Errata for Course Notes - J.S. Milne, Top
This file contains miscellaneous errata and additional remarks for my course notes that I haven't yet incorporated into the versions on the web.
Most are taken from e-mail messages -- I thank everyone who has contributed!

Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Complex Multiplication
Algebraic Groups
Basic Theory of Affine Group Schemes
Lie Algebras, Algebraic Groups, and Lie Groups
Reductive Groups
Algebraic groups, Lie groups, and their arithmetic subgroups

Group Theory v4.00 (GT)

p.69. Remove "not" from the last line of the proof of Lemma 4.36. (Deron Lessure).

p.29. In the proof of Theorem 1.64, I'm using that the characteristic polynomial has distinct roots. Specifically, an nxn matrix with coefficients in a field k is similar to a diagonal matrix if its characteristic polynomial has n distinct roots in k.

The e-reader version 3.11 is not being corrected to updated.

Fields and Galois Theory v4.61 (FT)

With the change in the definition of "separable polynomial" it no longer makes sense to speak of "inseparable polynomials" unless they are irreducible. This is only a problem in Example 4.5 (Weijie Zhang).

p.20. In the proof of 1.32, replace a superscript $n=1$ with $n-1$ (Matin Hajian).

p.48. In Corollary 4.2(b), it is necessary to assume that characteristic$(F)\neq 2$, otherwise even the first line of the proof fails (all elements of $S_n$ fix $\Delta(f)$). (Carlos Alberto Ajila Loayza)

p.127 in the solution for exercise 3-2 I write:
Thus for $p\equiv 1$ mod 4, $\alpha\beta=-((p-1)/2)^2/(p-1)=(p-1)/4$.
It should read $=-(p-1)/4$ (minus sign is missing). The minus sign is restored when I (correctly) write the polynomial satisfied by $\alpha$ and $\beta$ as $X^2+X-(p-1)/4$. (Ralf Goertz)

p.98. Exercise 7-1: replace $\mathbb{Z}_p$ and $\mathbb{Z}/p^m\mathbb{Z}$ with $\mathbb{Z}_p^{\times}$ and $(\mathbb{Z}/p^m\mathbb{Z})^{\times}$.

p.135. The solution to A-10 should say that $Gal(E_1E_2/F)$ is the fibred product of the two groups over $Gal(E_1\cap E_2/F)$. (Hongjun Lee)

I muddled part of the proof of Proposition 4.29, as explained in the following message from Kenneth Harris.
I have run into a problem at one step of your proof of Proposition 4.29 (based on van der Waerden) in the third paragraph of Lemma 4.34: "$\sigma_\alpha \in G_f \iff \text{ it maps }F(T)[\theta] \text{ isomorphically onto }F(T)[\sigma_\alpha \theta]$".
The direction => is clear, but the direction <= was a mystery to me (as was the point of the gimmick of introducing the new variables $T_1,...,T_m$!).
I went to van der Waerden and he states practically the same argument without follow-up for the reader. So, I backtracked to the beginning of the chapter on Galois theory in MA and realized he always assumed the splitting field of a polynomial was generated by a primitive element, which can be taken as a linear combination of the roots of the polynomial. van der Waerden also assumed the polynomial f(x) in the statement of the main theorem of Sec. 61 of MA was separable. Now, it is clear to me the <= direction of the argument and why the introduction of the new variables.
Your statement of Proposition 4.29 does not have this assumption (only that $\bar{f}$ is separable), but your conclusion includes that $f$ is separable (although the first line of the proof still has the assumption that $f$ is separable.) I don't see how to prove Lemma 4.34 without separability of $f$, but your argument that $\bar{f}$ separable implies $f$ separable did not depend on this Lemma.
On a very minor point: there are several places in the proof of Lemma 4.34 where you have $s_\alpha$, $s_T$ where you mean $\sigma_\alpha$, $\sigma_T$.

