Generalising the Cohen-Macaulay condition and other homological properties

This thesis concerns some homological properties for noetherian rings which are finite modules over their centres, but we look most particularly at the Cohen-Macaulay property. We look at ways if generalising these homological properties, either from the commutative to the noncommutative case, or from finite dimensional $k$-algebras to rings which are finite modules over a central subring. Chapter 1 contains known background material as well as some preliminary results to be used in later proofs. We consider, in Chapter 2, some generalisations of the well-known Cohen-Macaulay property for commutative rings. We focus on the centrally-Macaulay property as defined by Brown, Hajarnavis and MacEacharn and what we call Krull-Macaulay, a stronger and more homological condition. Centrally-Macaulay rings are defined in terms of a central subring and we consider the extent to which the property is dependent on the choice of central subring. We then consider generalisations of a commutative result relating the Cohen-Macaulay property to freeness over a regular subring before applying the Cohen-Macaulay condition to reconstruction algebras, as defined recently by Wemyss and Craw. Given that there is more than one generalisation of the Cohen-Macaulay property, we seek, in Chapter 3, to find the best generalisation to the particular class of rings we study. That is, noetherian rings which are finite modules over their centre. Thus we compare the two properties, centrally-Macaulay and Krull-Macaulay, and variants of them. In particular, we combine centrally-Macaulay with the symmetry of homological grade to obtain a property which is equivalent to Krull-Macaulay for equidimensional rings. We suggest that this property, which we call symmetrically-Macaulay, is the best generalisation in this case. We then go on to consider, in Chapter 4, generalisations of related properties for commutative rings, regular and Gorenstein and generalise the commutative hierarchy: regular implies Gorenstein which implies Cohen-Macaulay. Finally, in Chapter 5, we demonstrate the significance of the module $\Hom_C(R,C)$ for any central subring $C$ over which $R$ is finitely generated. This leads us to generalise the definition of a symmetric algebra and using our generalisation we are able to generalise some results of Braun in. We finish with showing when skew group algebras are symmetric