This is the first in a sequence of articles exploring the relationship between commutative algebras
and
-algebras in
characteristic
and
mixed characteristic. In this article we lay the groundwork by defining a new class of cohomology
operations over
called cotriple products, generalising Massey products. We compute the secondary
cohomology operations for a strictly commutative dg-algebra and the obstruction
theories these induce, constructing several counterexamples to characteristic-0
behaviour, one of which answers a question of Campos, Petersen, Robert-Nicoud and
Wierstra. We construct some families of higher cotriple products and comment on
their behaviour. Finally, we distinguish a subclass of cotriple products that we call
higher Steenrod operations and conclude with our main theorem, which says that
-algebras
can be rectified if and only if the higher Steenrod operations vanish coherently.
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