User:Maximilian Janisch/latexlist/Algebraic Groups/Mumford hypothesis

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The hypothesis that each semi-simple algebraic group $k$ is geometrically reductive, i.e., has the following property: For any rational representation of $k$ in a finite-dimensional vector space $V$ and any non-zero vector $v \in V$ fixed by $k$, there is a $k$-invariant homogeneous polynomial $f$ of positive degree on $V$ for which $f ( v ) \neq 0$.

This hypothesis was stated by D. Mumford [1] (in a different, but equivalent form) with the aim of finding a property of semi-simple groups, defined over an algebraically closed field of arbitrary characteristic, which, from the point of view of the geometric theory of invariants (cf. Invariants, theory of), would serve as a substitute for the classical property of complete reducibility of rational linear representations of semi-simple groups defined over fields of characteristic zero (this latter property not holding for ground fields of positive characteristic). It would allow the removal of restrictions on the characteristic of the ground field in a number of central results in the geometric theory of invariants, such as the theorem on finite generation of the algebra of invariants of reductive groups of automorphisms of an algebra of finite type over a field (see Hilbert theorem on invariants).

If the characteristic of the ground field $k$ is zero, then a proof of Mumford's hypothesis is given by Weyl's classical theorem on complete reducibility of rational representations of semi-simple groups (see [2]): in this case the invariant line $L = k v$ in $V$ has an invariant complement $I$ (an invariant hyperplane such that $L _ { \cap } \Gamma = 0$), and $f$ can be taken to be the linear form giving the equation of $I$. When $k$ has positive characteristic $D$, Mumford's hypothesis generalizes the fact that there is an invariant homogeneous hypersurface $I$ in $V$ for which $L _ { \cap } \Gamma = 0$ (with the degree of $I$ equal to $p ^ { n }$ for some integer $12$).

Mumford's hypothesis is also equivalent to the assertion that for any regular action of a semi-simple group $k$ on an affine algebraic variety $x$ and for any two closed non-intersecting invariant subsets $X _ { 1 }$ and $X _ { 2 }$ in $x$ there is an invariant regular function $h$ on $x$ for which $h ( X _ { 1 } ) = 0$ and $h ( X _ { 2 } ) = 1$ (i.e., $X _ { 1 }$ and $X _ { 2 }$ can be separated by regular invariants, see [3]).

Mumford's hypothesis was first proved in [4]; the proof was extended in [5] to the general case of reductive group schemes over a field.

The proof of Mumford's hypothesis, together with the results of [6] and [10], allow one, first, to give a final form to the generalization of Hilbert's theorem on invariants: If $R$ is an algebra of finite type over an algebraically closed field $k$, $k$ is a reductive group, acting as an automorphism group on $R ^ { n }$, and $R ^ { G }$ is the subalgebra of all $k$-invariant elements in $R$, then $R ^ { G }$ is also an algebra of finite type over $k$; and, secondly, to establish that a linear algebraic group over a field of arbitrary characteristic is geometrically reductive if and only if it is reductive (cf. Reductive group). Mumford's hypothesis has applications in the geometric theory of invariants and in moduli theory (see [7]–[9]).

References

[1]	D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[2]	J. Fogarty, "Invariant theory" , Benjamin (1969) MR0240104 Zbl 0191.51701
[3]	J. Dieudonné, J.B. Carrell, "Invariant theory: old and new" , Acad. Press (1971) MR0279102 Zbl 0258.14011
[4]	W.J. Haboush, "Reductive groups are geometrically reductive" Ann. of Math. , 102 (1975) pp. 67–83 MR0382294 Zbl 0316.14016
[5]	C.S. Seshadri, "Geometric reductivity over arbitrary base" Adv. Math. , 26 (1977) pp. 225–274 MR0466154 Zbl 0371.14009
[6]	M. Nagata, "Invariants of a group in an affine ring" J. Math. Kyoto Univ. , 3 (1964) pp. 369–377 (With appendix by M. Miyanishi) MR0179268 Zbl 0146.04501
[7]	C.S. Seshadri, "Mumford's conjecture for $GL$ and applications" , Algebraic geometry. Papers presented at the Bombay Colloq. 1968 , Oxford Univ. Press (1969) pp. 347–371 MR262248
[8]	H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977) MR0466143 Zbl 0359.14005
[9]	C.S. Seshadri, "Theory of moduli" R. Hartshorne (ed.) , Algebraic geometry (Arcata, 1974) , Proc. Symp. Pure Math. , 29 , Amer. Math. Soc. (1975) pp. 263–304 MR0396565 Zbl 0321.14005
[10]	M. Nagata, T. Miyata, "Note on semi-reductive groups" J. Math. Kyoto Univ. , 3 (1964) pp. 379–382 MR0179269 Zbl 0152.00902