The flat model structure on 𝐂𝐡(𝐑)

Gillespie, James

doi:10.1090/s0002-9947-04-03416-6

Cited by 182 publications

(219 citation statements)

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“…Proof. Since a Grothendieck category always has enough injectives, Theorem 3.12 from [Gil04] tells us that dg A ∩ E = A. By Proposition 3.6 we know (dg A, B) has enough injectives.…”

Section: Lemma 35 Let G Be a Grothendieck Category And Let (A B) Bmentioning

confidence: 92%

“…The assumption that G has enough A-objects is only used to guarantee that (dg A, B) is indeed a cotorsion pair. (See Corollary 3.8 of [Gil04]. )…”

Section: Lemma 35 Let G Be a Grothendieck Category And Let (A B) Bmentioning

confidence: 96%

“…Now let G be a Grothendieck category and let (A, B) be a cotorsion pair. Corollary 3.8 of [Gil04] If a cotorsion pair (A, B) is hereditary and our category D has enough projectives and injectives, then dg A ∩ E = A and dg B ∩ E = B, where E be the class of all exact chain complexes [Gil04]. In this case we say the induced cotorsion pairs are compatible, and we have cotorsion pairs (dg A, dg B ∩ E) and (dg A ∩ E, dg B).…”

Section: Preliminariesmentioning

confidence: 99%

“…Furthermore, in case D is a closed symmetric monoidal category, Hovey gives conditions on the classes Q and R which guarantee that the model structure is monoidal. So to find the flat model structure we will need to come up with the classes Q and R. This leads us to the following definition which first appeared in [Gil04]. Definition 2.2.…”

Section: Preliminariesmentioning

confidence: 99%

“…It was shown in [Gil04] that there is a new monoidal model structure on Ch(R) which we call the "flat model structure". The point of this paper is to generalize this result to Ch(O).…”

Section: Introductionmentioning

confidence: 99%

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The flat model structure on complexes of sheaves

Gillespie

2006

Trans. Amer. Math. Soc.

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Abstract. Let Ch(O) be the category of chain complexes of O-modules on a topological space T (where O is a sheaf of rings on T ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category Sh(O) of O-modules. I.e., we have a natural way to define the functors Ext and Tor in Sh(O).

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“…Proof. Since a Grothendieck category always has enough injectives, Theorem 3.12 from [Gil04] tells us that dg A ∩ E = A. By Proposition 3.6 we know (dg A, B) has enough injectives.…”

Section: Lemma 35 Let G Be a Grothendieck Category And Let (A B) Bmentioning

confidence: 92%

“…The assumption that G has enough A-objects is only used to guarantee that (dg A, B) is indeed a cotorsion pair. (See Corollary 3.8 of [Gil04]. )…”

Section: Lemma 35 Let G Be a Grothendieck Category And Let (A B) Bmentioning

confidence: 96%

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…It was shown in [Gil04] that there is a new monoidal model structure on Ch(R) which we call the "flat model structure". The point of this paper is to generalize this result to Ch(O).…”

Section: Introductionmentioning

confidence: 99%

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The flat model structure on complexes of sheaves

Gillespie

2006

Trans. Amer. Math. Soc.

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A model structure approach to the finitistic dimension conjectures

Cortés-Izurdiaga

Estrada

Asensio

2012

Mathematische Nachrichten

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We explore the interlacing between model category structures attained to classes of modules of finite Xdimension, for certain classes of modules X . As an application we give a model structure approach to the finitistic dimension conjectures and present a new conceptual framework in which these conjectures can be studied.Let Λ be a finite dimensional algebra over a field k (or more generally, let Λ be an Artin ring). The big finitistic dimension of Λ, Findim(Λ), is defined as the supremum of the projective dimensions of all modules having finite projective dimension. And the little finitistic dimension of Λ, findim(Λ), is defined in a similar way by restricting to the subclass of all finitely generated modules of finite projective dimension. In 1960, Bass stated the so-called Finitistic Dimension Conjectures: (I) Findim(Λ) = findim(Λ), and (II) findim(Λ) is finite. The first conjecture was proved to be false by Huisgen-Zimmermann in [19], but the second one still remains open. It has been proved to be true, for instance, for finite-dimensional monomial algebras [16], for Artin algebras with vanishing cube radical [20], or Artin algebras with representation dimension bounded by 3 [22].In [21] and [5], Huisgen-Zimmermann, Smalø, Auslander and Reiten proved that the finitistic dimension conjectures hold for Artin algebras in which the class P < ∞ of all finitely generated modules of finite projective dimension is contravariantly finite (equivalently, it is a precovering class in the sense of [9], [15]). In general, P < ∞ does not need to be contravariantly finite, even for Artin algebras satisfying the finitistic dimension conjectures. But, as Angeleri-Hügel and Trlifaj have noticed in [3], it cogenerates a cotorsion pair (F, C) in which the class F is precovering in R -Mod. By means of this idea, the authors are able to extend Auslander-Reiten's approach to arbitrary artinian rings and obtain a general criterium for an artinian ring to satisfy the finitistic dimension conjectures in terms of Tilting Theory (see [3]). This type of arguments has also been recently extended to more general homologies induced by arbitrary hereditary cotorsion pairs (see [1]).On the other hand, Hovey has recently shown in [18] that there exists a quite strong relation between the construction of hereditary cotorsion pairs in module categories and the existence of model structures in the sense of Quillen in the associated categories of unbounded chain complexes. Moreover, under very general hypotheses, the cohomology functors defined from these model structures coincide with the absolute cohomology functors defined from the injective model structure (in the sense of [18, Example 3.2])]. Recall that a model category is a category with three distinguished classes of morphisms (fibrations, cofibrations and weak equivalences) satisfying a certain number of axioms. We refer to [17] for a complete definition and main properties of model categories. One of the main advantages of these model categories is that they allow the construction of the ...

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