HH^ZZ SheafOfRings -- cohomology of a sheaf of rings on a projective variety

Synopsis

• Function: cohomology

• Usage:

  HH^d(R)

• Inputs:
  • i, an integer
  • R, a sheaf of rings, on a projective variety X
• Optional inputs:
  • Degree (missing documentation) => ..., default value 0,
• Outputs:
  • a module, the i-th cohomology group of R as a vector space over the coefficient field of X

Description

i1 : Cubic = Proj(QQ[x_0..x_2]/ideal(x_0^3+x_1^3+x_2^3))

o1 = Cubic

o1 : ProjectiveVariety

i2 : HH^1(OO_Cubic)
-- ker (114) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (114) returned CacheFunction: -*a cache function*-
-- ker (114) called with Matrix: 0
--                                        1
-- ker (114) returned Module: (QQ[x ..x ])
--                                 0   2
assert( ker(map((QQ[x_0..x_2])^0,(QQ[x_0..x_2])^1,0)) === ((QQ[x_0..x_2])^1))
-- ker (115) called with OptionTable: OptionTable{SubringLimit => infinity}
-- ker (115) returned CacheFunction: -*a cache function*-
-- ker (115) called with Matrix: 0
--                                        1
-- ker (115) returned Module: (QQ[x ..x ])
--                                 0   2
assert( ker(map((QQ[x_0..x_2])^0,(QQ[x_0..x_2])^{{-3}},0)) === ((QQ[x_0..x_2])^{{-3}}))

       1
o2 = QQ

o2 : QQ-module, free

See also

• coherent sheaves
• HH^ZZ SumOfTwists -- coherent sheaf cohomology module
• HH^ZZ CoherentSheaf -- cohomology of a coherent sheaf on a projective variety
• hh -- Hodge numbers of a smooth projective variety
• CoherentSheaf -- the class of all coherent sheaves