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  "body": "\n## Definition\n\nLet $k$ be a field and let $X$ be a [[scheme|5A35TYZD02]] over $k$.\nLet $\\mathrm{Art}_k$ be the category of local [[Artinian|ZV04WWYE2Y]] $k$-algebras $A$ with residue field $k$.\nFor $A \\in \\mathrm{Art}_k$, a *deformation of $X$ over $A$* is a pair $(\\mathcal{X}, \\phi)$ where:\n\n1. $\\mathcal{X}$ is a scheme with a [[flat|5M6KS8F47R]] morphism $\\mathcal{X} \\to \\operatorname{Spec} A$;\n2. $\\phi \\colon \\mathcal{X} \\times_{\\operatorname{Spec} A} \\operatorname{Spec} k \\to X$ is an isomorphism of $k$-schemes, where the left side is a [[fiber product|1DZMT4S9FB]].\n\nTwo deformations $(\\mathcal{X}, \\phi)$ and $(\\mathcal{X}', \\phi')$ are isomorphic if there is an isomorphism $\\mathcal{X} \\to \\mathcal{X}'$ over $A$ compatible with $\\phi$ and $\\phi'$.\nThe *deformation functor* of $X$ is $\\operatorname{Def}_X \\colon \\mathrm{Art}_k \\to \\mathrm{Set}$, sending $A$ to the set of isomorphism classes of deformations of $X$ over $A$.\nA *first-order deformation* is a deformation over the dual numbers $k[\\varepsilon] = k[\\varepsilon]/(\\varepsilon^2)$.\nDeformation theory is the study of $\\operatorname{Def}_X$ and of the analogous functors for other objects, such as sheaves, morphisms and subschemes.\n\n## Remarks\n\nLet $X$ be smooth over $k$ with tangent sheaf $T_X$.\nThen $\\operatorname{Def}_X(k[\\varepsilon]) \\cong H^1(X, T_X)$, in [[sheaf cohomology|PQ6Q143ARJ]].\nThe obstructions to lifting a deformation over $A$ to a small extension of $A$ lie in $H^2(X, T_X)$.\nExample: the trivial deformation $X \\times_k \\operatorname{Spec} A$ with the identity $\\phi$ satisfies both clauses, since a base change of $\\operatorname{Spec} A \\to \\operatorname{Spec} k$ is flat.\nFor $X = \\mathbb{P}^1$, $H^1(\\mathbb{P}^1, T) = H^1(\\mathbb{P}^1, \\mathcal{O}(2)) = 0$, so every first-order deformation of $\\mathbb{P}^1$ is trivial.\nNon-example: let $A = k[\\varepsilon]$ and $\\mathcal{X} = \\operatorname{Spec} k[\\varepsilon, x]/(\\varepsilon x)$.\nIts special fiber is $\\operatorname{Spec} k[x] = \\mathbb{A}^1_k$, so clause 2 holds.\nBut $\\mathcal{X}$ is not flat over $A$, since $x$ is killed by $\\varepsilon$ without being a multiple of $\\varepsilon$, so clause 1 fails.\n",
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