In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety X, the existence of a surjection with affine space fibers from an affine variety W to X.
Moreover, the variety W is homotopy-equivalent to X, and W has the technically advantageous property of being affine.
Jouanolou's original statement of the theorem required that X be quasi-projective over an affine scheme, but this has since been considerably weakened.
Jouanolou's original statement was: By the definition of a torsor, W comes with a surjective map to X and is Zariski-locally on X an affine space bundle.
Jouanolou's proof used an explicit construction.
as the space of (r + 1) × (r + 1) matrices over S. Within this affine space, there is a subvariety W consisting of idempotent matrices of rank one.
The image of such a matrix is therefore a point in X, and the map
that sends a matrix to the point corresponding to its image is the map claimed in the statement of the theorem.
To show that this map has the desired properties, Jouanolou notes that there is a short exact sequence of vector bundles: where the first map is defined by multiplication by a basis of sections of
Jouanolou deduces the theorem in general by reducing to the above case.
Pulling back the variety W constructed above for
along this immersion yields the desired variety W for X.
Finally, if X is quasi-projective, then it may be realized as an open subscheme of a projective S-scheme.
is a Cartier divisor, and therefore i is an affine morphism.
and pull back along i. Robert Thomason observed that, by making a less explicit construction, it was possible to obtain the same conclusion under significantly weaker hypotheses.
Thomason's construction first appeared in a paper of Weibel.
Thomason's theorem asserts: Having an ample family of line bundles was first defined in SGA 6 Exposé II Définition 2.2.4.
Any quasi-projective scheme over an affine scheme has an ample family of line bundles, as does any separated locally factorial Noetherian scheme.
Thomason's proof abstracts the key features of Jouanolou's.
By hypothesis, X admits a set of line bundles L0, ..., LN and sections s0, ..., sN whose non-vanishing loci are affine and cover X.
The sections define a morphism of vector bundles
to be the cokernel of s. On Xi, s is a split monomorphism since it is inverted by the inverse of si.
is a vector bundle over Xi, and because these open sets cover X,
, and from this description, it is easy to check that it is a torsor for
Each si determines a global section fi of W. The non-vanishing locus Wi of fi is contained in
Serre's criterion now implies that W is affine.