Abstract
In this chapter we assume by default that \(\mathbb{k}\) is an algebraically closed field.
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- 1.
See Sect. 5.1.2 of Algebra I.
- 2.
See Lemma 5.1 of Algebra I.
- 3.
Compare with Sect. 11.2.4 of Algebra I.
- 4.
See Problem 5.6 of Algebra I.
- 5.
See Sect. 5.2.2 of Algebra I.
- 6.
That is, has no nilpotent elements; see Sect. 2.4.2 of Algebra I.
- 7.
See Lemma 2.5 of Algebra I.
- 8.
If \(\mathbb{k}\) is not algebraically closed, then the map φ ↦ kerφ still embeds the set of homomorphisms \(\mathbb{k}[X] \rightarrow \mathbb{k}\) into \(\mathop{\mathrm{Spec}_{\mathrm{m}}}\nolimits A\). However, some maximal ideals \(\mathfrak{m} \subset A\) may not be represented as the kernels of homomorphisms \(A \rightarrow \mathbb{k}\). For example, the kernel of the evaluation \(\mathop{\mathrm{ev}}\nolimits _{i}: \mathbb{R}[x] \rightarrow \mathbb{C}\), f ↦ f(i), where \(i \in \mathbb{C}\), i 2 = −1, certainly is a maximal ideal in \(\mathbb{R}[x]\), but it cannot be realized as the kernel of a homomorphism \(\varphi: \mathbb{R}[x] \rightarrow \mathbb{R}\), because for the latter, \(\mathbb{R}[x]/\ker \varphi = \mathbb{R}\), whereas \(\mathbb{R}[x]/\ker \mathop{\mathrm{ev}}\nolimits _{i} = \mathbb{R}[x]/(x^{2} + 1) \simeq \mathbb{C}\).
- 9.
See Sect. 2.4.2 of Algebra I.
- 10.
Recall that an ideal \(\mathfrak{p} \subset A\) is called prime if the quotient ring \(A/\mathfrak{p}\) has no zero divisors; see Sect. 5.2.3 of Algebra I.
- 11.
In the sense of Example 9.13 on p. 203.
- 12.
In the sense of Example 9.14 on p. 204.
- 13.
See Sect. 5.4 of Algebra I.
- 14.
See Proposition 5.4 of Algebra I.
- 15.
Compare with Proposition 5.3 of Algebra I.
- 16.
This is the same notation as in Sect. 4.1 of Algebra I.
- 17.
Recall that it consists of all fractions f∕g with \(f \in \mathbb{k}[X]\), \(g \in \mathbb{k}[X]^{\circ }\), and f 1∕g 1 = f 2∕g 2 if and only if f 1g 2 = f 2g 1. (See Sect. 4.1 of Algebra I and compare it with Problem 9.10 on p. 224.)
- 18.
See Sect. 4.1.1 of Algebra I.
- 19.
That is, there exist \(f_{1},f_{2},\ldots,f_{m} \in \mathbb{k}[X]\) such that every \(h \in \mathbb{k}[X]\) can be written as h = ∑ φ ∗(g i) f i for appropriate \(g_{i} \in \mathbb{k}[Y ]\).
- 20.
- 21.
Recall that in the second statement, we assume \(\mathbb{k}[X]\) to be an integral domain.
- 22.
That is, φ(U) is open in Y for every open U ⊂ X.
- 23.
Whose closed sets are \(V (I) =\{ \mathfrak{m} \in \mathop{\mathrm{Spec}_{\mathrm{m}}}\nolimits C^{0}(X)\mid I \subset \mathfrak{m}\}\) for all ideals I ⊂ C 0(X).
References
Danilov, V.I., Koshevoy, G.A.: Arrays and the Combinatorics of Young Tableaux, Russian Math. Surveys 60:2 (2005), 269–334.
Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press, 1997.
Fulton, W., Harris, J.: Representation Theory: A First Course, Graduate Texts in Mathematics. Cambridge University Press, 1997.
Morris, S. A.: Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Society LNS 29. Cambridge University Press, 1977.
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Gorodentsev, A.L. (2017). Affine Algebraic Geometry. In: Algebra II. Springer, Cham. https://doi.org/10.1007/978-3-319-50853-5_11
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