Universal families of extensions

Abstract

If X denotes a projective variety over a field k and jr and .y coherent sheaves on X, the vector space Exti(Sr, *y) parametrizes the extensions of .F by .V’ on X and there is a universal extension of p:T by p:.Y on XX Exti(y, 5?) such that for every k-rational point u of Exti(%ir, .y) its restriction to X x (u) is just the extension represented by u. Moreover, P = P(Extk(ST, .V)) parametrizes the classes of nonsplitting extensions of 9 by .y on X module k* and there is a universal family of extensions of pT.F by pf5? @p; @$( 1) on X x P over P such that for every k-rational point p of P its restriction to X x {p} represents just the class of extensions given by p. It is the purpose of this paper not only to prove these two statements but to generalize them to the relative case of a flat projective morphism f: X + Y of noetherian schemes. They turn out to be a consequence of Grothendieck’s universal properties of vector bundles and projective bundles (cp. [2, Section I, 91). Special cases of these results appeared several times in the literature; to the best of my knowledge first in [5, Lemma 3.11 and second in [ 6, Lemma 2.41. The technical tool is the base change theory for relative Ext-sheaves, which is outlined in Section 1; in Section 2 some relations between relative Ext-sheaves and families of extensions are given, Section 3 contains the construction’ of the universal family of extensions and Section 4 the construction of the universal family of classes of nonsplitting extensions. Concerning the notation: f: X+ Y means a flat projective morphism of noetherian schemes and r and .Ce coherent Ox-modules, flat over Y. The proofs of Sections 2 to 4 are more generally correct if only there is a base change theory for relative Ext-sheaves. For example, if one uses the base change results of [ 1 ] valid for arbitrary proper flat morphisms of complex spaces, one gets the corresponding results in this case too. 101 OOZl-8693183 $3.00