The Jacobian ideal of a commutative ring and annihilators of cohomology

It is proved that for a ring $R$ that is either an affine algebra over a field, or an equicharacteristic complete local ring, some power of the Jacobian ideal of $R$ annihilates ${\mathrm{E}\mathrm{x}\mathrm{t}}_R^{d+1}(-,-)$, where $d$ is the Krull dimension of $R$. Sufficient conditions are identified under which the Jacobian ideal itself annihilates these Ext-modules, and examples are provided that show that this is not always the case. A crucial new idea is to consider a derived version of the Noether different of an algebra.