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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $X$ and $Y$ be [[algebraic varieties|HV6YBS0YDH]] over $k$.\nA *morphism* $\\varphi \\colon X \\to Y$ is a [[continuous map|0VGFHTM75Y]] for the Zariski topologies with the following property.\nFor every [[open set|02BH3MCBQB]] $V \\subseteq Y$ and every [[regular function|4M94FZH9Y8]] $f \\colon V \\to k$, the function $f \\circ \\varphi \\colon \\varphi^{-1}(V) \\to k$ is regular.\nAn *isomorphism* is a morphism that has an inverse morphism.\n\n## Remarks\n\nIf $X \\subseteq k^m$ and $Y \\subseteq k^n$ are [[affine varieties|JERVQT4C12]], a map $\\varphi \\colon X \\to Y$ is a morphism if and only if $\\varphi = (\\varphi_1, \\dots, \\varphi_n)$ with each $\\varphi_i$ a polynomial function on $X$.\nA morphism $\\varphi$ gives the [[ring homomorphism|0VTH8KPCG2]] $\\varphi^* \\colon k[Y] \\to k[X]$, $f \\mapsto f \\circ \\varphi$, on [[coordinate rings|G9WB27VWP7]].\nThis gives a bijection between morphisms $X \\to Y$ and $k$-algebra homomorphisms $k[Y] \\to k[X]$.\n\nExample: $\\varphi \\colon k \\to V(y - x^2)$, $t \\mapsto (t, t^2)$, is an isomorphism with inverse $(x, y) \\mapsto x$.\n\nNon-example: $\\psi \\colon k \\to C = V(y^2 - x^3)$, $t \\mapsto (t^2, t^3)$, is a bijective morphism but not an isomorphism.\nIts comorphism $k[C] \\to k[t]$ has image $k[t^2, t^3]$, which does not contain $t$, so $\\psi^*$ is not surjective and $\\psi$ has no inverse morphism.\n",
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