Algebraic Geometry v6.03 (AG)

p.50, last line, $X_m$ should be $X_n$ (Kai Wang).

p.179, the second line of the proof of Lemma 8.20 should send $(f,g)$ to the product $f|_{U'} \cdot g\circ\varphi|_{U'}$. Also, in the bottom row of the commutative diagram there is a missing product symbol $\prod_{i,j}$.
p.201, in Example 9.13 (b), flat maps of varieties not are defined until 9.22, two pages later.
p.208, I think Aside 9.39 begs for some extra explanations for the quotient of an affine variety by a finite subgroup of its automorphisms, for example that the quotient map is finite and its fibers are the orbits of the action. (Felipe Zaldivar)

Algebraic Number Theory v3.08 (ANT)

Solution to 2-7. For the converse, need to start with an element in $L$, not $S^{-1}B$ (Ying Zhao).

In Example 5.14, it is claimed that 22 is not a square modulo 29, however $14^2=196=22+6\times 29$.
Instead argue as follows. As $K$ is a subfield of $\mathbb{R}$ the first and last possibilities $\frac{\alpha-1}{\alpha+2}$, $-\alpha\frac{\alpha-1}{\alpha+2}$ are eliminated as they are negative hence not square.
Take $\mathfrak{q}=(29,\alpha-2)$, the third $\alpha\frac{\alpha-1}{\alpha+2}$ is excluded.
Take $\mathfrak{q}'=(29,\alpha-3)$, the second $-\frac{\alpha-1}{\alpha+2}$ is excluded. (Haohao Liu, Jingxin Wang)

In Remark 7.17, I say that "according to 7.3, the absolute values of $K$ are discrete", but in 7.3, I only prove that they are nonarchimedean. Sometime, I'll add a proof (not difficult) that they are, in fact, discrete (Haohao Liu).

Modular Forms and Modular Functions v1.31 (MF)

p78, in the normalized Eisenstein series $E_6(z)$, the fractional coefficient $54600/691$ should instead be $65520/691$ (Li Han).

p79 The claim in the proof of Proposition 5.26 that $E_2$ and $E_3$ generate the algebra of modular forms with integer coefficients $a_n$ is incorrect because the modular form $(E_2^3 - E_3^2)/1728$ has integer coefficients, but it is not a polynomial in $E_2$ and $E_3$ with integer coefficients.
The Eisenstein series do not form an integral basis of modular forms. For example, in the book Modular Forms by Cohen and Stromberg (section 10.6) they use a basis that also involves $(E_2^3 - E_3^2)/1728$. (Xevi Guitart)

p118, l. -7, "a \equiv 3 mod 3" doesn't look right (Timo Keller).

Abelian Varieties v2.00 (AV)

This draft is still very rough. Some proofs have been fixed in the corrected version of my 1986 article on Abelian Varieties (xnotes). There will be a new version sometime...

p1. In equation (1), $aXZ$ should be $aXZ^2$ (Helge Øystein Maakestad).

p2. The claim in the footnote that every abelian surface is a Jacobian variety is not quite true. See the preprints of E. Kani, "The moduli spaces of Jacobians isomorphic to a product of two elliptic curves" and "The existence of Jacobians isomorphic to a product of two elliptic curves". (Kuang-yen Shih)

p.113, for a finite group $M$, the Galois coverings with group $M$ are classified by the surjective homomorphisms $\pi_1(V,P)\to M$.

p. 136, l. 6: "there *is* an abelian variety ..." (Timo Keller).

p37, line 6. The regular map $\alpha$ should go from $T$ to the dual of $A$ (Bart Litjens).

From Bhupendra Nath Tiwari: For AV, CFT, and CM

From Tobias Barthel See pdf file (2 pages)

From Shaul Zemel In the proof of Theorem 10.15, p49, concerning the map from $\textrm{Hom}(A,B) \otimes Z_l$ to the module $\textrm{Hom}(T_l A,T_l B)$ (over $Z_l$), you start by proving that if $e_1,...,e_n$ are linearly independent over Z in Hom(A,B) then their images are linearly independent over Z_l in Hom(T_l A,T_l B). But this immediately proves that n cannot exceed the rank of the latter over Z_l, i.e., 4dimAdimB (as can be even more clearly seen in Hom(V_l A,V_l B) over Q_l, after tensoring the latter with Q_l). Hence you immediately obtain the finiteness of the rank of Hom(A,B), and the desired bound, without the need to involve decomposition into simple Abelian varieties, different topologies, and polynomials. This is in fact similar (as you have indicated there for something else) to the fact that showing that if a (clearly torsion-free) subgroup of a real vector space of dimension n is discrete then it's free of rank not exceeding n. [This is fixed in AVs I think.]

p47, Lemma 10.12. The statement should require that the degree of $f(xv+w)$ as a polynomial in $x$ is bounded, say, at most 2g (otherwise the first displayed sum in the proof may be infinite). (Angus Chung).

