View source for Mumford hypothesis

The hypothesis that each [[Semi-simple algebraic group|semi-simple algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655701.png" /> is geometrically reductive, i.e., has the following property: For any [[Rational representation|rational representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655702.png" /> in a finite-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655703.png" /> and any non-zero vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655704.png" /> fixed by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655705.png" />, there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655706.png" />-invariant homogeneous polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655707.png" /> of positive degree on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655708.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m0655709.png" />.

This hypothesis was stated by D. Mumford [[#References|[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|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|Hilbert theorem]] on invariants).

If the characteristic of the ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557010.png" /> 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 [[#References|[2]]]): in this case the invariant line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557012.png" /> has an invariant complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557013.png" /> (an invariant hyperplane such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557014.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557015.png" /> can be taken to be the linear form giving the equation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557016.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557017.png" /> has positive characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557018.png" />, Mumford's hypothesis generalizes the fact that there is an invariant homogeneous hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557019.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557020.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557021.png" /> (with the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557022.png" /> equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557023.png" /> for some integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557024.png" />).

Mumford's hypothesis is also equivalent to the assertion that for any regular action of a semi-simple group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557025.png" /> on an affine algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557026.png" /> and for any two closed non-intersecting invariant subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557029.png" /> there is an invariant regular function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557030.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557031.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557033.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557035.png" /> can be separated by regular invariants, see [[#References|[3]]]).

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

The proof of Mumford's hypothesis, together with the results of [[#References|[6]]] and [[#References|[10]]], allow one, first, to give a final form to the generalization of Hilbert's theorem on invariants: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557036.png" /> is an algebra of finite type over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557038.png" /> is a reductive group, acting as an automorphism group on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557039.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557040.png" /> is the subalgebra of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557041.png" />-invariant elements in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557042.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557043.png" /> is also an algebra of finite type over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557044.png" />; and, secondly, to establish that a [[Linear algebraic group|linear algebraic group]] over a field of arbitrary characteristic is geometrically reductive if and only if it is reductive (cf. [[Reductive group|Reductive group]]). Mumford's hypothesis has applications in the geometric theory of invariants and in moduli theory (see [[#References|[7]]]–[[#References|[9]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> D. Mumford, "Geometric invariant theory" , Springer (1965) {{MR|0214602}} {{ZBL|0147.39304}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Fogarty, "Invariant theory" , Benjamin (1969) {{MR|0240104}} {{ZBL|0191.51701}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J. Dieudonné, J.B. Carrell, "Invariant theory: old and new" , Acad. Press (1971) {{MR|0279102}} {{ZBL|0258.14011}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> W.J. Haboush, "Reductive groups are geometrically reductive" ''Ann. of Math.'' , '''102''' (1975) pp. 67–83 {{MR|0382294}} {{ZBL|0316.14016}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C.S. Seshadri, "Geometric reductivity over arbitrary base" ''Adv. Math.'' , '''26''' (1977) pp. 225–274 {{MR|0466154}} {{ZBL|0371.14009}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> M. Nagata, "Invariants of a group in an affine ring" ''J. Math. Kyoto Univ.'' , '''3''' (1964) pp. 369–377 (With appendix by M. Miyanishi) {{MR|0179268}} {{ZBL|0146.04501}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> C.S. Seshadri, "Mumford's conjecture for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065570/m06557045.png" /> and applications" , ''Algebraic geometry. Papers presented at the Bombay Colloq. 1968'' , Oxford Univ. Press (1969) pp. 347–371 {{MR|262248}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> H. Popp, "Moduli theory and classification theory of algebraic varieties" , Springer (1977) {{MR|0466143}} {{ZBL|0359.14005}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> 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 {{MR|0396565}} {{ZBL|0321.14005}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> M. Nagata, T. Miyata, "Note on semi-reductive groups" ''J. Math. Kyoto Univ.'' , '''3''' (1964) pp. 379–382 {{MR|0179269}} {{ZBL|0152.00902}} </TD></TR></table>