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  "body": "\n## Statement\n\nLet $R$ be a [[commutative ring|0310REJJY8]] and let $M$, $N$, $P$ be [[$R$-modules|0NZ8JA25WK]].\nLet $\\Hom_R(N, P)$ be the $R$-module of $R$-[[linear maps|0NRYHJPQG2]] $N \\to P$, with $(r\\psi)(n) = r\\psi(n)$.\nThen the map\n$$\\Phi \\colon \\Hom_R(M \\otimes_R N, P) \\to \\Hom_R(M, \\Hom_R(N, P)), \\qquad \\Phi(\\varphi)(m)(n) = \\varphi(m \\otimes n),$$\nis an isomorphism of $R$-modules, natural in $M$, $N$ and $P$.\nThus the functor $- \\otimes_R N$ is left adjoint to the hom functor $\\Hom_R(N, -)$.\n\n## Remarks\n\nThe inverse sends $\\psi \\colon M \\to \\Hom_R(N, P)$ to the linear map $M \\otimes_R N \\to P$ induced by the bilinear map $(m, n) \\mapsto \\psi(m)(n)$.\nThis is exactly the universal property of the [[tensor product|F0HQXQ0HCD]]: linear maps out of $M \\otimes_R N$ are bilinear maps on $M \\times N$.\nIn turn, a bilinear map is a linear map from $M$ to linear maps on $N$.\n\nThe adjunction has consequences for [[exact sequences|JB7ZBFBA9C]].\nThe functor $\\Hom_R(N, -)$ is left exact: if $0 \\to A \\to B \\to C$ is exact, then so is $0 \\to \\Hom_R(N, A) \\to \\Hom_R(N, B) \\to \\Hom_R(N, C)$.\nAs a left adjoint, $- \\otimes_R N$ is right exact: if $A \\to B \\to C \\to 0$ is exact, then so is $A \\otimes_R N \\to B \\otimes_R N \\to C \\otimes_R N \\to 0$.\nNeither functor is exact in general.\nFor $N = \\Z/2\\Z$ over $\\Z$, the surjection $\\Z \\to \\Z/2\\Z$ gives $\\Hom(\\Z/2\\Z, \\Z) = 0 \\to \\Hom(\\Z/2\\Z, \\Z/2\\Z) \\cong \\Z/2\\Z$, which is not surjective; and the injection $\\Z \\xrightarrow{2} \\Z$ becomes the zero map after $- \\otimes_\\Z \\Z/2\\Z$.\n\n## Explanations\n\n### L1 human\n\nA function of two variables can be seen as a function of the first variable whose values are functions of the second.\nThe tensor-hom adjunction is this idea for linear algebra over a ring.\nA linear map out of $M \\otimes N$ is the same thing as a linear map from $M$ into the maps out of $N$.\nIt lets one move a factor from one side of $\\Hom$ to the other.\n\n### L2 human\n\nThis is the prototype of an adjunction, and it explains at once why tensor products are right exact and $\\Hom$ is left exact.\nThe failure of exactness is measured by the derived functors $\\operatorname{Tor}$ and $\\operatorname{Ext}$.\n[[Flat modules|73RYYEVDKA]] are those $N$ for which $- \\otimes_R N$ is exact, and [[projective modules|BFFA03GE3V]] those for which $\\Hom_R(N, -)$ is exact.\nThe same adjunction holds for bimodules over noncommutative rings, and for sheaves of modules.\n",
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