Bivariant Chow operations and bivariant classes

Assume the Axiom of Choice (The Axiom of Choice) inherited from the proper/quasi-finite and scheme base-change suppliers. Fix a field $k$ and work in the category of schemes locally of finite type over $k$, with $k$-morphisms. For $f:Z\to T$ in this category, a degree-$d$ bivariant Chow operation assigns to every $k$-morphism $T^{\mathrm{\prime }}\to T$ in this category homomorphisms $c:A_m(T^{\mathrm{\prime }})\to A_{m-d}(Z{\times }_T{T}^{\mathrm{\prime }})$, commuting with proper pushforward, flat pullback of fixed relative dimension, and Cartier divisor Gysin. Restriction along any base morphism, composition, and proper pushforward of such operations are defined by their action. Flat pullback and Cartier divisor Gysin give bivariant classes. If $h:Z\to Z_1$ is proper over $T$, pushing a class for $Z\to T$ along $h$ gives a class for $Z_1\to T$; ordinary proper pushforward is a covariant Chow operation, not a class reversing that proper morphism. Equality can be tested on fundamental classes of integral schemes over $T$; it suffices to test after a proper birational modification of each such integral scheme.

Well-definedness. The three axioms are required for every base change $T^{\mathrm{\prime }}\to T$ in this category, and the operations are compared only on Chow groups, using Rational equivalence and the Chow group of cycles. Proper pushforward and flat pullback are the maps of Proper pushforward commutes with flat pullback; the Cartier divisor Gysin is the operation of Intersection with an invertible sheaf and the first Chern class. Restriction along a base morphism $T^{\mathrm{\prime }\mathrm{\prime }}\to T^{\mathrm{\prime }}$ simply changes the indexing family, and composition of operations is associative because each axiom is applied first to the inner and then to the outer factor; the compatibility of two Cartier Gysins is the tame-symbol reciprocity computation of Tame symbol reciprocity in dimension two (the Key Lemma), and if a cycle is contained in one of the two divisors the operation is $c_1$ of the corresponding invertible sheaf, whose commutation is the same rational-section argument. For a proper $h:Z\to Z_1$ over $T$ the definition $(h_{\ast }c{)}_{T^{\mathrm{\prime }}}=h_{T^{\mathrm{\prime }},\ast }{c}_{T^{\mathrm{\prime }}}$ satisfies the three axioms by the corresponding compatibilities of proper pushforward with flat pullback and Cartier Gysin. For the equality criterion, a finite cycle is a finite sum of pushforwards of integral fundamental classes. For a locally finite cycle ${\sum }_i{n}_i[V_i]$ on $T^{\mathrm{\prime }}$, the map $h:{\coprod }_i{V}_i\to T^{\mathrm{\prime }}$ is proper: over every quasi-compact open only finitely many supports occur, and their inclusions are closed immersions. Locally finite Chow groups on this disjoint union are the product of the Chow groups of its components, as are those of its base change to $Z$. Flat restriction to each open-and-closed component therefore determines an operation on the class $(n_i[V_i]{)}_i$. Equality on the integral classes gives equality there, and proper compatibility pushes it along $h$ to equality on the original cycle. Finally, for a proper birational $h:V^{\mathrm{\prime }}\to V$ of integral schemes one has $h_{\ast }[V^{\mathrm{\prime }}]=[V]$, so equality on $[V^{\mathrm{\prime }}]$ pushes forward to equality on $[V]$. This is a descent test for operations, not an assertion that Chow pullback to a modification is injective.