Timo Keller points out that, in the proof of Theorem 10.15, p49, M should be defined to be a Z-submodule of End^0(A), not End(T_l(A)) (the degree map P is defined on End^0(A), not on End(T_l(A)). Also, when I write "Now choose the e_i to be a Q-basis for End^0(A)." I seem to be assuming that End^0(A) is finite dimensional over Q, which is what I'm trying to prove. The proof should be replaced by this.

From Everett Howe In Prop. 13.2(b), I found a small typo, probably carried over from copying the result from [1986b] and not changing all of the notation: the "f" in the exponent should be an alpha.

Tim Dokchitser points out that I prove Zarhin's trick (13.12) only over an algebraically closed field , and then immediately apply it in (13.13) over a finite field. This is doubly confusing because (13.10) is certainly false over nonalgebraically closed field (over such a field an abelian variety need not be isogenous to a principally polarized abelian variety).
However, I believe everything is O.K. Specifically, the proof of Zarhin's trick requires only (13.8), and, because this holds over an algebraically closed field, it holds over every perfect field (see my 1986 Storrs article Abelian Varieties 16.11 and 16.14).

From Sunil Chetty. Near the start of I 14 (Rosati involution): in $(\alpha\beta)^\dagger = \beta\alpha$ there should be a dagger on each of $\beta$ and $\alpha$.

p.154, IV. Theorem 7.3. The property in (a) of the statement only holds for the points P outside a closed subset of codimension 1. As Martin Orr writes:

[The theorem] asserts that the Siegel moduli variety M_{g,d} over the complex numbers satisfies: for every point P in M_{g,d}, there is an open neighbourhood U of P and a family A of polarized abelian varieties over U such that the fibre A_Q represents j^{-1}(Q) for all Q in M_{g,d} (I assume this should say "all Q in U" at the end).
I don't see how such a neighbourhood can exist around the elliptic point 0 in the j-line, because of the standard monodromy argument that there is no family of elliptic curves on all of M_{1,1}. Specifically, if we had such a U then the period mapping (which is just the inclusion U -> M_{1,1}) would lift to a map from U' to the upper half plane H, for some open neighbourhood U' of 0 (wlog lifting 0 to i). Then the image of this lifting contains an open neighbourhood V of i in H, and the map H -> M_{1,1} is not injective on V.

From Roy Smith (on proofs of Torelli's theorem III 13)
You ask on your website for advice on conceptual proofs of Torelli. ... here goes.
There are many, and the one you give there is the least conceptual one, due I believe to Martens.
Of course you also wanted short, ....well maybe these are not all so short.
The one due to Weil is based on the fact that certain self intersections of a jacobian theta divisor are reducible, and is sketched in mumford's lectures on curves given at michigan. Indeed about 4 proofs are sketched there.
The most geometric one, due to Andreotti - Mayer and Green is to intersect at the origin of the jacobian, those quadric hypersurfaces occurring as tangent cones to the theta divisor at double points, thus recovering the canonical model of the curve as their base locus, with some few exceptions.
To show this works, one can appeal to the deformation theoretic results of Kempf. i.e. since the italians proved that a canonical curve is cut out by quadrics most of the time, one needs to know that the ideal of all quadrics containing the canonical curve is generated by the ones coming as tangent cones to theta. the ones which do arise that way cut out the directions in moduli of abelian varieties where theta remains singular in codimension three.
But these equisingular deformations of theta embed into the deformations of the resolution of theta by the symmetric product of the curve, which kempf showed are equal to the deformations of the curve itself. hence every equisingular deformation of theta(C) comes from a deformation of C, and these are cut out by the equations in moduli of abelian varieties defined by quadratic hypersurfaces containing C. hence the tangent cones to theta determine C.
This version of Green's result is in a paper of smith and varley, in compositio 1990.
Perhaps the shortest geometric proof is due to andreotti, who computed the branch locus of the canonical map on the theta divisor, and showed quite directly it equals the dual variety of the canonical curve. this is explained in andreotti's paper from about 1958, and quite nicely too, with some small errata, in arbarello, cornalba, griffiths, and harris' book on geometry of curves.
There are other short proofs that torelli holds for general curves, simply from the fact that the quadrics containing the canonical curve occur as the kernel of the dual of the derivative of the torelli map from moduli of curves to moduli of abelian varieties. this is described in the article on prym torelli by smith and varley in contemporary mat. vol. 312, in honor of c.h. clemens, 2002, AMS. there is also a special argument there for genus 4, essentially using zariski's main theorem on the map from moduli of curves to moduli of jacobians.
There are also inductive arguments, based on the fact that the boundary of moduli of curves of genus g contains singular curves of genus g-1, and allowing one to use lower genus torelli results to deduce degree torelli for later genera.
Then of course there is matsusaka's proof, derived from torelli's original proof that given an isomorphism of polarized jacobians, the theta divisor defines the graph of an isomorphism between their curves.
For shortest most conceptual, I recommend the proof in Arbarello, Cornalba, Griffiths Harris, i.e. Andreotti's, for conceptualness and completeness in a reasonably short argument..

