Benson's cofibrants, Gorenstein projectives and a related conjecture

In this short article, we will be principally investigating two classes of modules over any given group ring – the class of Gorenstein projectives and the class of Benson's cofibrants. We begin by studying various properties of these two classes and studying some of these properties comparatively against each other. There is a conjecture made by Fotini Dembegioti and Olympia Talelli that these two classes should coincide over the integral group ring for any group. We make this conjecture over group rings over commutative rings of finite global dimension and prove it for some classes of groups while also proving other related results involving the two classes of modules mentioned.

Gabriel–Roiter Measure, Representation Dimension and Rejective Chains

The Gabriel–Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel–Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel–Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel–Roiter measure $\mu$ in an abelian length category $\mathcal{A}$, there exists an object $X^{\prime}$ that depends on $X$ and $\mu$, such that $\Gamma =\operatorname{End}_{\mathcal{A}}(X\oplus X^{\prime})$ has finite global dimension. Analogously to Iyama’s original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

Rigidity dimension of algebras

A new homological dimension, called rigidity dimension, is introduced to measure the quality of resolutions of finite dimensional algebras (especially of infinite global dimension) by algebras of finite global dimension and big dominant dimension. Upper bounds of the dimension are established in terms of extensions and of Hochschild cohomology, and finiteness in general is derived from homological conjectures. In particular, the rigidity dimension of a non-semisimple group algebra is finite and bounded by the order of the group. Then invariance under stable equivalences is shown to hold, with some exceptions when there are nodes in case of additive equivalences, and without exceptions in case of triangulated equivalences. Stable equivalences of Morita type and derived equivalences, both between self-injective algebras, are shown to preserve rigidity dimension as well.

The finiteness of the Gorenstein dimension for Artin algebras

In [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478], the authors proved that an Artin algebra [Formula: see text] with infinite global dimension has an indecomposable module with infinite projective and infinite injective dimension, giving a new characterization of algebras with finite global dimension. We prove in this paper that an Artin algebra [Formula: see text] that is not Gorenstein has an indecomposable [Formula: see text]-module with infinite Gorenstein projective dimension and infinite Gorenstein injective dimension, which gives a new characterization of algebras with finite Gorenstein dimension. We show that this gives a proper generalization of the result in [A. Skowronski, S. Smalø and D. Zacharia, On the finiteness of the global dimension for Artinian rings, J. Algebra 251(1) (2002) 475–478] for Artin algebras.

Reduction of triangulated categories and maximal modification algebras for cAn singularities

In this paper we define and study triangulated categories in which the Hom-spaces have Krull dimension at most one over some base ring (hence they have a natural 2-step filtration), and each factor of the filtration satisfies some Calabi–Yau type property. If \mathcal{C} is such a category, we say that \mathcal{C} is Calabi–Yau with \dim\mathcal{C}\leq 1 . We extend the notion of Calabi–Yau reduction to this setting, and prove general results which are an analogue of known results in cluster theory. Such categories appear naturally in the setting of Gorenstein singularities in dimension three as the stable categories \mathop{\underline{\textup{CM}}}R of Cohen–Macaulay modules. We explain the connection between Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R and both partial crepant resolutions and \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R , and we show under quite general assumptions that Calabi–Yau reductions exist. In the remainder of the paper we focus on complete local cA_{n} singularities R. By using a purely algebraic argument based on Calabi–Yau reduction of \mathop{\underline{\textup{CM}}}R , we give a complete classification of maximal modifying modules in terms of the symmetric group, generalizing and strengthening results in [I. Burban, O. Iyama, B. Keller and I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 2008, 6, 2443–2484], [H. Dao and C. Huneke, Vanishing of Ext, cluster tilting and finite global dimension of endomorphism rings, Amer. J. Math. 135 2013, 2, 561–578], where we do not need any restriction on the ground field. We also describe the mutation of modifying modules at an arbitrary (not necessarily indecomposable) direct summand. As a corollary when k=\mathbb{C} we obtain many autoequivalences of the derived category of the \mathbb{Q} -factorial terminalizations of \operatorname{Spec}R .

Projective lattices of tiled orders

Tiled orders over discrete valuation ring have been studied since the 1970s by many mathematicians, in particular, by Yategaonkar V.A., Tarsy R.B., Roggenkamp K.W, Simson D., Drozd Y.A., Zavadsky A.G. and Kirichenko V.V. Yategaonkar V.A. proved that for every n > 2, there is, up to an isomorphism, a finite number of tiled orders over a discrete valuation ring O of finite global dimension which lie in $M_n(K)$ where K is a field of fractions of a commutatively discrete valuation ring O. The articles by R.B. Tarsy, V.A. Yategaonkar, H. Fujita, W. Rump and others are devoted to the study of the global dimension of tiled orders. H. Fujita described the reduced tiled orders in Mn(D) of finite global dimension for n = 4; 5. V.M. Zhuravlev and D.V. Zhuravlev described reduced tiled orders in Mn(D) of finite global dimension for n = 6: This paper examines the necessary condition for the finiteness of the global dimension of the tile order. Let A be a tiled order. The kernel of the projective resolvent of an irreducible lattice has the form M1f1 +M2f2 + ::: +Msfs, where Mi is irreducible lattice, fi is some vector. If the tile order has a finite global dimension, then there is a projective lattice that is the intersection of projective lattices. This condition is the one explored in the paper.

Global dimension of Harada rings and serial rings

Baba [On Harada rings and quasi-Harada rings with left global dimension at most 2. Comm. Algebra28(6) (2000) 2671–2684] proved that every left Harada rings with global dimension at most 2 is a serial ring. In this paper, improving the result, we show that every left Harada ring with global dimension at most 3 is a serial ring. We also prove that if a left Harada ring A of finite global dimension is of type (*) or has homogeneous right socle, then A is serial. Finally, we give an example of a non-serial left Harada ring of finite global dimension.