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Cohomological characterisation of monads
Author(s) -
Marques Pedro Macias,
Soares Helena
Publication year - 2014
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201300208
Subject(s) - mathematics , variety (cybernetics) , pure mathematics , projective variety , fano plane , cohomology , torsion (gastropod) , projective test , bounded function , derived category , algebra over a field , functor , mathematical analysis , medicine , statistics , surgery
Let X be an n ‐dimensional smooth projective variety with an n ‐block collection B = ( F 0 , ... , F n ) , withF i = F 1 i , ... , F α i i, of coherent sheaves on X that generate the bounded derived categoryD b ( X ) . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the formM • : 0 F i 0 i a F j 0 j b F k 0 k c 0 where 0 ≤ i < j < k ≤ n . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadricQ n ⊂ P n + 1or the Fano threefold V 5 . We construct a family of monads on a Segre variety and apply our main result to this family.