham complex associated to a morphism of schemes: more clearly, if f : Y → Z is a proper morphism of (semi-separated quasi-compact) k-schemes, then Rf * Ω Y /Z is perfect as an object in D (Z). Perfect complexes play a very important role in several parts of Algebraic Geometry -such as Hodge Theory, Deformation Theory, Enumerative Geometry, Symplectic Algebraic Geometry and Homological Mirror Symmetry -so it is very natural to ask whether they can be classified by some moduli stack; for this reason consider the functor The assumptions on the base scheme X in Theorem 2.17 -whose proof relies on Artin Representability Theorem (see [1]) -can be relaxed, but the key condition of Lieblich's result remains the vanishing of all negative Ext groups 13 ; in particular observe that such a condition ensures that Perf ≥0 X is a well-defined groupoid-valued functor: as a matter of fact the group Ext i (E, E), where E ∈ D (X) and i < 0, parametrizes i th -order autoequivalences of E, thus perfect complexes with trivial negative Ext groups do not carry any higher homotopy, but only usual automorphisms. By means of Derived Algebraic Geometry it is possible to outstandingly generalize Lieblich's result: indeed consider the functor where Perf (X) stands for the dg-category of perfect complexes on X,Â pe for the dg-category of perfect A-modules (see [41] for more details) and M ap for the mapping space of the model category of dg-categories (see [36], [38] and [39] for more details). 13 In [20] a perfect complex E ∈ D (X) such that Ext i (E , E ) = 0 for all i < 0 is called universally gluable; also in that paper the stack Perf ≥0 X is denoted by D b pug (X/k). 14 Recall that a derived stack F is said to be locally geometric if it is the union of open truncated derived geometric substacks.

Proof. See [41] Section 3; see also [40] Section 3.2.4 and Section 4.3.5 for a quicker explanation.
It is easily seen that there is a derived geometric 1-substack of RPerf X whose underived truncation is equivalent to Perf ≥0 X , so Theorem 2.17 is recovered as a corollary of Toën and Vaquié's work. Theorem 2.18 is a very powerful and elegant result, which has been highly inspiring in recent research: just to mention a few significant instances, it is one of the key ingredients in [35] where Simpson constructed a locally geometric stack of perfect complexes equipped with a λconnection, [32] where Schürg, Toën and Vaquié constructed a derived determinant map from the derived stack of perfect complexes to the derived Picard stack and studied various applications to Deformation Theory and Enumerative Geometry, [25] where Pantev, Toën, Vaquié and Vezzosi set Derived Symplectic Geometry. However the proof provided in [41] is quite abstract and complicated: as a matter of fact Toën and Vaquié actually constructed a derived stack parametrizing pseudo-perfect objects (see [41] for a definition) in a fixed dg-category of finite type (again see [41] for more details) and then proved by hand -i.e. without applying any representability result, but rather using just the definitions from [42] -that this is locally geometric and locally of finite type. Theorem 2.18 is then obtained just as an interesting application. In this section we will apply the representability and smoothness results discussed in Section 2.1 to obtain a simpler and more concrete proof of Theorem 2.18; actually we will follow the path marked by Pridham in [28], where he develops general methods to study derived moduli of schemes and sheaves. In a way the approach we propose is the derived counterpart of Lieblich's one, as the latter is based on Artin Representability Theorem rather than the definition of (underived) Artin stack. Moreover we will give a rather explicit description of the derived geometric stacks determining the local geometricity of RPerf X : again, such a picture is certainly present in Toën and Vaquié's work, but unravelling the language in order to clearly write down the relevant substacks might be non-trivial. Halpern-Leistner and Preygel have recently studied the stack RPerf X via representability as well, though their approach does not make use of Pridham's theory: for more details see [12] Section 2.5. Let X be a (possibly) derived scheme over R and recall that the Cěch nerve of X associated to a fixed affine open cover U := α U α is defined to be the simplicial affine schemě Definition 2.19. Define a derived module over X to be a cosimplicial O (X)-module in complexes.
We will denote by dgMod (X) the category of derived modules over X; just unravelling Definition 2.19 we see that an object M ∈ dgMod (X) is made of cochain complexes M m of O (X) m -modules related by maps satisfying the usual cosimplicial identities. Observe that the projective model structures on cochain complexes we discussed in Section 1.2 induces a model structure on dgMod (X), which we will still refer to as a projective model structure: in particular a morphism f : • a fibration if f m : M m → N m is degreewise surjective; • a cofibration if it has the left lifting property with respect to all fibrations (see [28] Section 4.1 for a rather explicit characterization of them).
