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Spring 2016 - Problem 2

category theory

Consider the functor FF from commutative rings to abelian groups that takes a commutative ring RR to the group R×R^\times of invertible elements. Does FF have a left adjoint? Does FF have a right adjoint? Justify your answers.
Solution.

FF is a forgetful functor, so its left adjoint is a universal construction. Hence, FF has the functor A↦Z[A]A \mapsto \Z\br{A} as a left adjoint. If FF were a left adjoint, then in particular FF must preserve colimits. However, if A=Z/2ZA = \Z/2\Z and B=Z/3ZB = \Z/3\Z, then

Z/2Z⨿CRingZ/3Z≃Z/2Z⊗ZZ/3Z≃Z/gcd⁡(2,3)Z≃0,\begin{aligned} \Z/2\Z \amalg_{\CRing} \Z/3\Z &\simeq \Z/2\Z \otimes_\Z \Z/3\Z \\ &\simeq \Z/{\gcd\p{2, 3}}\Z \\ &\simeq 0, \end{aligned}

but this would mean

1≃F(Z/2Z⨿CRingZ/3Z)≃F(Z/2Z)⨿AbF(Z/3Z)≃1⊕Z/2Z,\begin{aligned} 1 &\simeq F\p{\Z/2\Z \amalg_{\CRing} \Z/3\Z} \\ &\simeq F\p{\Z/2\Z} \amalg_{\Ab} F\p{\Z/3\Z} \\ &\simeq 1 \oplus \Z/2\Z, \end{aligned}

which is impossible. Hence, FF has no right adjoint.