Finite resolutions in a graded exact category

In a graded exact category the grading shift {1} and its inverse are conflation-exact, so they carry finite resolutions to finite resolutions. When the resolving class P is stable under the shift, the shift of a finite P-resolution of X is a finite P-resolution of X{1}, with the same length and the internal degrees of all its terms raised by one.

Consequently the objects of finite P-dimension form a shift-stable class as well. This is the hypothesis under which the full subcategory of such objects carries the induced graded exact structure of TauCeti.GradedExactStructure.fullSubcategory, so that the resolution theorem can be stated between graded Grothendieck groups.

Main results

• TauCeti.GradedExactStructure.admitsFiniteResolution_inverseImage_shift: if P is stable under the grading shift, then so is the property of admitting a finite P-resolution.

References

• Charles A. Weibel, The K-book: An Introduction to Algebraic K-theory, Chapter II, Section 7, for finite resolutions and the resolution theorem.
• Zsuzsanna Dancso and Anthony Licata, "Koszul algebras and flow lattices", Journal of Combinatorial Theory, Series A 185 (2022), Section 2.2, for grading shifts on exact categories.

theorem TauCeti.GradedExactStructure.admitsFiniteResolution_inverseImage_shift {C : Type u} [CategoryTheory.Category.{v, u} C] [CategoryTheory.Preadditive C] [CategoryTheory.Limits.HasZeroObject C] [CategoryTheory.Limits.HasBinaryBiproducts C] (E : GradedExactStructure C) {P : CategoryTheory.ObjectProperty C} [P.IsClosedUnderIsomorphisms] (hshift : P.inverseImage E.shift.functor = P) : (E.admitsFiniteResolution P).inverseImage E.shift.functor = E.admitsFiniteResolution P

Shift stability of finite P-dimension. If an object satisfies P exactly when its shift does, then an object admits a finite P-resolution exactly when its shift does: a resolution is carried across by the shift in one direction, and by the inverse shift followed by transport along the unit isomorphism in the other.