In the same way, the category dgMod (X) inherits a simplicial structure from the category of R-modules in complexes: more clearly for any M, N ∈ dgMod (X) consider the chain complex Definition 2.20. A derived quasi-coherent sheaf over X is a derived module M for which all face maps ∂ i are weak equivalences.
Let dgMod (X) cart to be the full subcategory of dgMod (X) consisting of derived quasicoherent sheaves: this inherits a simplicial structure from the larger category and -even if it has not enough limits and thus cannot be a model category -it also inherits a reasonably well behaved subcategory of weak equivalences, so there is a homotopy category Ho (dgMod (X) cart ) of quasi-coherent modules over X simply obtained by localizing dgMod (X) cart at weak equivalences.
Remark 2.21. The constructions above make sense in a much wider generality: as a matter of fact in [28] Pridham defined derived quasi-coherent modules over any homotopy derived Artin hypergroupoid (see [26]) and through these objects he recovered the notion of homotopy-Cartesian module over a derived geometric stack which had previously been investigated by Toën and Vezzosi in [42]; also Corollary 2.23 -which is the main tool to deal with derived moduli of sheaves -holds in this much vaster generality. We have chosen to discuss derived quasi-coherent modules only for derived schemes since our goal is to study perfect complexes on a proper scheme, for which the full power of Pridham's theory of Artin hypergroupoids is not really needed. In particular bear in mind that the Cěch nerve of a derived scheme associated to an affine open cover is an example of homotopy Zariski 1-hypergroupoid.
From now on fix R to be an ordinary (underived) k-algebra and X to be a quasi-compact semi-separated scheme over R; note that in in [18] Hütterman showed that Ho (dgMod cart (X)) D (QCoh (X)) so in this case derived quasi-coherent modules are precisely what one would like them to be. Now define the functor where dgMod cart X ⊗ L R A c is the full simplicial subcategory of dgMod cart X ⊗ L R A on cofibrant objects, i.e. it is the (simplicial) category of cofibrant derived quasi-coherent modules on the derived scheme X ⊗ L R A. Proof. This is [28] Proposition 4.11, which relies on the arguments of [28] Proposition 3.7; we will discuss Pridham's proof here for the reader's convenience. We first prove that dCART X is 2-homogeneous; let A → B be a square-zero extension and C → B a morphism in dg b Nil ≤0 R and fix F , F ∈ dCART X (A × B C). Since by definition F and F are cofibrant (i.e. degreewise projective by Proposition 1.12) we immediately have that the commutative square of simplicial sets is actually a Cartesian diagram. Moreover fix F A ∈ dCART X (A) and F C ∈ dCART X (C) and let α : and also observe that F is cofibrant, i.e. F ∈ dCART X (A × B C). This shows that Hom dCART X is homogeneous, which means that dCART X is a 2-homogeneous functor. Now we prove that dCART X is formally 2-quasi-smooth; again let I → A B be a squarezero extension and pick F , F ∈ dCART X (A). Observe that, since F is cofibrant as a quasicoherent module over X ⊗ L R A, we have that the induced map F → F ⊗ A B is still a square-zero extension; furthermore if A → B is also a quasi-isomorphism, then so is F → F ⊗ A B: as a matter of fact notice, as a consequence of Proposition 1.12, that Now it follows that the natural chain map is degreewise surjective and a quasi-isomorphism whenever so is A → B. Now, by just applying truncation and Dold-Kan denormalization, we get that the morphism of simplicial sets is a fibration, which is trivial in case the square-zero extension A → B is a quasi-isomorphism. This shows that Hom dCART X is formally quasi-smooth, so in order to finish the proof we only need to prove that the base-change morphism is a 2-fibration, which is trivial whenever the extension A → B is acyclic. The computations in [26] Section 7 imply that obstructions to lifting a quasi-coherent module F ∈ dCART X (B) to dCART X (A) lie in the group . so in particular if H * (I) = 0 then map (2.6) is a trivial 2-fibration. Now fix F ∈ dCART X (A), denoteF := F ⊗ A B and let θ :F → G be a homotopy equivalence in dCART X (B). By cofibrancy, there exist a unique liftG of G to A as a cosimplicial graded module and, in the same fashion, we can lift θ to a graded morphismθ : F →G : we want to prove that there also exist compatible lifts of the differential. The obstruction to lift the differential d of G to a differential δ onG is given by a pair . It follows that the obstruction to lifting θ and G lies in Since θ is a homotopy equivalence we have that θ * is a quasi-isomorphism: in particular the cohomology group (2.7) is 0, which means that suitable lifts exist. This completes the proof.

