The integral motivic dual Steenrod algebra

Abstract We calculate the motivic integral dual Steenrod algebra $$\mathbb {M}^S_*\mathbb {Z}(\mathbb {M}^S \mathbb {Z})$$ M ∗ S Z ( M S Z ) over base schemes S for which the mod-p motivic dual Steenrod algebra $$\mathbb {M}^S_*\mathbb {F}_p(\mathbb {M}^S \mathbb {F}_p)$$ M ∗ S F p ( M S F p ) conforms with Voevodsky’s formula. Then the kernel of the augmentation $$\mathbb {M}^S_*\mathbb {Z}(\mathbb {M}^S \mathbb {Z})\rightarrow \mathbb {M}^S_*\mathbb {Z}$$ M ∗ S Z ( M S Z ) → M ∗ S Z is simple p -torsion; in more detail, after localizing at p the projection $$\mathbb {Z}\rightarrow \mathbb {F}_p$$ Z → F p gives rise to a pullback square of commutative $$\mathbb {M}^S_*\mathbb {Z}$$ M ∗ S Z -algebras "Equation missing" where the $$\mathbb {M}^S_*\mathbb {F}_p$$ M ∗ S F p -algebra $$\ker \beta \subseteq \mathbb {M}^S_*\mathbb {F}_p(\mathbb {M}^S \mathbb {Z})$$ ker β ⊆ M ∗ S F p