§ Mnemonic for hom-tensor and left-right adjoints
- Remember the phrase
tensor-hom adjunction, thus tensor is left adjoint.
- Remember that the type of an adjunction is
(f x -> y) -> (x -> g y) and here,
f is left adjoint,
g is right adjoint. Then see that currying is
((p, x) -> y) ->(x -> (p -> y)). Thus tensor is left adjoint, hom is rightadjoint.
- Remember that RAPL (right adjoints preserve limits); Then recall thattensoring a direct limit (a colimit) preserves the tensor, as a colimit retainstorsion (example: prufer group has torsion, its components also have torsion; tensor can detect this bytensoring with ).On the other hand, tensoring of an inverse-limit (a limit) is not preserved:think of p-adics. Each of the components have torsion, but the p-adics do not.Thus, tensor DOES NOT preserve limits (inverse limits). And so, tensor CANNOTbe right adjoint; tensor must be left adjoint.
- Since tensor is right exact, because it kills stuff, and could this destroyinjectivity, it is left adjoint.