v0.0 Posted during the lectures at CMS, Zhejiang University, Hangzhou,
Feb 28, 2005 --- May 7, 2005.
Note: When the course is ended I'll post a slightly revised version, so unless you
wish to follow the course in "real time" you should wait for that version.
Erratum
There are many "cosmetic problems" that need to be fixed --- missing punctuation, inconsistent spelling, inconsistent notation,...
Somewhere I should warn that algebraic groups over nonalgebraically closed fields are not "homogeneous".
For example, their different connected components may look quite different.
This is seen already with mu_3 over Q.
p6, in the section on nonconnected groups, the G should be an M.
p7, in the table, the group at left has a composition series whose quotients
are the groups on the right.
p11, Example 2.8. SL_V(R) should be automorphism of R\otimes V etc.,
and similarly for the other groups.
p14, in the description of S for GL_n replace y^{-1} with y.
p15, there are minor errors in the diagrams for identity/co-identity, most of which
are fixed on p23.
p15, in 2.11 should require the bialgebras to be finitely generated as k-algebras (or perhaps make this part of the definition of a k-bialgebra).
p16, in order of k[X]/(X^n) to be not reduced, need n>1 (not n>0).
p16, it is not true that the map Delta:A -> AotimesA defines a map B ->BotimesB. This can
be seen already for G=G_a. Fortunately, the error can be corrected by working mainly
with the ring A, or as in the original paper Oort, Inventiones Math. 2 (1966), 79-80.
p23, in the diagram for S, replace A at bottom-left with k.
p24, line 2, should be ... xy'.
p24, line 3, should be: This is an algebraic group because it is represented
by k[X,Y,Z], and it is noncommutative.
p24, line 5. Omit the second matrix.
p28, Theorem 4.8. Replace "there exists a unique smooth" with "there exists a unique reduced" (if k is not
perfect, it need not be smooth). For the uniqueness part of the statement, should assume that k is
algebraically closed.
p37. It is not difficult to prove 5.18 directly. First show that a linear combination of reduced monomials is in the centre if and only if each monomial is, and then find the monomials that commute with each e_j.
p39, line 14. In the definition of the Clifford group, the condition should be
gamma(t)Vt^{-1}=V.
p42, "is the sequence" is too strong --- with a little effort they can be shown to be isomorphic.
p45, 6.19b. The proof in fact shows that G->Q is a quotient map if G(kbar)->Q(kbar) is surjective and Q is smooth.
p48, 6.25b HcapN is a normal algebraic subgroup of H (not G).
p48, exercise 6-1. Need to assume k is perfect in order to be sure Abar\otimes Abar is reduced.
Also need to add a condition on G_{red} for uniqueness! For example, it is the largest smooth subgroup;
or the unique smooth subgroup H with spm(k[H])=spm(k[G]).
p49, n>1 (not n>=1).
p51, 7.7b: Better to say ... the algebraic group G is etale if and only if
the morphism of schemes Spec k[G] -> Spec k is etale.
p52, 8.3. Here is a proof. Replace A with A/rad(a). Then we
have to show that if a is not nilpotent then it is not contained in some
maximal ideal m. But a will not be nilpotent in A\otimes kbar, and
so there exists a k-algebra homomorphism f:A\otimes kbar -> kbar with
f(a) nonzero (Nullstellensatz). The image of f:A -> kbar is a field (exercise),
and so its kernel is a maximal ideal not containing a.
p53, the argument for the converse in the proof of 8.7 doesn't directly generalize to n.
You can do it two at a time, or use a different argument to show that taking products of rings corresponds to taking disjoint unions of the spm's.
p54, the proof of 8.10 is still not persuasive. Because spmA is noetherian, it is a finite union
of its irreducible components (AG 2.21). Each connected component is a finite union of irreducible components,
and so there are only a finite number, say r, of connected components. The number of e_i is at most r.
p63, the proof that the sum is direct is incomplete, but is completed in (11.21).
p67. In 10.5 the subscripts n should be u.
p69, in the second last line of the proof of 10.11, it should be the inclusion \rho:V...
p69, the proof of 10.12(a) is incomplete --- see p72.
p85, in a noncommutative situation, the Leibniz rule should be written D(fg)=fD(g)+D(f)g.
p87,91, in 12.13, 12.22 and probably elsewhere, p is the characteristic of the field.
p87, Example 12.14 (and elsewhere). The notation id_V+\epsilon\alpha for the elements of Lie(GL_V) is bad
--- the id_V should be id_{V+Vepsilon}.
The element id+\epsilon\alpha sends x+\epsilon y to x+\epsilon y + \epsilon\alpha(x).
p121, roughly speaking, the reason the Weyl group W is constant
is that W(k) equals the Weyl group of the root datum, which doesn't
change when the field is enlarged.
p121, the first sentence of the proof of 17.9 should read:
The key point is that the derived group of G_\alpha has rank one
and T is a maximal torus in G_\alpha.
p122, in 17.10, T_alpha = 1 (not T).
p122, in 17.11, n_alpha represents the unique nontrivial
element s_alpha of W(G_alpha,T) (not T_alpha).
p123, line 3, the 0 should be the neutral element of k^\times,
usually denoted 1.
p124, line -4, the map is not chi_i but \sum a_i lambda_i.
p124, 125. There is a minus sign missing in all the
descriptions of the Lie algebras. For example, on line 10 the
equation should be phi(Ax,y)= (minus) phi(x,Ay).