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We investigate gravity as a gauge theory in the language of fiber bundleswith tools from algebraic geometry. Compelled by the construction of theEilenberg-MacLane classifying space via Fox derivations in an integral groupring, the origin of locally curved spacetime as a an abelian group extension issought algebraically by means of a suitable derivation rule that conforms withthe algebraic content of Einstein's equation. Accordingly, a peculiar flavor ofbackground independence is advanced, curved spacetime should be thought of asan abelian proxy for the Lorentz group, causal and spin throughout. Thus,diffeomorphism invariance amounts to "choosing" an appropriatealgebro-geometric--differential tool as to make algebraic sense of Einstein'sequation or of a sensible evolution of such an equation. In this regard, it issuggested that a left-$\alpha$ derivation rule may facilitate the construction.On the other hand, we are led to interpret "spacetime curvature" in the broadsense of the exterior covariant derivative of a Cartan connection whichincludes both the torsion of the vierbein as well as the standard curvature ofthe spin connection. Making manifest use of the index structure in the standardHilbert action--in particular, the distinction between internal and externalindices--inclusion of the torsion 2-form is found to be inconsistent withcontraction of the spacetime curvature as a scalar density. Aiming to erect ageometrically justified gravitational action that incorporates torsion, we layout several fundamental geometric, differential and algebraic building blockswhile highlighting descent relations among the different pieces. The groundworkis thus laid out for the search of the sought after action functional.

Baby Steps in Quantum Ring Theory: towards a background independent framework for Quantum Gravity