Chalupnik's formality conjecture for derived Frobenius twists

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Let kk be a field of characteristic pp, let rr be a nonnegative integer, and let GG belong to the category of homogeneous strict polynomial functors of degree dd. Let G(r)G^{(r)} denote its rr-th Frobenius twist, let Kr\boldsymbol{K}^{\boldsymbol r} be the relevant derived functor, and let ErE_r be the graded vector space of dimension prp^r that is equal to kk in dimensions 2i2i for 0≤i<pr0\leq i<p^r and zero otherwise. Chalupnik's formality conjecture.

Kr(G(r))≅GEr\boldsymbol{K}^{\boldsymbol r}(G^{(r)})\cong G_{E_r}

in the bounded derived category of strict polynomial functors. In particular, Kr(G(r))\boldsymbol{K}^{\boldsymbol r}(G^{(r)}) is formal. The source does not provide evidence resolving this conjecture, so its status remains open.

References

Primary source

Wilberd van der Kallen, “Lectures on bifunctors and finite generation of rational cohomology algebras”, arXiv:1208.3097 (2014).

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