We have seen in the previous chapters that holonomic modules are preserved by inverse images under projections and by direct images under embeddings. However, as we also saw, inverse images under embeddings and direct images under projections do not preserve the fact that a module is finitely generated. Fortunately, though, holonomic modules are preserved by all kinds of inverse and direct images. The proof of this result will use all the machinery that we have developed so far. It gives yet one more way to construct examples of holonomic modules. We retain the notations of 14.1.2.

INVERSE IMAGES

The key to the results in this chapter is a decomposition of polynomial maps in terms of embeddings and projections. The idea goes back to A. Grothendieck.

Let F : X→ Y be a polynomial map. We may decompose F as a composition of three polynomial maps: a projection, an embedding and an isomorphism. The maps are the following. The projection is π : X × Y → Y, defined by π(X, Y) = Y. The isomorphism is G : X × Y → X × Y where G(X, Y) = (X, Y+F(X)). Finally, the embedding is i : X → X × Y, defined by i(X) = (X, 0). One can immediately check that F = π · G · i.