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In JP May's paper A general algebraic approach to Steenrod operations, Steenrod operations are constructed in wide generality. In this context, it is not necessarily true that negative Steenrod operations are zero. However, I could not find any examples (not in May's paper, not even elsewhere), where one actually proves in some specific situation that some negative Steenrod operation is non-zero. It appears that sheaves of differential graded algebras $A$ (not concentrated in degree $0$) on a topological space $X$ should provide a counterexample, but I would like to see a specific example of $X$ and $A$, possibly with a proof.

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Notice the switch in grading from homology to cohomology on May's page 182: $P^s(x) = P_{-s}(x)$. Operations that raise degree when graded homologically lower degree when graded cohomologically. A large part of the motivation for the paper was to give a common framework for the Dyer-Lashof operations in the homology of iterated loop spaces and the Steenrod operations in the cohomology of spaces. But if one grades these the same way, either homologically or cohomologically, one has positive operations and the other has negative operations. The grading May found most sensible for the Steenrod operations on the cohomology of cocommutative Hopf algebras (p. 226) also gives examples. There are a number of more recent papers that make serious use of the large Steenrod algebra (sometimes called the Kudo-Araki-May algebra) constructed using all operations.

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    $\begingroup$ Thank you very much for your reply. I am still curious about the last sentence of my question, i.e. the case of hypercohomology of sheaves of dgas (using cohomological grading), so let me elaborate a bit. I read here mysite.science.uottawa.ca/pparent/May.pdf (Remark 7 page 12, 'Not true for hypercohomology') that there should be (positive and) negative Steenrod operations, contrary to the case of sheaf of algebras (as studied by Epstein in the Inventiones paper). Do you have a specific example in mind? $\endgroup$ – V. Pofek Jun 7 at 3:20
  • $\begingroup$ What a terrible talk, beginning with a terrible typo. That talk was when I was trying to work out an alternative construction of Voevodsky's Steenrod operations in motivic cohomology. The attempt failed (see the last few pages). I'd still like to see an operadic construction of those operations, but this is not the place to explain the difficulties I was seeing then. There are surely examples that others will know off the top of their head, but they are not on the top of mine. $\endgroup$ – Peter May Jun 7 at 17:42

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