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  "body": "\n## Statement\n\nLet $k$ be a [[field|04C5ZBG1WT]] and let $G$ be a linearly reductive linear algebraic group over $k$.\nLet $V$ be a finite-dimensional rational representation of $G$.\nLet $k[V]$ be the [[polynomial ring|0SDC934R52]] of polynomial functions on $V$, with the induced action of $G$.\nThen the [[ring of invariants|6VTR5EBNYC]] $k[V]^G$ is a [[Cohen–Macaulay ring|N1QJQEGBFC]].\n\n## Remarks\n\nA group is linearly reductive when every rational representation is a direct sum of irreducible ones.\nIn characteristic $0$ this holds exactly for the reductive groups; in characteristic $p \u003e 0$ it holds for tori and for finite groups of order prime to $p$, for example.\nProof idea: linear reductivity gives the Reynolds operator, a $k[V]^G$-linear projection $k[V] \\to k[V]^G$, so $k[V]^G$ is a direct summand of the regular ring $k[V]$.\nHochster and Roberts showed that such summands are Cohen–Macaulay here, first in characteristic $p$ with the Frobenius map, and then in characteristic $0$ by reduction to characteristic $p$.\n\nExample: let $G = \\set{1, g}$ act on $k[x, y]$ by $g \\cdot x = -x$, $g \\cdot y = -y$, where the characteristic of $k$ is not $2$.\nThen $k[x, y]^G = k[x^2, xy, y^2]$ is a free module over the polynomial ring $k[x^2, y^2]$ with basis $1, xy$, which is the Cohen–Macaulay property in graded form.\n\nLinear reductivity cannot be dropped: for a finite group whose order is divisible by the characteristic of $k$, the ring of invariants of a linear action need not be Cohen–Macaulay.\n\n## Explanations\n\n### L1 human\n\nInvariant rings of nice groups are well behaved.\nThe theorem says that one can choose finitely many algebraically independent homogeneous invariants such that the whole invariant ring is a free module over the polynomial ring they generate.\nSo every invariant has a unique expression of a fixed simple form.\n\n### L2 human\n\nThe Cohen–Macaulay property makes the Hilbert series of $k[V]^G$ a rational function with a numerator with nonnegative coefficients, and it controls the free resolution of the invariant ring.\nThe proof is an early major use of reduction to positive characteristic, the method that later gave [[tight closure|KJ9MMP99G6]].\nThe result fits with the finite generation of rings of invariants of reductive groups, the positive side of [[Hilbert's fourteenth problem|M5N08HD75V]].\nBoutot later proved the stronger fact that, in characteristic $0$, these invariant rings have rational singularities.\n",
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