M is open in the functor
4. For all finitely generated A ∈ Alg H 0 (R) and all E ∈ M (A), the functors . Then the functorWM is (the restriction to dg b Nil ≤0 R of ) a derived geometric n-stack.
Proof. This is [28] Theorem 4.12; we just sketch the main ideas of the proof for the reader's convenience. We basically need to verify that the various conditions in the statement imply that the simplicial presheafWM satisfies Pridham Nilpotent Representability Criterion (Theorem 2.6). First observe that by Condition (2) we have that 15 (2.8) 15 The symbol ≈ stands for "weakly equivalent".
As a matter of fact, the openness of M inside π 0 W dgMod X ⊗ L R − cart says that the inclusion for all nilpotent dgca's A ∈ dg b Nil ≤0 R , all complexes E ∈WM (A) and dg A-modules M . Now Proposition 2.22 and Proposition 2.12 tell us that Condition (4) and Condition (5) imply the homotopy-theoretic properties required by Pridham Nilpotent Representability Criterion, while the description of cohomology theories given by (2.9) ensures the compatibility of such modules with filtered colimits and base-change. In the end the weak completeness condition given by Condition (9) of Theorem 2.6 follows from Condition (7) through a few standard Mittag-Leffler computations: for more details see [28]  Theorem 2.24. In the above notations, assume that the scheme X is also proper; then functor (2.10) induces a derived geometric n-stack RPerf n X .
Proof. We have to prove that functor (2.10) satisfies the conditions of Corollary 2.23. First of all, notice that the vanishing condition on higher negative Ext groups guarantees that the simplicial presheaf M n is n-truncated, which is exactly Condition (1). Now we look at Condition (2), hence we need to prove the openness of M n as a subfunctor of π 0 W dgMod X ⊗ L R − cart ; it is immediate to see that M n (A) is a full simplicial subcategory of π 0 W dgMod X ⊗ L R A cart , so we only need to check that the map is homotopy formally étale, i.e. that the morphism of formal groupoids 16 π 0 M n → π 0 π 0 W dgMod X ⊗ L R − cart is formally étale. By classical Formal Deformation Theory this amounts to check that the map which morphism (2.11) induces on tangent spaces is an isomorphism and the one on obstruction spaces is injective (see for example [33] Section 2.1 and [24] Section V.8), so fix a square-zero extension I → A B and a perfect complex E ∈ M n (B). By Lieblich's work (see [20] Section 3) we have that • the tangent space to the functor π 0 M n at E is given by the group Ext 1 On the other hand, it is well known (for instance see the proof of [28] Theorem 4.12) that • the tangent space to the functor π 0 π 0 W dgMod X ⊗ L R − cart at E is given by the group It follows that the group homomorphism induced by map (2.11) on first-order deformations and obstruction theories is just the identity, so Condition (2) holds. Now let us look at Condition (3): take an étale cover {f α : A → B α } α in Alg R and let E be an object in π 0 π 0 W dgMod X ⊗ L R A cart such that the derived modules (f α ) * E over X ⊗ L R B α are perfect; then the derived quasi-coherent module E has to be perfect as well, because perfectness is a local property which is preserved under pull-back. It follows that Condition (3) holds. In order to check Condition (4), fix a finitely generated R-algebra A and a perfect complex E of O X ⊗ L R A -modules and consider an inductive system {B α } α of A-algebras. The perfectness assumption on E allows us to substitute this with a bounded complex F of flat O X ⊗ L R Amodules, so we get that Ext i

filtered colimits if and only if so does
, which is just the classical Ext functor. Now a few standard results in Homological Algebra imply the following canonical isomorphisms ∀i ≥ 0 In particular in the first isomorphism we are using the fact that filtered colimits commute with exact functors (and so is the tensor product as F is flat in each degree), while in the second one we are using the fact that filtered colimits commute with all Ext functors, since E is a finitely presented object as by perfectness this is locally quasi-isomorphic to a bounded complex of vector bundles. Ultimately the key idea in this argument is that the assumptions on the complexes we are classifying allow us to compute the Ext groups by choosing a "projective resolution" in the first variable and a "flat resolution" in the second one, so that all necessary finiteness conditions to make Ext i X⊗ L R A and lim −→ α commute are verified (see [43] Section 2.6). It follows that Condition (4) holds. 16 Notice that (homotopy) forma étaleness is a local property, so we can restrict map (2.11) to formal objects.