Lectures on Etale Cohomology v2.21 (LEC)

From Section 9 on, I often abbreviate $H^i(X_{et},F)$ to $H^i(X,F)$ (Hongbo Yin).
On page 147 in theorem 25.1 (The Lefschetz fixed-point formula) the trace term should have a matching right parenthesis. (Andrew Salmon)

From Zheng Yang.
In S. 2, p. 20, there is a typo in the definition of an unramified morphism of schemes, namely, it should be "$O_{X, \varphi(y)} \rightarrow O_{Y, y} ...$"
In S. 6, p. 45, under "The sheaf defined by a scheme $Z$" the functor given is notated $\mathcal{F}$ but in subsequent appearances it is written as $\mathcal{F}_Z$. It may be better to correct it for this first instance.
In S. 6, Example 6.13 (b) the stalks of finite type schemes $Z$ over $X$ are computed. It might be helpful to include a reference for this fact, for instance Lemma 3.3 in your book "Étale Cohomology."
In S. 10, Theorem 10.7 there is a typo in the statement of the Grothendieck spectral sequence, namely $FG$ on the right should be $GF$.
In S. 13, p. 84, the very first sentence references "We saw in the last section..." but it should instead reference S. 11.
In S. 14, p. 89, after "We saw in the last section..." the $0$-th cohomology group of $U$ should be $\Gamma(U, O_U^{\times})$. (Which matches with the notation in Thm. 13.7).
Fix the following problems: the heading of S. 27 "Proof of the Weil Conjectures..." is off (p. 154) and there is some overlap with the section name of S. 29 "The Lefschetz fixed point formula..." with the page numbers.

(Bence Forrás)
(Apparently) I should use caron instead of breve in Cech.
In the diagram on page 22, the subscript $b$ should be before the last bracket.
In the proof of Proposition 8.12, $\mathcal{P}^*$ is not defined. Define it to be $\prod i_{x*}i^{x*}\mathcal{F}$. There are canonical injective homomorphisms $\mathcal{F}\to \mathcal{P}^*\to \mathcal{I}$.
The first sentence on page 118, should read "as it is in topology" instead of "as it is topology".
The second word in Example 21.4 should be "hypothesis" instead of "hypotheses"?

p.56. "Grothendieck has banished [\'espace \'etal\'e] from mathematics." Grothendieck uses them in his 1957 Tohoku paper (p. 154-155) but (somewhat) banishes them in 3.1.6, p.25, of Chapter 0 in EGA I.

(Nikita Kozin)
p.78: "Proof: Let M be..." - change M (mathfrak?) to another font
p.80: "Remark 11.7: ... associated with the preseaf ... H^r" - must be "H^s"?
p.81: Diagram - must be "Sh(Y_et)" in left low corner?

