## Koszul duality and extensions of exponential functors.(English)Zbl 1148.18008

The aim of this interesting paper is to study Koszul duals of some important functors and to apply the results to compute Ext-groups between exponential functors. One of the main results is Theorem 3.2. In this theorem the author computes the cohomology of the Koszul dual for the twisted divided power functor. Using this result, in the last section of the paper, explicit formulas for some Ext-groups are given (Corollaries 4.3, 4.4 and 4.5).

### MSC:

 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects) 16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. 18E30 Derived categories, triangulated categories (MSC2010) 18G40 Spectral sequences, hypercohomology 20G10 Cohomology theory for linear algebraic groups

### Keywords:

Koszul duality; Ext-groups; exponential functors
Full Text:

### References:

 [1] Akin, K., Extensions of symmetric tensors by alternating tensors, J. algebra, 121, 358-363, (1989) · Zbl 0682.14033 [2] Akin, K.; Buchsbaum, D.; Weyman, J., Schur functors and Schur complexes, Adv. math., 44, 207-278, (1982) · Zbl 0497.15020 [3] Chałupnik, M., Schur-de-Rham complex and its cohomology, J. algebra, 282, 699-727, (2004) · Zbl 1085.18012 [4] Chałupnik, M., Extensions of strict polynomial functors, Ann. sci. école norm. sup. (4), 38, 5, 773-792, (2005) · Zbl 1089.20029 [5] Donkin, S., On tilting modules for algebraic groups, Math. Z., 212, 39-60, (1993) · Zbl 0798.20035 [6] Franjou, V.; Friedlander, E.; Scorichenko, A.; Suslin, A., General linear and functor cohomology over finite fields, Ann. of math., 150, 2, 663-728, (1999) · Zbl 0952.20035 [7] Franjou, V.; Lannes, J.; Schwartz, L., Autour de la cohomologie de maclanedes corps finis, Invent. math., 115, 513-538, (1994) · Zbl 0798.18009 [8] Friedlander, E.; Suslin, A., Cohomology of finite group schemes over a field, Invent. math., 127, 209-270, (1997) · Zbl 0945.14028 [9] Kuhn, N., Generic representations of the finite general linear groups and the Steenrod algebra III, K-theory, 9, 273-303, (1995) · Zbl 0831.20057 [10] Totaro, B., Projective resolutions of representations of $$\operatorname{GL}(n)$$, J. reine angew. math., 482, 1-13, (1997) · Zbl 0859.20034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.