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  "body": "\n## Summary\n\nAn Azumaya algebra over a [[commutative ring|0310REJJY8]] $R$ is an [[associative algebra|3CTW1P4HBP]] that behaves like a [[central simple algebra|5HRKJBAY2V]] over every point of $R$.\nIt is a faithful, finitely generated projective $R$-module $A$ for which $A \\otimes_R A^{\\mathrm{op}} \\to \\End_R(A)$ is an isomorphism.\nOver a field the Azumaya algebras are exactly the central simple algebras.\nAzumaya algebras define the [[Brauer group|BEW48TA9WZ]] of a commutative ring and, in geometry, of a scheme.\n\n## Explanations\n\n### L1 human\n\nA central simple algebra over a field looks like a matrix algebra after a change of field.\nAn Azumaya algebra is the same idea with the field replaced by a commutative ring.\nIts fibre at each maximal ideal $\\mathfrak{m}$ of $R$ is a central simple algebra over the field $R/\\mathfrak{m}$, and these fibres vary nicely.\nThe simplest example is the ring $M_n(R)$ of $n \\times n$ matrices over $R$.\n",
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