Maps from acyclic complexes to complexes of injectives #
Let E be an exact structure. A map φ : A ⟶ I from an E-acyclic complex to a bounded below
complex of E-injective objects is null-homotopic. This is the dual of Bühler's Corollary 12.5 (no
maps from right bounded complexes of projectives to acyclic complexes), and the basic input for
computing morphisms into complexes of injectives in the derived category of E.
The null-homotopy hⁿ : Aⁿ⁺¹ ⟶ Iⁿ is built upwards from the lower bound of I. Write the
differentials of A as Aⁿ ↠ Zⁿ⁺¹ ↣ Aⁿ⁺¹. If φⁿ⁺¹ - hⁿ ≫ d vanishes on Zⁿ⁺¹, it descends to
Zⁿ⁺² ⟶ Iⁿ⁺¹, whose extension across Zⁿ⁺² ↣ Aⁿ⁺² into the injective Iⁿ⁺¹ is hⁿ⁺¹; the next
difference φⁿ⁺² - hⁿ⁺¹ ≫ d again vanishes on Zⁿ⁺², since Aⁿ⁺¹ ↠ Zⁿ⁺² is an epimorphism.
Main definitions #
TauCeti.ExactStructure.AcyclicData.homotopyStep: one step of a null-homotopy from an acyclic complex into an injective term, withd_comp_homotopyStepandi_comp_sub_homotopyStep_comp_d. It is shared with the comparison theorem for complete resolutions.TauCeti.ExactStructure.AcyclicData.homotopyZero: the null-homotopy, forIstrictly bounded below at0.
Main results #
TauCeti.ExactStructure.isInjective_mappingCone_X: the mapping cone of a map between complexes of injectives is a complex of injectives.TauCeti.ExactStructure.AcyclicData.quotient_map_eq_zero: a map from an acyclic complex to a bounded below complex of injectives vanishes in the homotopy category.
References #
- Theo Bühler, Exact categories, Expositiones Mathematicae 28 (2010), 1--69, https://arxiv.org/abs/0811.1480, Corollary 12.5, of which this is the dual.
The terms of the mapping cone of a map between complexes of injectives are injective.
One step of a null-homotopy into an injective object. Let h : Aⁿ⁺¹ ⟶ Iⁿ be a component
such that the correction φⁿ⁺¹ - h ≫ d vanishes on the cycles Zⁿ⁺¹ ↣ Aⁿ⁺¹. The correction then
descends along Aⁿ⁺¹ ↠ Zⁿ⁺², and its extension across Zⁿ⁺² ↣ Aⁿ⁺² into the injective Iⁿ⁺¹ is
the next component Aⁿ⁺² ⟶ Iⁿ⁺¹. Iterating this step builds null-homotopies of maps from acyclic
complexes into complexes of injectives.
Instances For
The homotopy identity of a step: d ≫ hⁿ⁺¹ + hⁿ ≫ d = φⁿ⁺¹.
The homotopy identity of a step: d ≫ hⁿ⁺¹ + hⁿ ≫ d = φⁿ⁺¹.
The next correction φⁿ⁺² - hⁿ⁺¹ ≫ d again vanishes on the cycles, so the step can be
repeated.
Maps from acyclic complexes to complexes of injectives are null-homotopic: a map from an
E-acyclic complex to a complex of E-injective objects that vanishes in negative degrees.
Instances For
Maps from acyclic complexes to bounded below complexes of injectives vanish in the homotopy
category: the dual of Bühler's Corollary 12.5. The bound is moved to 0 by shifting.