Errata for Expository Notes

This file contains miscellaneous errata and additional remarks for my expository notes that I haven't yet incorporated into the versions on the web.

Corrections for SVI, SVH, and MOT from Bhupendra Nath Tiwari

A Primer of Commutative Algebra v4.02, 2017 (CA)

p.34, Corollary 7.12. The prime ideals $p$ are in $A$, not $B$.

Motives --- Grothendieck's Dream v2.04, 2012

p.10, line 2: should read "it is possible" (the "is" is omitted).

p.15, Literature: the book of Murre et al. has now been published:
Murre, Jacob P.; Nagel, Jan; Peters, Chris A. M. Lectures on the theory of pure motives. University Lecture Series, 61. American Mathematical Society, Providence, RI, 2013.

Introduction to Shimura Varieties (as posted on my site)

p.11. From Alexey Beshenov: On p. 11 you give example of P^1 (C) as a Hermitian symmetric space and you say "reflection along a geodesic through a point is a symmetry". I think you actually want to give a symmetry with isolated fixed points, and it could be rather the rotation of the sphere by \pi along the axis, which leaves the poles fixed.

p. 23, line 13: t is missing on the right hand side
p. 26, line 23: the pair (r,s) should be (p,q) (Christian Weiss)
p. 60, in the first display, z/\bar{z} and \bar{z}/z should be interchanged.

Lemma 5.22 This is misstated: in general, T(Q) is not closed in T(A_f) (unless (G,X) satisfies SV5) and so T(Q)\T(A_f) is not Hausdorff (hence not compact). The last step of the proof "An arbitrary torus ..." fails when T(Q)\T(A_f) is not compact. The proof of the finiteness of T(Q)\T(A_f)/nu(K) needs to be rewritten. (Bas Edixhoven)

p72. Lucio Guerberoff points out that the uniqueness assertion in Proposition 8.14 fails and that the condition (**) in Theorem 8.17 is inadequate. He writes (slightly edited): this.

Poonen has sent me a long list of comments and corrections. These will be incorporated into a new version later this month (July, 2017).

Shimura Varieties and Moduli

A few minor misprints were fixed in the published version. In Definition 3.7, delete "algebraic group!of"

Tannakian Categories

No known errors.

The Work of John Tate

The following were corrected on the version on my website 03.12.12

1968a, p63. In fact, linearity fails even for two finite potent operators on an infinite-dimensional vector space. See: A Negative Answer to the Question Of the Linearity of Tate's Trace For the Sum of Two Endomorphisms, Julia Ramos Gonzalez and Fernando Pablos Romo

From Matthias Schuett

p.31 at least in my pdf retrieved from arXiv, there is the same k for k and its alg closure [The bars are there, but a bit weird; they are better on the copy on my website.]
p.37 in footnote 92, Carayol... is missing a comma
5.3 ( [Don't know where this got lost; it's OK on the copy on my website.]
p.45, footnote 123 the -> that
p.60 This is not [a] major result.
1966f refers to 1963a which does not exist

The following were corrected on all versions 23.09.12.

p. 13, eq. (8), should read H^{2-r}(G, M') (Timo Keller)

p16, in the second line after the heading ``The Tate module\ldots'', there is a $k$ missing from one of the $A(k^{sep})$'s.

p.22, Section 2.5. The Mumford-Tate group need not be semisimple, only reductive.

There is also the following article, which, not being clairvoyant, I didn't know about: Tate, John, Stark's basic conjecture. Arithmetic of L-functions, 7 31, IAS/Park City Math. Ser., 18, Amer. Math. Soc., Providence, RI, 2011.

I should have mentioned the work of Tate on liftings of Galois representations, as included in Part II of: Serre, J.-P. Modular forms of weight one and Galois representations. Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 193--268. Academic Press, London, 1977.
See also: Variations on a theorem of Tate. Stefan Patrikis. arXiv:1207.6724.

Also: An oft cited (1979) letter from Tate to Serre on computing local heights on elliptic curves. arXiv:1207.5765 (posted by Silverman).

From Matthias Künzer
p. 3, l. -17: are classified by subgroups of ray class groups ?
p. 6, l. -3: x + Z
p. 8, l. -10: \sum_{\sigma\in G}
p. 9, l. -10: \hat H^r(G,C(\phi))
p. 11, l. 16: the composite I can derive from the ses in l. 14 is trivial - or?
p. 13: it seems that in the displays in l. 24, 27, 34, some modules M should be M'
p. 14, third display, 9-term exact sequence in (9): last but one term should contain H^2, not H^0
p. 17, (14) and p. 20, (17): A', not A^t
p. 17, l. -16: \phi_f
p. 18, l. 1: definition of h_{T,q(P)}(t) ?
p. 23, l. -24: "Much is known about the conjecture." - Which one?
p. 26, l. -12: months
p. 31, l. -3: a great
p. 32, (26): bracket missing on lhs
p. 37, l. -9: "Hodge 1950" - reference missing
p. 39, l. 2: endomorphism f of F
p. 39, l. -7: q is a power of p ?
p. 41, l. 5: space that is
p. 42, l. 17: natural to replace
p. 42, l. -13: groups generalize
p. 42, l. -11: an n-dimensional
p. 42, l. -11: n-dimensional commutative formal Lie group (cf. p. 42, l. -6)
p. 42, l. -1: of a p-divisible group
p. 48, l. 22, the relation for the commutator: x_{il}(rs) on the rhs
p. 49, l.1: the free abelian group ?
p. 49, l. 3,4,5: brackets {,}, not (,)
p. 49, l. -13: if a > 0 or b > 0
p. 52, l. -20: extension of fields
p. 53, l. -3: power of A(\chi,f) is in Q
p. 54, l. -18: and a character
p. 56, §9.1: What motivates the definition of a regular algebra? If I'm interested in Azumaya algebras (cf. l. -23), what leads me to regular algebras?
p. 62, l. 1, 2: "resp.", not "or" (I'd write)
p. 62, l. 9, 25: p-subgroup
p. 68, l. -14: K_2
p. 70, l. 10: GL_n