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 $B$.
p.86, Proof of 18.4. In the display, ">" should be "<" (Roy Smith).

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 (revised version 2017)

All known errors in he original 2004 version have been fixed. In particular, the statements of Lemma 5.22 and Theorem 8.17 have been corrected.

Shimura Varieties and Moduli

A few minor misprints were fixed in the published version. In Definition 3.7, delete "algebraic group!of"
p.20 Paragraph 3, for a proof that $\Gamma$ is a lattice if $\Gamma\backslash D$ is an algebraic variety, see Proposition 2.8 of my manuscript Kazhdan's theorem on arithmetic varieties www.jmilne.org/math/articles/1984T.pdf

Tannakian Categories

Remark 6.19 should read: The proposition shows that the category of Artin motives over k is equivalent to the category of sheaves of finite-dimensional Q-vector spaces with finite-dimensional stalk... (Julian Rosen).

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