Annihilation of Cohomology over Gorenstein Rings

One of the fundamental links between geometry and homological algebra is that smooth affine schemes have coordinate rings of finite global dimension. The roots of this link goes back to Hilbert’s syzygy theorem and later to the work of Auslander and Buchsbaum and also of Serre and Zariski. Having finite global dimension can be characterized by Ext-modules. Namely, a ring $R$ has finite global dimension if and only if there is a natural number $n$ such that $\Ext_R^n(M,N) = 0$ for every pair $M,N$ of $R$-modules. Hence, in the singular case, there are nonzero Ext-modules for arbitrarily large $n$. So, for a commutative Noetherian ring $R$, one is interested in the cohomology annihilator ideal which consists of the ring elements that annihilate all $\Ext$-modules for arbitrarily large $n$. The main theme of this thesis is to study the cohomology annihilator ideal over Gorenstein rings. Over Gorenstein rings, the cohomology annihilator ideal can be seen as the annihilator of the stable category of maximal Cohen-Macaulay modules. The first main result concerns the cohomology annihilator ideal of a complete local coordinate ring of a reduced algebraic plane curve singularity. We show that the cohomology annihilator ideal coincides with the conductor ideal in this case. We use this to investigate the relation between the Jacobian ideal and the cohomology annihilator ideal. The second main result shows that if the Krull dimension of $R$ is at most $2$, then the cohomology annihilator ideal is equal to the stable annihilator ideal of a non-singular $R$-order. We also give several generalizations of this which brings us to the second part and the closing section of this thesis. Namely, we study the dominant dimension of orders over Cohen-Macaulay rings. We provide examples and prove results on tilting modules for orders with positive dominant dimension.Ph.D