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This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring. [ABSTRACT FROM AUTHOR]

This work concerns a map \varphi \colon R\to S of commutative noetherian rings, locally of finite flat dimension. It is proved that the André-Quillen homology functors are rigid, namely, if \mathrm {D}_n(S/R;-)=0 for some n\ge 1, then \mathrm {D}_i(S/R;-)=0 for all i\ge 2 and {\varphi } is locally complete intersection. This extends Avramov's theorem that draws the same conclusion assuming \mathrm {D}_n(S/R;-) vanishes for all n\gg 0, confirming a conjecture of Quillen. The rigidity of André-Quillen functors is deduced from a more general result about the higher cotangent modules which answers a question raised by Avramov and Herzog, and subsumes a conjecture of Vasconcelos that was proved recently by the first author. The new insight leading to these results concerns the equivariance of a map from André-Quillen cohomology to Hochschild cohomology defined using the universal Atiyah class of \varphi. [ABSTRACT FROM AUTHOR]

We develop a support theory for elementary supergroup schemes, over a field of positive characteristic p ≥ 3, starting with a definition of a π-point generalising cyclic shifted subgroups of Carlson for elementary abelian groups and π-points of Friedlander and Pevtsova for finite group schemes. These are defined in terms of maps from the graded algebra k[t,τ]/(tp−τ2), where t has even degree and τ has odd degree. The strength of the theory is demonstrated by classifying the parity change invariant localising subcategories of the stable module category of an elementary supergroup scheme. [ABSTRACT FROM AUTHOR]

A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike previous proofs of these results. [ABSTRACT FROM AUTHOR]