Appendix A Commutative Algebra In this chapter we will summarise various results from commutative algebra for the reader’s convenience. Some are well known, but others are not easily available despite being standard in the specialist literature. Our main reference is Eisenbud’s book [Eis95]. A.1 Inductive Limits We refer to [Eis95, Appendices 5 and 6] for an introduction to categories and limits: the reader should be aware that our inductive limits are called filtered colimits in Eisenbud’s book. We recall some basic definitions. Definition A.1.1 (Inductive system)LetC be a category and let (J, ) be a partially ordered set such that ∀(i, j) ∈ J 2, ∃k ∈ J | i k and j k. We then call (J, ) a filtered set,adirected set or a directed preorder.Aninductive system indexed by J is the data of a family {M j } j∈J of objects of C and morphisms ϕij : Mi → M j for all pairs of indices (i, j) ∈ J 2 such that i j which satisfy ∀ ∈ = 1. j J, ϕ jj idM j ; 3 2. ∀(i, j, k) ∈ J , i j k =⇒ ϕ jk ◦ ϕij = ϕik. Definition A.1.2 (Inductive limit)Let{M j } j∈J be an inductive system in a category C. A object M in C is the inductive limit, direct limit or colimit of a filtered set of the system {M j } if it is equipped with morphisms ϕ j : M j → M satisfying the com- patibility relations ϕi = ϕ j ◦ ϕij for every i j and having the following universal property: if N is an object in C equipped with morphisms ψ j : M j → N which are compatible with the inductive system structure then there is a unique map M → N such that for all j the morphism ψ j factors © Springer Nature Switzerland AG 2020 313 F. Mangolte, Real Algebraic Varieties, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-43104-4 314 Appendix A: Commutative Algebra ψ j M j N ϕ j M When the inductive limit of the system {M } ∈ exists we denote it by lim M . j j J −→ j∈J j Example A.1.3 If the category C is the category of groups, rings or A-modules/ algebras for some given ring A then the inductive limit exists: it is simply the quotient of the disjoint union of the M j s modulo an equivalence relation = ∼ −lim→ M j M j j∈J j∈J where xi ∈ Mi is equivalent to x j ∈ M j if and only if there is a k ∈ J such that ϕik(xi ) = ϕ jk(x j ). Example A.1.4 Let F be a sheaf (see Appendix C) of elements of C over a topo- logical space X. For any given x ∈ X the set of open neighbourhoods of x ordered by inclusion (U V if and only if U ⊇ V ) is a filtered set and {F(U)}U x is an inductive system. The limit of this system is called the stalk of F at x and is denoted Fx . For every open neighbourhood of x the canonical morphism F(U) → Fx sends a section s of F over U to the germ sx ∈ Fx of s at x. A.2 Rings, Prime Ideals, Maximal Ideals and Modules By convention all our rings are assumed to be commutative with a multiplicative unit, and ring morphisms are required to send the multiplicative unit to the multiplicative unit. The set of invertible elements of A is denoted U(A).IfK is a ring then a K -algebra A is a ring equipped with a ring morphism K → A. For example, K [X1,...,Xn] is the K -algebra of polynomials in n variables with coefficients in K and K (X1,...,Xn) is the K -algebra of rational functions in n variables with coefficients in K . Definition A.2.1 A non zero element a in a ring A is said to be a zero divisor in A if and only if there is a non zero element b ∈ A such that ab = 0. Aringissaidtobeanintegral domain if and only if it has at least two elements and does not contain a zero divisor. A field is a ring with at least two elements such that all its non zero elements are invertible. Appendix A: Commutative Algebra 315 Definition A.2.2 Let A be a ring. 1. An ideal I in A is said to be prime if and only if it satisfies the following properties: (a) I is not equal to A (b) If a and b are elements of A such that ab ∈ I then a ∈ I or b ∈ I . 