08 Fakultät Mathematik und Physik
Permanent URI for this collectionhttps://elib.uni-stuttgart.de/handle/11682/9
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Item Open Access From simplicial groups to crossed squares(2022) Asiki, Natalia-MariaSimplicial groups are defined to be contravariant functors from the simplex category to the category of groups. The truncation functor maps a simplicial group to a [2,0]-simplicial group, satisfying the Conduché condition. A functor from the category of [2,0]-simplicial groups to the category of crossed squares is constructed, following Porter. It is shown that the latter functor is not an equivalence of categories. In addition, Loday's variant of the resulting crossed square is constructed and shown to be isomorphic to Porter's variant.Item Open Access Gendo-Frobenius algebras and comultiplication(2022) Yırtıcı, ÇiğdemGendo-Frobenius algebras are a common generalisation of Frobenius algebras and of gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius and on gendo-symmetric algebras. In addition, Frobenius algebras are shown to be precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication. Moreover, a new characterisation of gendo-Frobenius algebras is given. This new characterisation is a key for constructing the comultiplication of gendo-Frobenius algebras.Item Open Access 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.Item Open Access Desarguesian and geometric right ℓ-groups(2021) Dietzel, Carsten; König, Steffen (Prof. Dr.)Item Open Access On quiver Grassmannians and orbit closures for gen-finite modules(2021) Pressland, Matthew; Sauter, JuliaWe show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.Item Open Access The stable module category inside the homotopy category, perfect exact sequences and equivalences(2021) Nitsche, Sebastian; König, Steffen (Prof. Dr.)We consider the functor from the stable module category to the homotopy category constructed by Kato. This functor gives an equivalence between the stable module category and a full subcategory L of the unbounded homotopy category of projective modules. Moreover, the functor induces a correspondence between distinguished triangles in the homotopy category and perfect exact sequences in the module category. In general, the stable module category and the category L are not triangulated. We provide a description of a triangulated hull of L inside the homotopy category and discuss its Grothendieck group. We also construct a larger subcategory which is shown to be characteristic inside the homotopy category under suitable assumptions. Both subcategories coincide with L if and only if the algebra is self-injective. Furthermore, stable equivalence of Morita type are shown to preserve both subcategories. Another focus is put on the relationship between stable equivalences and perfect exact sequences. On the one hand, we give sufficient conditions for a stable equivalence to preserve perfect exact sequences up to projective direct summands. A stable equivalence which preserves perfect exact sequences in this way is shown to induce a triangulated equivalence between the categories of stable Gorenstein-projective modules. On the other hand, given a stable equivalence that is induced by an exact functor, we provide various sufficient conditions under which the equivalence is a stable equivalence of Morita type. In particular, stable equivalences of Morita type arise from equivalences that are given by tensoring with an arbitrary bimodule on the level of the category L. Finally, we give a description of all algebras that can be obtained by deleting or inserting nodes via stable equivalences constructed by Koenig and Liu.Item Open Access Dominant and global dimension of blocks of quantised Schur algebras(2021) Fang, Ming; Hu, Wei; Koenig, SteffenGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and Sq(n,r) with n⩾r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59-85, 2021).Item Open Access Brauer graph algebras are closed under derived equivalence(2022) Antipov, Mikhail; Zvonareva, AlexandraIn this paper the class of Brauer graph algebras is proved to be closed under derived equivalence. For that we use the rank of the maximal torus of the identity component Out0(A)of the group of outer automorphisms of a symmetric stably biserial algebra A .Item Open Access Gendo-Frobenius algebras and comultiplication(2020) Yirtici, Cigdem; Koenig, Steffen (Prof. Dr.)Two large classes of algebras, Frobenius algebras and gendo-symmetric algebras, are characterised by thenexistence of a comultiplication with some special properties. Symmetric algebras are both Frobenius and gendo-symmetric. Kerner and Yamagata investigated two variations of gendo-symmetric algebras and in fact these two variations contain gendo-symmetric and Frobenius algebras. We call one of these variations gendo-Frobenius algebras. In this thesis, we construct a comultiplication for gendo-Frobenius algebras, which specialises to the known comultiplications on Frobenius algebras and on gendo-symmetric algebras. Moreover, we show that Frobenius algebras are precisely those gendo-Frobenius algebras that have a counit compatible with this comultiplication.Item Open Access Lifting and restricting t‐structures(2022) Marks, Frederik; Zvonareva, AlexandraWe explore the interplay between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring 𝐴 using homotopy colimits. More precisely, we show that every intermediate t-structure in 𝐷𝑏(mod(𝐴)) can be lifted to a compactly generated t-structure in 𝐷(Mod(𝐴)), by closing the aisle and the coaisle of the t-structure under directed homotopy colimits. Conversely, we provide necessary and sufficient conditions for a compactly generated t-structure in 𝐷(Mod(𝐴)) to restrict to an intermediate t-structure in 𝐷𝑏(mod(𝐴)), thus describing which t-structures can be obtained via lifting. We apply our results to the special case of HRS t-structures. Finally, we discuss various applications to silting theory in the context of finite dimensional algebras.