Projective stable quotients of exact categories #
For an exact structure E, this file packages the objects that are both relatively projective
and relatively injective. It also defines E.ProjectiveStableCategory, the additive quotient by
morphisms factoring through relative projectives. When E is Frobenius, the projectives are
exactly the injectives, so this is the stable category obtained by killing the
projective-injective objects.
The quotient is formed with the general morphism-ideal API. In particular, it has the same objects as the original category, while a morphism becomes zero precisely when it factors through a relative projective. An object becomes zero precisely when it is relatively projective; under the Frobenius hypothesis this is equivalently relative injectivity or membership in the projective-injective class.
This additive category is the input to Happel's construction of the suspension and distinguished triangles. No triangulated structure is asserted here: that construction requires the full Frobenius hypothesis and choices of projective-injective conflations.
Main definitions #
TauCeti.ExactStructure.projectiveInjective: the objects that are both projective and injective relative to an exact structure.TauCeti.ExactStructure.ProjectiveInjective: their full subcategory.TauCeti.ExactStructure.projectiveStableIdeal: the ideal of maps generated by factorizations through relative projectives.TauCeti.ExactStructure.ProjectiveStableCategory: the quotient byprojectiveStableIdeal.TauCeti.ExactStructure.projectiveStableFunctor: the quotient functor to the projective stable category.TauCeti.ExactStructure.projectiveStableIsoBiprod: the isomorphismY ≅ P ⊞ Yin the stable category attached to a relatively projectiveP.
References #
- Dieter Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Chapter I, Section 2.
- Theo Bühler, Exact Categories, Expositiones Mathematicae 28 (2010), 1–69, https://arxiv.org/abs/0811.1480, Sections 11–13.
The objects that are both projective and injective relative to E. For a Frobenius exact
structure this agrees with either one of those two object properties.
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- E.projectiveInjective = E.isProjective ⊓ E.isInjective
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Membership in the projective-injective class means simultaneous relative projectivity and relative injectivity.
The projective-injective objects form a replete class stable under retracts.
The projective-injective class contains a zero object.
Binary biproducts of projective-injective objects are projective-injective.
The projective-injective class is closed under all finite products, hence under finite biproducts in the ambient preadditive category.
In a Frobenius exact structure, the projective-injective class is the class of relative projectives.
In a Frobenius exact structure, the projective-injective class is the class of relative injectives.
The full subcategory of objects that are both projective and injective relative to E.
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The ideal generated by morphisms factoring through relative projective objects. For a Frobenius exact structure these are exactly the projective-injective objects.
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The projective stable category of E, formed by quotienting morphisms by those factoring
through relative projectives. This is the projective-injective stable category when E is
Frobenius.
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The quotient functor from an exact category to its projective stable category.
Equations
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Membership in the projective stable ideal is equivalent to factoring through a relative projective.
In a Frobenius exact structure, the projective stable ideal is generated by factorizations through the projective-injective objects.
A morphism becomes zero in the projective stable category exactly when it factors through a relative projective.
For a Frobenius exact structure, a morphism becomes zero in the projective stable category exactly when it factors through a projective-injective object.
An object becomes zero in the projective stable category exactly when it is relatively projective.
For a Frobenius exact structure, an object becomes zero in the projective stable category exactly when it is projective-injective.
For a Frobenius exact structure, an object becomes zero in the projective stable category exactly when it is relatively injective.
A relatively projective summand is invisible in the projective stable category: the
biproduct inclusion Y ⟶ P ⊞ Y becomes an isomorphism there, with inverse the projection.
Equations
- One or more equations did not get rendered due to their size.
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The comparison with a relatively projective summand is the biproduct inclusion.
The inverse comparison with a relatively projective summand is the biproduct projection.
Two morphisms φ ψ : S ⟶ T of short complexes out of a conflation S, agreeing on first
terms, induce the same map on third terms in the projective stable category once the middle term
of T is relatively projective: φ.τ₃ - ψ.τ₃ factors through T.X₂.
Two morphisms φ ψ : S ⟶ T of short complexes into a conflation T, agreeing on third
terms, induce the same map on first terms in the projective stable category once the middle term
of S is relatively projective: φ.τ₁ - ψ.τ₁ factors through S.X₂.
A square on the first two terms which commutes in the stable quotient extends to a morphism of short complexes after changing only the middle map by a projectively trivial map. The source must be a conflation; the target need only be a short complex. It suffices that every relatively projective object is relatively injective.