38 M. L. RACINE
and will be referred to as the transformation of Theorem II. 1.
COROLLARY 3. Any maximal *-stabl e order F or C is isomorphic to
ti
s I O
u
where, if F = E(L) nE(L)* a s in the preceding theorem, s is the number of
elementary divisors of H which are equal 1.
PROOF. Argue as in the proof of Lemma 11 .
Let L be a lattice of lr. A sublattice K of L is said to be a
component of L if L = K 0 K , where K = (&K) nL. An element x e L is
primitive if x I pL; L is said to be i-modular, i e S , if h(x, L) = * P for all
primitive vectors x = L; L is said to be modular if it is i-modular for some i.
The rank of L is the $ dimension of $L. Introduce the following notation:
L = L1 _ L L
?
means L = L7 0 L
?
and L. are components i = 1,2, : L ^ L',
L is isometric to L'; L ~ (a..) means the matrix of h with respect to a bas e
of L is (a,,); L ~ (a) , L is a line and the restriction of h to L ha s
matrix (a); L ~ (a.,) J_ (b ) means L = L. ± L
0
and L. ~ (a..), L . ~ (b ).
i j ' v rs 1 2 1 -
IJ
2 rs
Scaling by "K means multiplying h by 7\.
LEMMA 13. Any lattice L is a direct sum of modular components of
rank 1 or 2 ( i . e . L is an orthogonal sum of modular lines and planes).
0 0 . . . 0
© o . . . r | s
y © . . . ©
? © . . . © ,
s
Previous Page Next Page