Oriented bivariant theory, II: Algebraic cobordism of S-schemes

This is a sequel to our previous paper “Oriented bivariant theory, I”. In 2001, Levine and Morel constructed algebraic cobordism for (reduced) schemes [Formula: see text] of finite type over a base field [Formula: see text] in an abstract way and later Levine and Pandharipande reconstructed it more geometrically, using “double point degeneration”. In this paper in a similar manner to the construction of Levine–Morel, we construct an algebraic cobordism for a scheme [Formula: see text] over a fixed scheme [Formula: see text] in such a way that if the target scheme [Formula: see text] is the point [Formula: see text], then our algebraic cobordism is isomorphic to Levine–Morel’s algebraic cobordism. Our algebraic cobordism can be interpreted as “a family of algebraic cobordism” parametrized by the base scheme [Formula: see text].

ORIENTED BIVARIANT THEORIES, I

In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory (BT), which is a sophisticated unification of covariant theories and contravariant theories. This is for the study of singular spaces. In 2001 M. Levine and F. Morel introduced the notion of algebraic cobordism, which is a universal oriented Borel–Moore functor with products (OBMF) of geometric type, in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism. In this paper we introduce a notion of oriented bivariant theory (OBT), a special case of which is nothing but the oriented Borel–Moore functor with products. The present paper is a first one of the series to try to understand Levine–Morel's algebraic cobordism from a bivariant theoretical viewpoint, and its first step is to introduce OBT as a unification of BT and OBMF.