Errata for Linear algebraic groups by Springer

Question

Does any one prepared a list of errata for Linear algebraic groups by Springer.

I could not find any in Google search.

First typo that i came across is in page 6, Regular functions and ringed spaces:

If $U $ and $V$ are open subsets and $U\subset V$, restriction defines a $k$ - algebra homomorphism $\mathcal{O}(U) \rightarrow \mathcal{O}(V)$.

I believe it has to be

If $U $ and $V$ are open subsets and $U\subset V$, restriction defines a $k$ - algebra homomorphism $\mathcal{O}(V) \rightarrow \mathcal{O}(U)$.

Am i misunderstanding something or is it really a typo.

I would be grateful if some one can provide list of typos in that book.

Answer 1

The answer 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.

Answer 2

Here is a different one, on page 299: Proposition 17.3.13(iii) giving the relative root system for quasi-simple groups of type $D_{n}$ is not correct as stated. I believe the following is correct, but please comment if not!

If $n > rd$, then the relative root system has type $B_r$ if $d = 1$ and type $BC_r$ if $d > 1$. If $n = rd$, then the relative root system has type $D_r$ if $d = 1$ and type $C_r$ if $d > 1$. The case $n = rd$ only occurs when the group is an inner form (type ${}^1 D_{n,r}^{(d)}$ in Tits' notation).

(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.)

Updated: also part (i) is incorrect if $rd = n-1$. In this case the group is an outer form and the last element $a_{rd}$ in Springer's list should be replaced by the conjugate pair $a_{rd}$, $a_{rd+1}$ (i.e. $a_{n-1}$, $a_n$). See Tits' Boulder article from 1965.