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We consider the Arveson-Douglas conjecture on the essential normality ofhomogeneous submodules corresponding to algebraic subvarieties of the unitball. We prove that the property of essential normality is preserved byisomorphisms between varieties, and we establish a similar result for mapsbetween varieties that are not necessarily invertible. We also relate thedecomposability of an algebraic variety to the problem of establishing theessential normality of the corresponding submodule. These results are appliedto prove that the Arveson-Douglas conjecture holds for submodules correspondingto varieties that decompose into linear subspaces, and varieties that decomposeinto components with mutually disjoint linear spans.

Essential normality and the decomposability of algebraic varieties