Browsing by Author "Santos Cruz, Tiago Miguel dos"

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Algebraic analogues of resolution of singularities, quasi-hereditary covers and Schur algebras
(2021) Santos Cruz, Tiago Miguel dos; Koenig, Steffen (Prof. Dr.)
In algebraic geometry, a resolution of singularities is, roughly speaking, a replacement of a local commutative Noetherian ring of infinite global dimension by a local commutative Noetherian ring of finite global dimension. In representation theory, an analogous problem is asking to resolve algebras of infinite global dimension by algebras of finite global dimension. In addition, such resolutions should have nicer properties to help us study the representation theory of algebras of infinite global dimension. This motivates us to take split quasi-hereditary covers as these algebraic analogues of resolutions of singularities and measure their quality using generalisations of dominant dimension and deformation results based on change of rings techniques. The Schur algebra, SR(n,d) with n ≥ d, together with the Schur functor is a classical example of a split quasi-hereditary cover of the group algebra of the symmetric group, RSd , for every commutative Noetherian ring R. The block algebras of the classical category O, together with their projective-injective module, are split quasi-hereditary covers of subalgebras of coinvariant algebras. In this thesis, we study split quasi-hereditary covers, and their quality, of some cellular algebras over commutative Noetherian rings. The quality of a split quasi-hereditary cover can be measured by the fully faithfulness of the Schur functor on standard modules and on m-fold extensions of standard modules. Over fields, the dominant dimension controls the quality of the split quasi-hereditary cover of KSd formed by the Schur algebra SK(n,d) and the Schur functor. In particular, this quality improves by increasing the characteristic of the ground field. To understand the integral cases, the classical concept of dominant dimension is not useful since in most cases there are no projective-injective modules. Using relative homological algebra, we develop and study a new concept of dominant dimension, which we call relative dominant dimension, for Noetherian algebras which are projective over the ground ring making this concept suitable for the integral setup. For simplicity, we call Noetherian algebras which are projective over the ground ring just projective Noetherian algebras. While developing the theory of relative dominant dimension, we generalize the Morita-Tachikawa correspondence for projective Noetherian algebras and we prove that computations of relative dominant dimension over projective Noetherian algebras can be reduced to computations of dominant dimension over finite-dimensional algebras over algebraically closed fields. Using relative dominant dimension, concepts like Morita algebras and gendo-symmetric algebras can be defined for Noetherian algebras. We compute the relative dominant dimension of Schur algebras SR(n,d) for every commutative Noetherian ring R. Using such computations together with deformation results that involve the spectrum of the ground ring R we determine the quality of the split quasi-hereditary covers of RSd , (SR(n,d), V⊗d) formed by the Schur algebra SR(n,d) and the Schur functor HomSR(n,d)(V⊗d,−): SR(n,d)-mod → RSd-mod for all regular Noetherian rings. Over local commutative regular rings R, the quality of (SR(n,d), V⊗d) depends only on the relative dominant dimension and on R containing a field or not. For this cover, the quality improves compared with the finite-dimensional case whenever the local commutative Noetherian ring does not contain a field. This theory is also applied to q-Schur algebras and Iwahori-Hecke algebras of the symmetric group. In full generality, we prove that the quality of a split quasi-hereditary cover of a finite-dimensional algebra B is bounded above by the number of non-isomorphic simple B-modules. Other split quasi-hereditary algebras that we study in this thesis are deformations of block algebras of the Bernstein-Gelfand-Gelfand category O of a semi-simple Lie algebra. These deformations provide split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras. We compute the relative dominant dimensions of these block algebras and we determine the quality of these covers. In these deformations, the quality dramatically improves compared with the finite-dimensional case. Using approximation theory to generalize once more the concept of dominant dimension to relative dominant dimension with respect to direct summands of the characteristic tilting module, we find new split quasi-hereditary covers. In particular, the relative dominant dimension of a characteristic tilting module of SR(n,d) with respect to V⊗d is a lower bound of the quality of a split quasi-hereditary cover of the cellular algebra EndSR(n,d)(V⊗d)op, independent of the natural numbers n and d. This split quasi-hereditary cover involves the Ringel dual of the Schur algebra SR(n,d). Using this technology for deformations of block algebras of the classical BGG category O of a semi-simple Lie algebra, we obtain a new proof for Ringel self-duality of the blocks of the classical BGG category O of a complex semi-simple Lie algebra. Here, the uniqueness of split quasi-hereditary covers of deformations of subalgebras of coinvariant algebras with higher quality is the crucial factor to deduce Ringel self-duality.