sentences:  Global Verdier duality states that for a continuous map formula_1, the derived functor of the direct image with proper supports formula_2 has a right adjoint formula_3 in the derived category of sheaves, in other words, for a sheaf formula_4 on formula_5 and formula_6 on formula_7 we have 
  
  The exclamation mark is often pronounced "shriek" (slang for exclamation mark), and the maps called "formula_9 shriek" or ""formula_9" lower shriek" and ""f" upper shriek" – see also shriek map.Local Verdier duality states that
  in the derived category of sheaves of "k" modules over "Y".It is important to note that the distinction between the global and local versions is that the former relates maps between sheaves, whereas the latter relates (complexes of) sheaves directly and so can be evaluated locally.