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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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References:
 [1] Breen L., Publ. Sci. IHES 48 pp 39– (1978) · Zbl 0404.14018 [2] Cartan H., Homological Algebra (1956) [3] DOI: 10.1016/0021-8693(86)90218-8 · Zbl 0606.20038 [4] DOI: 10.2307/121092 · Zbl 0952.20035 [5] DOI: 10.1007/BF01231771 · Zbl 0798.18009 [6] DOI: 10.1007/s002220050119 · Zbl 0945.14028 [7] Godement R., Topologie Algébrique et Théorie des Faisceaux (1958) [8] Henn H.-W., Alg. Top. Proc. Northwestern University, Cont. 96 pp 197– (1989) [9] James G., The Representation Theory of the Symmetric Group (1981) [10] Jantzen J., Representations of Algebraic Groups (1987) [11] DOI: 10.1016/0021-8693(91)90093-N · Zbl 0724.18005 [12] DOI: 10.2307/2374932 · Zbl 0813.20049 [13] DOI: 10.1007/BF00961668 · Zbl 0831.20057 [14] Pirashvili, T. ”Vanishing of Ext-Groups in Functor Categories”. Bonn: MPI. · Zbl 0793.18009 [15] DOI: 10.1023/A:1007700132156 · Zbl 0918.20036 [16] Schwartz, L. 1994. ”Unstable Modules over the Steenrod Algebra and Sullivan’s Fixed Point Set Conjecture”. The Univ. of Chicago Press. · Zbl 0871.55001 [17] DOI: 10.1515/crll.1997.482.1 · Zbl 0859.20034
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