p. 176. Lemma 30.2 is true for semisimple endomorphisms, but is false in general, as the following example illustrates. Let $K=\mathbb{Q}[a]$, where $a=\sqrt{2}$, and let \[ A=% \begin{pmatrix} a & 1 & 0 & 0\\ 0 & a & 0 & 0\\ 0 & 0 & -a & 0\\ 0 & 0 & 0 & -a \end{pmatrix} \in M_{4}(K). \] Then $A$ has characteristic polynomial $(X-a)^{2}(X+a)^{2}=(X^{2}-2)^{2}% \in\mathbb{Q}{}[X]$, and so, according to the lemma, $A=CBC^{-1}$ with $B\in M_{4}(\mathbb{Q}{})$ and $C\in\mathrm{GL}_{4}(K)$. Let $\sigma$ be the automorphism of $K$ sending $a$ to $-a$. Then $\sigma A=(\sigma C)B(\sigma C)^{-1}$, and so $\sigma A$ is similar to $A$. But \[ \sigma A=% \begin{pmatrix} -a & 1 & 0 & 0\\ 0 & -a & 0 & 0\\ 0 & 0 & a & 0\\ 0 & 0 & 0 & a \end{pmatrix}, \] which is not similar to $A$ by the uniqueness of Jordan forms. (The point is that the characteristic polynomial doesn't say anything about the Jordan blocks.)

From a different perspective, there are four isomorphism classes of $K[X]$-modules with characteristic polynomial $(X^{2}-2)^{2}$, but only two isomorphism classes of $\mathbb{Q}{}[X]$-modules with this characteristic polynomial. Thus, not every $K[X]$-module with characteristic polynomial $(X^{2}-2)^{2}$ can be obtained by extension of scalars from a $\mathbb{Q}% {}[X]$-module, contradicting the lemma.

This necessitates some later changes (I think minor). (From Vladimir Uspenskiy.)

Class Field Theory v4.03 (CFT)

p.20, Corollary 1.2, (d) should read Nm($(L\cap L')^{\times}$), not $L\cap L$.

p.32, line -5. Delete the third $\pi$: ... $=\pi^{r+1}Q(X_1\ldots)$ (David Craven).

p.41. In step 3, $\theta F_f(\theta^{-1}(X),\theta^{-1}(Y))$ has the properties characterizing $F_g(X,Y)$ (and not $F_f(X,Y)$) (Ashutosh Jangle).

IV, Example 1.11. Replace the last sentence with a reference to GT 7.19(a).

Proof of Theorem II.3.10. In the paragraph "To proceed further", there is a bit of confusion created by the use of $H^r$ instead of $H^r_T$, which then propagates into the next paragraph and the use of the Sylow theorems: the reduction to the solvable case is only valid for $r>0$, whereas you also need the case $r=0$ in order to carry out the dimension-shifting argument to cover $r<0$.
This is of course easy to fix: one just has to observe that since $H^0_T(G_p, M) = 0$, we can deduce directly that $H^0_T(G, M)$ is killed by $[G:G_p]$. Hence again, varying over all $p$ tells us that this group vanishes; and then the dimension-shifting argument kicks in. (Kiran Kedlaya)

Example 2.1 in Chapter IV (page 126) reads: "...note that a ring $B$ containing a subring $R$ in its centre is isomorphic to $M_n(R)$ if and only if it admits a basis $(e_{ij})$ as an $R$-module such that..."
The condition of $R$ being in the centre of $B$ is too restrictive, since it in particular implies that $R$ itself is commutative. Hence it isn't of use in applications concerning Brauer Groups, where we have $R=A$ a $k$-algebra which is not necessarily commutative.
The statement should be: "...note that a ring $B$ containing a subring $R$ is isomorphic to $M_n(R)$ if and only if it admits a basis $(e_{ij})_{1 \leq i,j \leq n}$ as a left $R$-module such that $e_{ij}\cdot e_{lm} = e_{im}$ if $j=l$ and $=0$ otherwise, and every $e_{ij}$ commutes with the elements of $R$." (Milan Malcic)

At the bottom of page 154 and the beginning of page 155, replace "$1-s$" with "$s-1$" (three times), (Yiming Tang).

From Raghuram Sundararajan

They are labelled as (typographical), (mathematical) and (suggestion) for typographical/grammatical errata, mathematical errata and suggestions to make proofs easier to follow, respectively.

Complex multiplication v0.10 (CM)

No known errors.

Algebraic Groups v1.00 (iAG)

These notes are obsolete --- read my book, Algebraic Groups, Cambridge University Press, 2017 instead.

19.56 is misstated: It should say that every split reductive group... T is a split torus.

