Profundidad de las estanterías en las tiendas

We briefly sketch the definition of coderived categories of linear cofactorizations and cocurved mixed complexes, following [Pos11, §4]. As for algebras, these exist for cate- gories of comodules over arbitrary curved differential graded (cdg) k-coalgebras and we can construct them within the framework of abelian model categories.

In the following, let (Γ, | · |) be a grading group as in Notation II.2.3.1. A curved differential graded (cdg)k-coalgebra is a Γ-graded k-coalgebra C together with a map d : C →ΣC of Γ-gradedk-modules and an element ω ∈ (C−2₎∗_{such that d}2_{(x) = x∗ω−ω∗x}

for all x ∈ C. Given such a cdg k-coalgebra C, a (cdg) comodule over C is a Γ-graded comodule M over the Γ-graded k-coalgebra C] _{underlying C together with a degree 1}

endomorphism d : M → ΣM which is a coderivation with respect to the coaction of C and satisfies d2 = ω ∗ −. The forgetful functor C -coMod → C]_{-coMod is cocontinuous}

and monadic and has exact left and right adjoints G±as in Proposition II.2.3.2 from Part II. In particular, C -coMod is a locally Noetherian Grothendieck category as follows from Lemma II.B.3 and the fact that C]_{-coMod is locally Noetherian Grothendieck. Finally,}

denote C -coModinj the class of those C-comodules whose underlying C]-comodules are

Chapter I.7. Variations

Proposition I.7.1.11. Let k be a field and C be a Γ-graded cdg k-coalgebra. Then there exists a unique abelian model structure Mco(C) = (C, W, F) on C -coMod with C = C -coMod and F = C -coModinj. Moreover, Mco(C) is cofibrantly generated.

What’s more, Positselski’s constructive description of coacyclic modules – which was available in the ring-theoretic context only under some Noetherianness assumption on the ring (see Proposition II.2.3.13 from Part II) – holds without any conditions on C: Proposition I.7.1.12. In the context of Proposition I.7.1.11 the class of coacyclic cdg C-comodules is the smallest thick subcategory of C-coMod which is closed under coprod- ucts and contains the totalizations of short exact sequences of C-comodules as well as all contractible C-comodules.

Proof. See [Pos11, §3.6 and §4.4].

In Examples I.7.1.13, I.7.1.14 we fix a cocommutativek-coalgebra C and ω ∈ C∗. Example I.7.1.13. The following is analogous to the paragraph following Definition I.2.1.1: Linear cofactorizations of type (C, ω) can be identified with comodules over the Z/2Z-graded curved k-coalgebra Cω given by (Cω)0 := Cω, (Cω)1 := 0 and cocur-

vature ω. In particular, there is a unique cofibrantly generated projective abelian model structure on cLF(C, ω) with cMF(C, ω) as the class of fibrant objects. Its ho- motopy category will be called the coderived category of linear cofactorizations, denoted

Dco_{cLF(C, ω). ♦}

Example I.7.1.14. The following is analogous to the paragraph following Definition I.2.1.7: cocurved mixed complexes of type (C, ω) are comodules over the Z-graded Koszul-coalgebra K(C, ω) that we describe now: Firstly, the Z-graded k-module un- derlying K(C, ω) is given by K(C, ω)0 _{:= C, K(C, ω)}1 _{:= C and K(C, ω)}i _{:= 0 for}

i 6= 0, 1. Secondly, the comultiplication ∆ : K(C, ω) → K(C, ω) ⊗_kK(C, ω) is given by ∆0 := K(C, ω)0= C −∆→ C ⊗_kC= K(C, ω)0⊗_kK(C, ω)0= (K(C, ω) ⊗_kK(C, ω))0, ∆1 := K(C, ω)1= C  ∆ ∆  −−−−−→ (C ⊗_kC) ⊕ (C ⊗_kC) = K(C, ω)1_⊗ kK(C, ω)0
⊕ K(C, ω)0⊗kK(C, ω)1
⊂ (K(C, ω) ⊗kK(C, ω))1,

and finally, the differential is given by d0 : K(C, ω)0 _{= C −→ C = K(C, ω)}∗ω 1_{. We}

elaborate on K(C, ω) -coMod ∼= cMC(C, ω); it is only a matter of writing out definitions, but since working with comodules doesn’t seem to be very common, we give some details: First, we claim that if X is a Z-graded k-module, endowing it with the structure of a comodule over K(C, ω)] _{is equivalent to providing each component of X with the}

structure of a C-comodule and to choosing a C-colinear map s : X → ΩX with s2 _{= 0.}

I.7.1. Avoiding technicalities II: Working with comodules For this, suppose X is endowed with a K(C, ω)]_{-comodule structure. Then, since the}

projection of K(C, ω)]_{onto its degree 0 component C is a map of Z-graded k-coalgebras,}

any component of X inherits a C-comodule structure by composing the coaction of K(C, ω)] _{with K(C, ω)}] _{→ C; this is analogous to restriction of module structures}

to subrings. Further, the coassociativity of the K(C, ω)]_{-coaction implies that X →}