In order to check Condition (5), fix a finitely generated R-algebra A and a perfect complex E of O X ⊗ L R A -modules and again choose F to be a bounded complex of flat O X ⊗ L R A -modules being quasi-isomorphic to E . Consider the derived endomorphism complex of E over X ⊗ L R A: we have that . Notice that, again, we have computed the complex RHom O X ⊗ L R A (E , E ) by choosing a "flat resolution" in the second entry and a "projective resolution" in the first one; now consider the cohomology sheaves and note that these are coherent O X ⊗ L R A-modules. The local-to-global spectral sequence relates the cohomology of the Ext sheaves to the Ext groups and is well-known to converge: since the sheaves Ext i are coherent and finitely many, formula (2.12) implies that the groups Ext p+q X⊗ L R A (E , E ) are finitely generated as A-modules, thus Condition (5) holds. Now we look at Condition (6); fix an inductive system {A α } α of R-algebras and let A := lim where for any R-algebra B c (π 0 M n (B)) := isomorphism classes of perfect complexes of O X ⊗ L R B -modules .
Because being a perfect complex is local property, it suffices to show that formula (2.13) holds locally, i.e replacing X with an open affine subscheme U ; in particular, as flat modules are locally free, observe that a class [M ] ∈ c (π 0 M n (B)) is locally determined by an equivalence class of bounded complexes where s is some natural number and M i is a free O X (U ) ⊗ L R B-module for all i; again we have used the property that perfect complexes are quasi-isomorphic to bounded and degreewise flat ones. Now denote by i k the rank of the module M k in representative (2.14) and consider the scheme defined for all B ∈ Alg R through the functor of points (2.15) Formula (2.15) determines a closed subscheme of k=1,...,s−1 and provides a local description of c (π 0 M n (B)); clearly k=1,...,s−1 and since the subscheme S → k=1,...,s−1 Mat i k ,i k+1 is defined by finitely many equations, formula (2.16) descends to S (A), meaning that Formula (2.17) implies formula (2.13), so Condition (6) holds. Lastly, we have to check Condition (7), so fix a complete discrete local Noetherian R-algebra A and a perfect complex E of O X ⊗ L R A; again the assumptions on E allow us to substitute it with a bounded complex F of flat O X ⊗ L R A-modules. We first prove the compatibility of the Ext functors; the properties of A imply that the canonical morphism A −→Â to the pronilpotent completion is an isomorphism, which we can use to induce ∀i > 0 a canonical isomorphism (2.19) Again, we compute the Ext groups by using E (which is degreewise projective) in the first variable and F (which is degreewise flat) in the second variable. The obstruction for the functors (see [28] Section 4.2 for details). In particular we get At last, we show the compatibility condition on components, i.e. we want to prove that the push-forward map is bijective. This basically means to show that any inverse system determines uniquely a perfect O X ⊗ L R A-module in complexes via map (2.22); such a statement is precisely the version of Grothendieck Existence Theorem for perfect complexes: for a proof see [23]

Derived Moduli of Filtered Perfect Complexes
This section is devoted to the main result of this paper, that is the construction of a derived moduli stack RFilt X classifying filtered perfect complexes of O X -modules over some reasonable k-scheme X; (local) geometricity of such a stack will be ensured by some quite natural cohomological finiteness conditions given in terms of the Rees construction (see Section 1.4): actually the very homotopy-theoretical features of the Rees functor collected in Theorem 1.29 will allow us to mimic most of the results and arguments of Section 2.2, which deal with the corresponding unfiltered situation. In full analogy with what we did in Section 2.2, associate to any given derived scheme X over R the cosimplicial differential graded commutative R-algebra O (X) defined by formula (2.4).
Definition 2.26. Define a filtered derived module over X to be a cosimplicial filtered O (X)module in complexes.
