We calculate the motivic integral dual Steenrod algebra \(\mathbb {M}^S_*\mathbb {Z}(\mathbb {M}^S \mathbb {Z})\) over base schemes S for which the mod-p motivic dual Steenrod algebra \(\mathbb {M}^S_*\mathbb {F}_p(\mathbb {M}^S \mathbb {F}_p)\) conforms with Voevodsky’s formula. Then the kernel of the augmentation \(\mathbb {M}^S_*\mathbb {Z}(\mathbb {M}^S \mathbb {Z})\rightarrow \mathbb {M}^S_*\mathbb {Z}\) is simple p-torsion; in more detail, after localizing at p the projection \(\mathbb {Z}\rightarrow \mathbb {F}_p\) gives rise to a pullback square of commutative \(\mathbb {M}^S_*\mathbb {Z}\) -algebras where the \(\mathbb {M}^S_*\mathbb {F}_p\) -algebra \(\ker \beta \subseteq \mathbb {M}^S_*\mathbb {F}_p(\mathbb {M}^S \mathbb {Z})\) is the kernel of the Bockstein homomorphism. Concrete calculations follow, for instance, if F is a field of characteristic different from p where all p-torsion in the motivic homology \(\mathbb {M}^F_*\mathbb {Z}\) of \(\operatorname {Spec}(F)\) is p-divisible (meaning that if \(px=0\) then \(x=py\) for some y—as is the case if F is algebraically closed) then the motivic p-adic dual Steenrod algebra is expressed explicitly as an algebra in terms of generators and relations. Some other cases of interest are also explored more fully, most notably the base cases of finite fields and \(\mathbb {Z}[1/2]\) .
