Constructive category theory and tilting equivalences via strong exceptional sequences

In this thesis we establish a constructive framework for homological algebra with a special focus on (complete) strong exceptional sequences in bounded homotopy categories and their induced exact equivalences, alongside with a CAP and homalg based implementation of this framework in the computer algebra system GAP.

First, we assemble the key concepts in homological algebra in a constructive style that is suitable for a direct computer implementation. This includes constructing bounded complexes, homotopy and derived categories in which we can perform computations like projective and injective resolutions of bounded complexes and derived functors. Then, we set the stage for performing computations in triangulated categories. This is accomplished by stating all the existential quantifiers and disjunctions in the defining axioms of a triangulated category as concrete algorithms. The two primary examples of a triangulated category in this thesis are the stable category of a Frobenius category and the bounded homotopy category of an additive category.

Given a field k, a k-linear Hom-finite additive category 𝓒 and a strong exceptional sequence 𝓔 in the bounded homotopy category 𝓚ᵇ(𝓒), we develop an algorithm to check the membership of objects in the triangulated hull 𝓔^△ ⊆ 𝓚ᵇ(𝓒). In particular, if 𝓚ᵇ(𝓒) is finitely generated as a triangulated category, one can employ that to algorithmically decide the completeness of the strong exceptional sequence 𝓔, i.e., decide whether 𝓔^△=𝓚ᵇ(𝓒). For a complete strong exceptional sequence 𝓔, we use the so-called Postnikov systems to provide an explicit construction of exact equivalences

𝓓ᵇ(End(T_𝓔)) ≃ 𝓚ᵇ(𝓔^⊕) ≃ 𝓚ᵇ(𝓒)

where T_𝓔 ≔ E₁⊕⋯⊕Eₙ,Eᵢ∈ 𝓔, 𝓓ᵇ(End(T_𝓔)) denotes the bounded derived category of the category End(T_𝓔)-mod of finitely generated End(T_𝓔)-modules, and 𝓔^⊕ is the universal additive closure category of 𝓔.

These techniques enable us to make the following special case of Happel's theorem for derived equivalences constructive:
Let 𝔸 be a finite dimensional k-algebra and T a tilting 𝔸-module whose indecomposable summands form a complete strong exceptional sequence in 𝔸-mod.
Then we can compute the induced adjoint derived equivalences

-⊗^𝕃 T : 𝓓ᵇ(End(T)) ⇄ 𝓓ᵇ(𝔸): ℝHom(T,-)

The categorical framework along with all algorithms presented in this thesis are implemented in the GAP meta-package HigherHomologicalAlgebra.