More concretely Definition 2.26 says that a filtered derived module over X is a pair (M, F ) made of filtered cochain complexes (M m , F ) of O (X) m -modules related by maps satisfying the usual cosimplicial identities and such that the diagrams . .

?
O O commute; in other words a derived filtered module (M, F ) is just a nested sequence Notice that a filtered derived module is equipped with three different indexings, one coming from the filtration, one from the differential graded structure and the last one from the cosimplicial structure: a morphism of derived filtered modules will be an arrow preserving all of them, so there is a category of derived filtered modules on X, which we will denote by FdgMod (X). Just like the unfiltered situation analysed in Section 2.2, observe that the projective model structure on filtered cochain complexes given by Theorem 1.18 induces a projective model structure on FdgMod (X); in particular a morphism f : • a cofibration if it has the left lifting property with respect to all fibrations.
There is also a natural simplicial structure on the category FdgMod (X) again coming from the simplicial structure on and define the Hom spaces just by taking good truncation and denormalization, i.e. set In a similar way, notice that the HOM complex for filtered derived modules defined by formula (2.23) sheafifies, so we have a well-defined Hom-sheaf bifunctor and consequently a derived Hom sheaf, given by the bifunctor Definition 2.28. Define a filtered derived quasi-coherent sheaf over X to be a filtered derived module (M, F ) for which and F p M ∈ dgMod cart (X) for all p.
Denote by FdgMod cart (X) the full subcategory of FdgMod (X) consisting of filtered quasicoherent derived sheaves: the homotopy-theoretic properties of FdgMod (X) induce a simplicial structure and a well-behaved subcategory of weak equivalences on it.
Remark 2.29. The Rees functor (2.26) respects quasi-coherence, meaning that it restricts to a functor Rees : which obviously still maps weak equivalences to weak equivalences. Now our goal is to study derived moduli of filtered derived quasi-coherent sheaves by means of Lurie-Pridham representability: in order to reach this we will literally follow the strategy described in Section 2.2 when tackling moduli of unfiltered complexes; in particular we will prove filtered analogues of Proposition 2.22, Corollary 2.23 and Theorem 2.24. In the following, given any filtered derived quasi-coherent sheaf (E , F ) over some derived geometric stack denote byF the filtration induced by (derived) base-change and byF the one induced on quotients. From now on fix R to be an ordinary (underived) k-algebra and X to be a quasi-compact semi-separated scheme over R; define the functor where again FdgMod cart X ⊗ L R A c is the full simplicial subcategory of FdgMod cart X ⊗ L R A on cofibrant objects.
is a square-zero extension in dgAlg ≤0 . Moreover f is acyclic whenever so is f .
Proof. Denote I := ker (f ); then ker (f) = I [t], where In particular I Proof. The argument of Proposition 2.22 applies to this context as well, we sketch the main adjustments.
In order to verify that F dCART X is 2-homogeneous take a square-zero extension A → B and a morphism C → B in dg b Nil ≤0 R and fix (E , F ) , (E , F ) ∈ F dCART X (A × B C). Cofibrancy of such pairs -which by Proposition 1.21 implies filtration-levelwise degreewise projectivityensures that the commutative square of simplicial sets The filtered derived module (E , F ) is actually a cofibrant filtered derived quasi-coherent sheaf on X ⊗ R (A ⊗ B C), namely (E , F ) ∈ F dCART X (A × B C); we also have that and this completes the proof that F dCART X is a 2-homogeneous functor. Now we want to prove that the functor F dCART X is formally 2-quasi-smooth, so we start by showing that Hom F dCART X is formally quasi-smooth; for this reason take a square-zero extension A → B in dg b Nil ≤0 R and consider the induced R [t]-linear morphism A → B, as done in Lemma 2.30. Let (E , F ) , (E , F ) ∈ F dCART (A) and look at the induced morphism of simplicial sets Gm is a (trivial) fibration, which in turn is equivalent to say that is a 2-fibration, which is trivial whenever the square-zero extension A → B is acyclic. Note first that the computations in [26] Section 7, together with the definition of Ext groups for filtered derived modules given by formula (2.27) and the isomorphism provided by formula (2.28), imply that obstructions to lifting a filtered quasi-coherent module (E , F ) ∈ F dCART X (B) to F dCART X (A) lie in the group  1. M is a n-truncated hypersheaf;

M is open in the functor
4. For all finitely generated A ∈ Alg H 0 (R) and all (E , F ) ∈ M (A), the functors are isomorphisms.

LetM
: dg b Nil ≤0 R −→ sCat be the full simplicial subcategory of W (F dCART X (A)) consisting of objects (F , F ) for which the pair F ⊗ A H 0 (A) ,F is weakly equivalent in FdgMod cart X ⊗ L R H 0 (A) to an object of M H 0 (A) . Then the functorWM is (the restriction to dg b Nil ≤0 R of ) a derived geometric n-stack.
