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.

A Primer of Commutative Algebra v4.03, 2020 (CA)

p. 10. Proof of (b). Let $S'$ be a finite set of generators for $M'$ (not $M$).

p. 21, Example 5.7, 4th line it should be: A prime ideal is disjoint from $S_p$ if and only if it is contained in $p$. The 'in' is missing.

p. 41, under the section Tensor products of algebras second paragraph last line $(f,g)\circ i=f$ (and not $\alpha$) and $(f,g)\circ j=g$ (and not $\beta$). (Chirantan Mukherjee)

p. 62. The folklore in the footnote is apocryphal --- the Rabinowitsch of Rabinowitsch's trick is not the Rabinowitsch who became Rainich. For the history of Hilbert's Nullstellensatz, see arXiv:2309.14024. Rabinowitsch's proof is exactly our proof, but is less detailed (it is only 11 lines).

p. 81. The polynomial $F-F(x)$ (12th line from bottom) may not have coefficients in $A$ as claimed. Instead write $F(X)=\sum a_i X^i$, write $F(x)=\sum b_i x^i$ with $b_i\in \mathfrak{p}$, and consider $\sum(a_i - b_i)X^i$ instead of $F-F(x)$ (Yutaka Matsuura).

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 were fixed. In particular, the statements of Lemma 5.22 and Theorem 8.17 were corrected.

Corrections from Jungin Lee for 2017 version.

In 5.30, I should have said that the local rings of $S$ are noetherian and regular (instead of that $S$ is locally noetherian and regular) (Qiu, Congling).

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

p.69 In Theorem 1.16, the condition should be $Z(\mathbb{Q})$ is discrete in $Z(\mathbb{A}_f)$, not $Z(\mathbb{R})$. (Condition SV5 of Introduction to Shimura varieties.)

Tannakian Categories (fixed on current version).

A comment, not a correction: the definition of the rank of an object only really makes sense in characteristic zero. For a discussion of this, see sx3775904
page 3: line 3: I would say 'for some $G$'.
page 7: line -1: There is no '.' at the end of (1.6.1).
page 19: "finite-dimensional" in the proof of 2.7 should be "finitely generated". To see that the image $G_X$ of $G$ in $\mathrm{GL}_X$ is an algebraic subgroup (hence closed) of $\mathrm{GL}_X$, use that every homomorphism from $G$ to an algebraic group factors through an algebraic quotient (see the proof of 2.7).
page 29: line -4: caegory. (Enis Kaya).

p.23, Proof of 2.16. The pair doesn't satisfy the conditions of 2.14 (Kapil Paranjape). Assume, for simplicity, that $B$ is finite (otherwise need to pass to limits). Let $\omega^{\prime}$ be the functor \[ (X,Y)\mapsto\omega(X)\otimes_k\omega(Y)\colon\mathrm{Comod}_{B}\times\mathrm{Comod}_{B} \rightarrow\mathrm{Vec}_{k}. \] First show that there is a canonical isomorphism $\mathrm{End}(\omega^{\prime})\simeq(B\otimes_k B)^{\vee}$ (cf. Saavedra 1972, II, 2.5.1.1). Now let $\phi$ be a functor $\mathrm{Comod}_{B}\times\mathrm{Comod}_{B}\rightarrow\mathrm{Comod}_{B}$ such that $\phi(X,Y)=X\otimes_{k}Y$ as a $k$-vector space. Such a $\phi$ defines a homomorphism $\mathrm{End}(\omega)\rightarrow\mathrm{End}(\omega^{\prime})$, i.e., a homomorphism $B^{\vee}\rightarrow(B\otimes_k B)^{\vee}$, and hence a homomorphism $u\colon B\otimes_k B\rightarrow B$. Now $\phi\mapsto u$ is inverse to $u\mapsto \phi^{u}$.

p.30. The sentence "For any $R$-algebra ..." in the proof of Theorem 3.2 requires clarification (Parul Keshari).

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 $\mathbb{Q}$-vector spaces with finite-dimensional stalk... (Julian Rosen).

p.67. In the definition of fibred category, $F$ should be in the category $F_{V}$, not $F_{U}$ (Parul Keshari).

The Work of John Tate

In 3.1, replace $|j|<1$ with $|j|>1$ (twice).

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