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Some calculations of cohomology of composed symmetric powers. (Quelques calculs de cohomologie de compositions de puissances symétriques.) (French) Zbl 1005.18010
Summary: The main purpose of this paper is to compute the Ext-groups \(\text{Ext}_{\mathcal F}^* (\text{Id},S^p\circ S^{p^h})\) and \(\text{Ext}^*_{ \mathcal F} (\text{Id},S^{p^h}\circ S^p)\), in the category \({\mathcal F}\) of functors from finite \(\mathbb{F}_p\)-vector spaces to all \(\mathbb{F}_p\)-vector spaces. This computation gives an approach to the more general computation of \(\text{Ext}^*_{\mathcal F}(\text{Id}, S^{p^h}\circ S^{p^k})\). The two main tools are a comparison theorem between Ext-groups in the category \({\mathcal F}\) and Ext-groups in the category \({\mathcal P}\) of strict polynomial functors, and the “exactness modulo \(\text{Ext}^*_{\mathcal F}(\text{Id},-)\)” of the post-composition. This means that the homology appearing when applying a functor \(F\) to a short exact sequence is in the kernel of \(\text{Ext}^*_{\mathcal F}(\text{Id},-)\).
[A. Troesch, C.R. Acad. Sci. Paris, Sér. I, Math. 333, No. 6, 509-512 (2001; Zbl 0990.18009)].

MSC:
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
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