Proof. The same argument used to prove Corollary 2.23 carry over to this context, using Proposition 2.31 in place of Proposition 2.22 and observing -as done in the proof of Proposition 2.31 itself -that Also Condition (2) tells us that which is the filtered analogue of formula (2.8).
The only claim which still needs to be verified is the one saying thatW W (M) is an étale hypersheaf: observe that, by combining Condition (3) and Proposition 2.16, this amounts to check thatW W (F dCART X ) is a hypersheaf for the étale topology, thus fix an étale hypercover B → B • and consider the induced map  which classifies filtered perfect O X -modules in complexes with trivial Ext groups in higher negative degrees.
Theorem 2.33. In the above notations, assume that the scheme X is also proper; then functor (2.36) induces a derived geometric n-stack RFilt n X .
Proof. We have to prove that functor (2.36) satisfies the conditions of Corollary 2.32: again our strategy consists of adapting the proof of Theorem 2.24 to the filtered case by means of the homotopy-theoretical properties of the Rees construction. First of all, notice the vanishing assumption about the Ext groups given by Axiom (c) corresponds exactly to the n-truncation of the presheaf M n filt , which gives us Condition (1). As regards Condition (2), let us show the openness of M n filt inside π 0 W FdgMod X ⊗ L R − cart , which essentially amounts to prove that the morphism of formal groupoids • the tangent space to the functor π 0 M n filt at (E , F ) is given by • a functorial obstruction space for π 0 M n filt at (E , F ) is given by • the tangent space to the functor π 0 π 0 W FdgMod X ⊗ L R − cart at (E , F ) is given by the group Gm so the group homomorphisms induced on first-order deformations and obstruction theories is just identities, therefore Condition (2) holds. In terms of Condition (3), notice that the argument showing the analogous claim in the proof of Theorem 2.24 also holds in this context, since the filtered complexes we are parametrizing are perfect in each level of the filtration; thus Condition (3) holds.
In order to check Condition (4), fix a finitely generated R-algebra A and a pair (E , F ) ∈ M n filt (A) and consider an inductive system {B α } α of A-algebras. Since F m E is perfect for any m, we can choose a "flat" resolution (see Theorem 2.24 for more explanation) F ,Ḟ for the filtered complex (E , F ); therefore there is a chain of isomorphisms where we have used the various properties collected in Theorem 1.29, the induced description of the Ext groups determined by formula (2.28), the exactness of the functor (−) Gm and the filtration-levelwise degreewise flatness of the representative F ,Ḟ . It follows that Condition (4) holds. The way we prove Condition (5) is exactly the same utilized to show the corresponding claim in Theorem 2.24: indeed, note that such an argument carries over to this context, provided that we use the "filtered version" of the local-to-global spectral sequence given by Remark 2.27 in place of the classical one; thus Condition (5) holds. Now we look at Condition (6); fix an inductive system {A α } α of R-algebras and let A := lim where for any R-algebra B c (π 0 M n filt (B)) := isomorphism classes of filtered perfect complexes of O X ⊗ L R B -modules .
where for all p and all α F p E α is a perfect complex of O X ⊗ L R A α -modules. In the proof of Theorem 2.24 we have shown that each system {[F p E α ]} α determines uniquely an isomorphism class of perfect O X ⊗ L R A-module in complexes and notice that inclusions are preserved under inductive limits, thus the object described by formula (2.43) determines a unique class in c (π 0 M n filt (A)), which means that formula (2.39) is verified. It follows that Condition (6) holds. Lastly, we have to check Condition (7), so fix a complete discrete local Noetherian R-algebra A and a pair (E , F ) ∈ M n filt (A) . Consider for all i < 0 the canonical map (Rees ((E , F )) , Rees ((E , F ))) Gm Rees ((E , F )) , Rees E/m r A ,F Gm which is an isomorphism, as follows by combining the exactness of the functor (−) Gm and the computations in the proof of Theorem 2.24. At last, the compatibility condition on the components is easily checked by using techniques similar to the ones utilized to verify Condition (6). As a matter of fact take any inverse system of filtered perfect complexes of O X ⊗ L R A /m r A -modules and note that the proof of the corresponding statement in Theorem 2.24 allows us to lift each level F p E r to a perfect complex of O X ⊗ L R Amodules; moreover countable limits preserve inclusions: this concludes the verification of the claim. It follows that Condition (7) holds, so the proof is complete.
