Chalupnik's formality conjecture for derived Frobenius twists

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 0i<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.

Sources & referencesView supporting material

Primary source

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

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