K(C, ω)]_⊗

kX → K(C, ω)1⊗kΩX = C ⊗kΩX is C-colinear (here the right hand side is

endowed with the cofree C-comodule structure), hence of the form X −→ ΩXs −∆→ C ⊗_kΩX for some unique k-linear s : X → ΩX. Since ∆ : ΩX → C ⊗_kΩX is C-colinear and injective, it follows that s : X → ΩX is even C-colinear. The coaction of K(C, ω)] _is

therefore given by the composition X −−−→ (C ⊗(∆ s) k X) ⊕ ΩX (id ∆⊗id) −−−−−−→ (C ⊗ k X) ⊕ (C ⊗k ΩX) = K(C, ω) ] _⊗ k X, (I.7.1.1)

the coassociativity of which implies that s2 _{= 0. Conversely, any family of C-comodule}

structures on the components on X together with a C-colinear map s : X → ΩX satisfying s2_{= 0 gives rise to a K(C, ω)}]_{-coaction through (I.7.1.1). Finally, given such a}

K(C, ω)]_{-comodule X, ak-linear map d : X → ΣX is a K(C, ω)}]_{-coderivation if and only}

if d s + s d = ∗ω (details omitted), finishing the sketch of K(C, ω) -coMod ∼= cMC(C, ω). In particular, we get a projective abelian model structureMco_{cMC(C, ω) on cMC(C, ω)}

in which the fibrant objects are those ω-cocurved mixed complexes (X, d, s) for which (X, s) is an injective K(C, ω)]_{-module, i.e. (by the analogue of Lemma II.2.3.3 from}

Part II for cdgk-coalgebras) for which (X, s) is contractible with injective components. The homotopy category of Mco_{cMC(C, ω) will be called the coderived category of ω-}

cocurved mixed complexes, denoted Dco_{cMC(C, ω). Analogously to Proposition I.4.1.10}

we have that bounded below acyclic cocurved mixed complexes are coacyclic, see [Pos11,

Theorem 4.3.1]. ♦

Now we can ripe the fruits of our detour to comodules. To begin, the cotensor product functors for linear cofactorizations and cocurved mixed complexes give rise to functors

− R

C − : DcocLF(C, ω) × DcocLF(C, ω0) → DcocLF(C, ω + ω0), (I.7.1.2)

− RC − : DcocMC(C, ω) × DcocMC(C, ω0) → DcocMC(C, ω + ω0), (I.7.1.3)

defined by taking fibrant resolutions in one of the two factors. This is well-defined since, as explained in [Pos11, §4.7], the coflatness of cofree comodules and the explicit description of coacyclic comodules from Proposition I.7.1.12 show that the cotensor product of a coacyclic C-comodule with a C]_{-injective C-comodule is again coacyclic.}

Next, we come to the main point why we switched to comodules: the folding by sums functor can be computed naively and commutes with coderived tensor products:

Chapter I.7. Variations

Proposition I.7.1.15. The folding by sums functor fold⊕ : cMC(C, ω) → cLF(C, ω) is a left Quillen functor McocMC(C, ω) → Mco_{cLF(C, ω). In particular, it descends}

naively to a functor Lfold⊕: Dco_{cMC(C, ω) → D}co_{cLF(C, ω).}

Proof. We have to show that fold⊕ preserves cofibrations and trivial cofibrations. By definition, the cofibrations inMco_{cMC(C, ω) and M}co_{cLF(C, ω) are just the monomor-}

phisms, while the trivial cofibrations are the monomorphisms with coacyclic cokernel. Since fold⊕ is exact it is therefore sufficient to show that it maps coacyclic cocurved mixed complexes to coacyclic linear cofactorizations, which in turn follows from the ex- plicit description of coacyclic comodules from Proposition I.7.1.12 and the observation that fold⊕ commutes both with coproducts and with totalizations.

As in the case of modules, Koszul cocurved complexes and Koszul cofactorizations can often be described in terms of single comodules, up to coderived equivalence. For this, we need the analogue of the notion of a regular sequence in the context of comodules: Definition I.7.1.16. Let C be a cocommutative k-coalgebra and M be a C-comodule. An element α ∈ C∗ is called M-coregular if M −→ M is surjective. Inductively, a∗α sequence α1, . . . , αn ∈ C∗ is M -coregular if either n = 0 or if α1 is M -coregular and

α2, . . . , αn is coregular for the C-comodule ker(M ∗α1

−−→ M ).

Remark I.7.1.17. Definition I.7.1.16 is in agreement with the notion of coregular se- quences introduced in [Tan04, §2] for Artinian modules: namely, if C is thek-coalgebra from Example I.7.1.2 the dual of which is the power series ringkJt1, . . . , tnK, then a se- quence α1, . . . , αn∈ C∗ =kJt1, . . . , tnK is C -coregular in the sense of Definition I.7.1.16 if and only if it is coregular for the Artinian C∗-module C ∼= E(k) ∼= Hnm(kJt1, . . . , tnK)

in the sense of [Tan04]. ♦

Fact I.7.1.18. If in the situation of Definition I.7.1.16 the sequence α= (α1, . . . , αn)

is M -coregular and ifK(M ; α) :=Nn

i=1



M −−→ M∗αi is the Koszul complex of α, then the canonical morphism n T i=1 (M ∗αi −−→ M ) → K(M ; α) is a quasi-isomorphism.

Proposition I.7.1.19. Let C be a cocommutativek-coalgebra, α = (α1, . . . , αn) be a C-

coregular sequence in C∗and β= (β1, . . . , βn) be any sequence in C∗. Then, putting ω:=

P

iαiβi ∈ C∗, the canonical morphismω

 _n T i=1  C−−→ C∗αi   → {α, β} is an isomorphism in DcocLF(C, ω) and Dco_{cMC(C, ω).}