2. An ideal I in A is said to be maximal if and only if it is different from A and the only ideals in A containing I are I and A. √ Definition A.2.3 For any ideal I in a ring A,theradical I of I in A is the ideal of roots of elements of I . √ I := {a ∈ A | there is a natural number n 1, an ∈ I } . √ An ideal I ⊂ A is said to be radical if and only if I = I . Exercise A.2.4 (See Remark 1.2.29)LetK be a field. Prove that if F is a Zariski closed subset of An(K ) then I(F) is radical. Definition A.2.5 An element a inaringA is said to be nilpotent if and only if there is a natural number n > 1 such that an = 0. The nilradical of a ring A is the set of its nilpotent elements. A ring is said to be reduced if and only if its nilradical is the zero ideal, or in other words if it has no non zero nilpotent elements. Exercise A.2.6 The nilradical of a ring A is an ideal, namely the radical ideal of the zero ideal of A. Proposition A.2.7 Let A be a ring and let I be an ideal of A. 1. The ideal I is radical if and only if the quotient ring A/Iisreduced. 2. The ideal I is prime if and only if the quotient ring A/Iisanintegral domain. 3. The ideal I is maximal if and only if the quotient ring A/Iisafield. Proof Easy exercise. Proposition A.2.8 (Correspondence theorem) Let A be a ring and let I ⊂ Abe an ideal. The canonical surjection A → A/I induces a one-to-one correspondence between prime ideals of A/I and prime ideals of A containing I and a similar one-to-one correspondence between maximal ideals of A/I and maximal ideals of A containing I . Proof Easy exercise. The following lemma is extremely useful despite its simplicity. Lemma A.2.9 Let A be a ring and let B ⊂ A be a sub-ring. If I is a prime ideal of A then I ∩ B is a prime ideal of B. 316 Appendix A: Commutative Algebra Proof Let a and b be elements of B such that ab ∈ I ∩ B and a ∈/ I ∩ B.Asa ∈ B and a ∈/ I ∩ B, a does not belong to I .Itfollowsthatb ∈ I because I is a prime ideal of A and hence b belongs to I ∩ B. Example A.2.10 We calculate the dimension of the ideal I := (x2 + y2) from Example 1.5.20. There is a unique chain (Definition 1.5.2) of prime ideals in R[x, y] containing (x2 + y2) which is of maximal length (x2 + y2) ⊂ (x, y). There is therefore only one chain of prime ideals of R[x, y]/(x2 + y2) of maximal length. The dimension of the ring R[x, y]/(x2 + y2) is therefore equal to 1 and according to Definition 1.5.9 dim I = 1. Lemma A.2.11 (Nakayama’s Lemma) Let A be a ring, let a ⊂ A be an ideal and let M be a finitely generated A-module such that M = aM. There is then an element a ∈ 1 + a such that aM = 0: in particular, if A is local and a is its maximal ideal then M = 0. Proof See [Eis95, Corollary 4.8]. Definition A.2.12 AringS is said to be graded if and only if it has a decomposi- tion S =⊕d0 Sd as a direct sum of abelian groups Sd such that for any d, e 0, Sd · Se ⊂ Sd+e. An ideal I ⊂ S is said to be a homogeneous ideal if and only if I =⊕d0(I ∩ Sd ). A.3 Localisation Definition A.3.1 Let A be a ring, let M be an A-module and let S ⊂ A be a multi- plicative subset1 or in other words a subset stable under multiplication. The localised module (or localisation)ofM in S, denoted S−1 M, is the set of equivalence classes of pairs (m, s) ∈ M × S for the relation (m, s) ∼ (m, s) if and only if there is an element t ∈ S such that t(sm − sm) = 0. This set is equipped with an obvious A-module structure. The equivalence class of (m, s) is denoted by m/s. When M = A, this construction yields the localisation S−1 A of A at S. If f is an element of A,thesetS ={1, f, f 2,..., f k ,...,} is a multiplica- −1 tive subset of A and we denote these special localisations by A f := S A and −1 M f := S M.If f is nilpotent then A f is the zero ring.