• pairs made of a dg-vector space over k and a subcomplex (again, possibly without higher autoequivalences). where the top map is the natural forgetful morphism while "const W " denotes the constant morphism sending any filtered complex to W . and formula (2.44) shows in particular that DFlag n k (W ) is a derived geometric n-stack over k.
Proposition 2.37. Assume W is a graded vector space. We have that: • the formal neighbourhood of π 0 π ≤0 DFlag k (W ) at the point (W, F ) identified by the filtration F is the deformation functor determined by the total flag variety associated to W ; • if furthermore W is concentrated in degree 0 then DFlag k (W ) is a derived scheme and the formal neighbourhood of its underived truncation at (W, F ) is the deformation functor determined by the flag variety associated to W .
Proof. This is proven in [5] Section 2.4 and Section 3.4, where the local theory of the stack DFlag k (W ) is deeply analysed; see also [9] Section 6, where -as far as we are aware -the name "homotopy flag variety" appeared for the first time.
In the same fashion we can define Grassmannians in Derived Algebraic Geometry.
Definition 2.38. Define the total derived Grassmannian over k associated to W to be the derived stack given by the homotopy fibre and formula (2.45) shows in particular that DGrass n k (W ) is a derived geometric n-stack over k.
Proposition 2.40. Assume W is a graded vector space. We have that: • the formal neighbourhood of π 0 π ≤0 DGrass k (W ) at a pair U ⊆ W is the deformation functor determined by the total Grassmannian associated to W ; • if furthermore W is concentrated in degree 0 then DGrass k (W ) is a derived scheme and the formal neighbourhood of its underived truncation at U ⊆ W is the deformation functor determined by the Grassmannian associated to W .
Proof. Take the derived stack DGrass k (W ), consider its formal neighbourhood at the pair (W, U ) and restrict to its underived 0-truncation π 0 π ≤0 DGrass (W,U ) ; the latter deformation functor is precisely the coarse dg-Grassmannian described in [8] Section 8 and Section 9 or in [5] Section 2.2, i.e. it is the deformation functor associated to the total Grassmannian variety: this observation completes the proof.

Notations and conventions
• diag (−) = diagonal of a bisimplicial set • k = fixed field of characteristic 0, unless otherwise stated • If A is a (possibly differential graded) local ring, m A will be its unique maximal (possibly differential graded) ideal • R = (possibly differential graded) commutative unital ring or k-algebra, unless otherwise stated • Alg k = category of commutative associative unital algebras over k • Alg R = category of commutative associative unital algebras over R • Alg H 0 (R) = category of commutative associative unital algebras over H 0 (R) • Ch ≥0 (Vect k ) = model category of chain complexes of k-vector spaces • dgAlg ≤0 k = model category of (cochain) differential graded commutative algebras over k in non-positive degrees • dgAlg ≤0 R = model category of (cochain) differential graded commutative algebras over R in non-positive degrees • dgAlg ≤0 R[t] = model category of (cochain) differential graded commutative algebras over R [t] in non-positive degrees • dgArt ≤0 k = model category of (cochain) differential graded local Artin algebras over k in non-positive degrees • dgMod R = model category of R-modules in (cochain) complexes • dgMod (X) = model category of derived modules over X • dgMod (X) cart = ∞-category of derived quasi-coherent sheaves over X • dg b Nil ≤0 R = ∞-category of bounded below differential graded commutative R-algebras in non-positive degrees such that the canonical map A → H 0 (A) is nilpotent H 0 (R) = ∞-category of bounded below differential graded commutative H 0 (R)algebras in non-positive degrees such that the canonical map A → H 0 (A) is nilpotent • dgVect ≤0 k = model category of (cochain) differential graded vector spaces over k in nonpositive degrees • FdgMod R = model category of filtered R-modules in (cochain) complexes