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We equate various Euler classes of algebraic vector bundles, including thoseof [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P.Serre. We establish integrality results for this Euler class, and give formulasfor local indices at isolated zeros, both in terms of 6-functor formalism ofcoherent sheaves and as an explicit recipe in commutative algebra of Scheja andStorch. As an application, we compute the Euler classes associated toarithmetic counts of d-planes on complete intersections in P^n in terms oftopological Euler numbers over R and C.

A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections