The decomposition of global conformal invariants IV: A proposition on local Riemannian invariants

This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of "global conformal invariants"; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in the first two papers in this series, see Alexakis (2007, 2009) [2,3]. These results may be of independent interest, applicable to related problems. (C) 2010 Elsevier Inc. All rights reserved.