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We define a trace map for every cohomological correspondence in the motivicstable homotopy category over a general base scheme, which takes values in thetwisted bivariant groups. Local contributions to the trace map give rise toquadratic refinements of the classical local terms, and some$\mathbb{A}^1$-enumerative invariants, such as the local $\mathbb{A}^1$-Brouwerdegree and the Euler class with support, can be interpreted as local terms. Weprove an analogue of a theorem of Varshavsky, which states that for acontracting correspondence, the local terms agree with the naive local terms.

Trace maps in motivic homotopy and local terms