Basic Theory of Affine Group Schemes v1.00 (AGS)

p.30. The definition of a sub-coalgebra doesn't make sense unless, e.g., k is a field, because in general D\otimes D is not a submodule of C\otimes C. (Roland Loetscher).

p.60, 5.3. This is completely muddled --- the homomorphism $i$ goes in the other direction and is surjective etc. I was probably thinking of a group $G$ over $k$ (not $k^{\prime}$) etc.... (Giulia Battiston)

p.261. For an improved exposition of the proof of Theorem 5.1, see RG I, 1.29.

p.259. There is a problem is with my definition of "almost-simple". Certainly "almost-simple" should imply "semisimple", so we should define an almost-simple algebraic group to be a noncommutative smooth connected algebraic group that is semisimple (so its radical over the algebraic closure is trivial) and has no nontrivial proper smooth connected normal algebraic subgroup. The quotient of such a group by its centre should be called simple. (The centre of a reductive group is the kernel of the adjoint representation on the Lie algebra. It suffices to check this over an algebraically closed field, see RG 2.12).

If we don't require that G be semisimple, then we get to the land of pseudo-reductive groups (Conrad-Gabber-Prasad), which is very complicated. Tits defines a pseudo-simple algebraic group to be a noncommutative smooth connected affine algebraic group with no nontrivial proper smooth connected normal subgroup. (Sebastian Petersen)

Lie Algebras, Algebraic Groups, and Lie Groups, v2.00 (LAG)

p.15. In the final paragraph before "The isomorphism theorems", it states that if a is a characteristic ideal of g, then every ideal in a is also an ideal in g. It should read: If a is an ideal in g, then every characteristic ideal in a is also an ideal in g (which is characteristic if a is characteristic).
p.39. From Rui Xiong: At the end of the proof of Theorem 3.17, the fact that $[\overline{x}_s,\mathfrak{g}]\subset \mathfrak{g}$ follows not by $\overline{x}_s$ being a polynomial of $x$. It follows by the fact that $\operatorname{ad} \overline{x}_s$ is a polynomial of $\operatorname{ad} x$, which can be checked by using standard basis $E_{ij}$ with $1$ at $i,j$-indices and $0$ at others. I learnt this proof from Fulton's Representation Theory Page 479.

From David Calderbank
Page 52, Weyl's Theorem 5.20. Part (a) states "If ad is semisimple, then g is semisimple". However, semisimplicity of the adjoint representation is characteristic of reductive Lie algebras (Proposition 6.2), not just semisimple ones. The proof uses more than semisimplicity of the adjoint representation.

Page 56, Proposition 5.29. In the non-algebraically closed case, I don't think it is necessarily true that a Lie algebra with all elements semisimple is commutative. Indeed, so(3) over R would be a counterexample.

Page 57, Proposition 6.4. I don't see how the proof establishes the implication "g has a faithful semisimple representation => g is reductive".

Reductive Groups, v1.00 (RG)

p.14, following the statement of 1.29: The centre (and radical) of GL_n consists of the scalar matrices, not the diagonal matrices. (Justin Campbell)

From Timo Keller
p. 21: ... with [A,B] = AB - BA,and <- space missing
p. 25, l.-3: ) missing at the end of the equation

Algebraic groups, Lie groups, and their arithmetic groups v3.00 (ALA)

p. 340 footnote 4. From Brian Conrad: you point out that a root datum is really an ordered 6-tuple, not an ordered 4-tuple. In case it may be of interest, the last bit of information in the 6-tuple (the bijective map a |---> a^{\vee} from roots to coroots) is uniquely determined by everything else, due to the requirements in the axioms of a root datum -- see Lemma 3.2.4 in the book "Pseudo-reductive groups".

I, Theorem 16.21. The centre of a reductive group is of multiplicative type, but need not be reduced (e.g., SL_p). The correct statement is that the radical of G (which is a torus) is the reduced connected group attached to the centre. (There may be other errors of this type, where I incorrectly translated from the characteristic zero case to the general case.) There will be a new version of the notes probably in March 2012.


p. 372 footnote 16. Chapter II?? should be Chapter III.
p. 172, l.1: "and so is split be a finite" should be "and so is split by a finite" (Timo Keller).