Algebra Chapter 0 Errata

[Ben Hollinbeck]

Theanswer to your specific question is that it's really stated backwards. More generally, your question about lists of errata comes up fairly often here and is hard to answer in detail. It's a legitimate question to ask when looking at relatively advanced books in mathematics. (Maybe a special tag is needed?) But unfortunately, queries to both authors and publishers are apt to be ignored.

As noted in the comments, Tonny Springer himself is no longer alive. His years were 1926-2011, and his death followed a sudden aneurysm. He was a gifted and creative Dutch mathematician whose work has continued to be highly influential (as seen for instance in the title of a well-attended workshop here at UMass in October 2015). His 1978 Notre Dame lectures on linear algebraic groups led to his first edition (Birkhauser, 1981). But in attempting to avoid some of the tricky aspects of algebraic geometry in prime characteristic, he didn't quite succeed at first. He also found a need to add some topics to the book, so a second edition followed in 1998. I'm assuming this is the edition you are looking at.

[By the way, the publisher Birkhauser (with an Umlaut over the a) is the same, but their Boston branch was originally an offshoot of the Swiss publishing house and was later acquired by Springer-Verlag: no relation to T.A. Springer. Indeed, when there was concern decades ago about the confusion of Springer-Verlag with the right-wing Axel Springer publishing company, Tonny Springer posted on his office door in Utrecht a headline Springer $\neq$ Springer from the Springer-Verlag disclaimer published in mathematical journals at the time.]

Concerning errata in books, these very often occur but are inadequately tracked by publishers or living authors. An exception is the American Mathematical Society, which offers to authors an online bookpage where errata and other supplementary materials can be posted. I've maintained similar lists on my own webpage, never having tried out my own books on actual students.

Errata come in all sizes, ranging from obvious misprints to mistaken assertions to exercises poorly stated or misplaced. Those I've noticed and marked in my own copy of Springer's second edition tend to be minor, but I try to keep a list of page numbers. It's amusing to spot the errors (sometimes more than one) on his pages 195, 218, 265, 284, 304, 325, 332. Probably the most serious oversight is on page 321, where one index is apparently inexact. There are also some items such as $F$-reductive missing from the list of terminology at the end of the book.

(I don't know a reference. I didn't read this section of Springer, but I worked out the relative root datum from the absolute one in each case. Over a non-archimedean local field the answer agrees with Tits' Corvallis table.)

The errata below are for the updated first edition of Modern Robotics (as well as the practice exercises and linear algebra refresher appendix). The updated first edition (also called "version 2") was originally published by Cambridge University Press in late 2019 (marked "3rd printing 2019" or later) and the corresponding online preprint is dated December 2019. The updated first edition includes several corrections and minor additions to the original first edition, which was originally published by Cambridge in May 2017, with a corresponding online preprint dated May 2017.

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.

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 themodular form $(E_2^3 - E_3^2)/1728$ has integer coefficients,but it is not a polynomial in $E_2$ and $E_3$ with integercoefficients.

The Eisenstein series do not form anintegral basis of modular forms. For example, in the bookModular Forms by Cohen and Stromberg (section 10.6) they use a basis thatalso involves $(E_2^3 - E_3^2)/1728$. (Xevi Guitart)

Timo Keller points out that, in the proof of Theorem 10.15, p49, M should be defined to be aZ-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 bea Q-basis for End^0(A)." I seem to be assuming that End^0(A) isfinite dimensional over Q, which is what I'm trying to prove. The proof should be replaced bythis. From Everett Howe In Prop. 13.2(b), I found a small typo, probably carried overfrom copying the result from [1986b] and not changing all of the notation: the "f" in theexponent should be an alpha.

Tim Dokchitser points out that I prove Zarhin's trick (13.12) onlyover 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 fieldan 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 trickrequires only (13.8), and, because this holds over an algebraically closed field, it holdsover every perfect field (see my 1986 Storrs article Abelian Varieties 16.11 and16.14).From Sunil Chetty.Near the start of I 14 (Rosati involution): in $(\alpha\beta)^\dagger = \beta\alpha$ thereshould be a dagger on each of $\beta$ and $\alpha$.p.154, IV. Theorem 7.3. The property in (a) of the statement onlyholds 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 selfintersections of a jacobian theta divisor are reducible, and issketched in mumford's lectures on curves given at michigan. Indeedabout 4 proofs are sketched there.

The most geometric one, due to Andreotti - Mayer and Green is tointersect at the origin of the jacobian, those quadric hypersurfacesoccurring as tangent cones to the theta divisor at double points, thusrecovering the canonical model of the curve as their base locus, withsome few exceptions.

To show this works, one can appeal to the deformation theoreticresults of Kempf. i.e. since the italians proved that a canonicalcurve is cut out by quadrics most of the time, one needs to know thatthe ideal of all quadrics containing the canonical curve is generatedby the ones coming as tangent cones to theta. the ones which do arisethat way cut out the directions in moduli of abelian varieties wheretheta remains singular in codimension three.

But these equisingular deformations of theta embed into thedeformations of the resolution of theta by the symmetric product ofthe curve, which kempf showed are equal to the deformations of thecurve itself. hence every equisingular deformation of theta(C) comesfrom a deformation of C, and these are cut out by the equations inmoduli of abelian varieties defined by quadratic hypersurfacescontaining 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, whocomputed the branch locus of the canonical map on the theta divisor,and showed quite directly it equals the dual variety of the canonicalcurve. this is explained in andreotti's paper from about 1958, andquite 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 generalcurves, simply from the fact that the quadrics containing thecanonical curve occur as the kernel of the dual of the derivative ofthe torelli map from moduli of curves to moduli of abelian varieties. this is described in the article on prym torelli by smith and varleyin contemporary mat. vol. 312, in honor of c.h. clemens, 2002, AMS. there is also a special argument there for genus 4, essentially usingzariski's main theorem on the map from moduli of curves to moduli ofjacobians.

There are also inductive arguments, based on the fact that theboundary of moduli of curves of genus g contains singular curves ofgenus g-1, and allowing one to use lower genus torelli results todeduce degree torelli for later genera.

Then of course there is matsusaka's proof, derived from torelli'soriginal proof that given an isomorphism of polarized jacobians, thetheta divisor defines the graph of an isomorphism between theircurves.

For shortest most conceptual, I recommend the proof in Arbarello,Cornalba, Griffiths Harris, i.e. Andreotti's, for conceptualness andcompleteness 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 $\mathcalF$ but in subsequent appearances it is written as $\mathcalF_Z$. 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 "tale 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.

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