1. Rings of Fractions

1.0. Rings and algebras (1.0.1) All the rings considered in this work possess a unit element; all the modules over such a ring are assumed unital; homomorphisms of rings are assumed to take unit element to unit element; and unless explicitly mentioned otherwise, all subrings of a ring A are assumed to contain the unit element of A. We generally consider commutative rings, and when we say ring without specifying, we understand this to mean a commutative ring. If A is a not necessarily commutative ring, we consider every A module to be a left A- module, unless we expressly say otherwise.

(1.0.2) Let A and B be not necessarily commutative rings, φ : A → B a homomorphism. Every left (respectively right) B-module M inherits the structure of a left (resp. right) A-module by the formula a.m = φ(a).m (resp m.a = m.φ(a)). When it is necessary to distinguish the A-module structure from the B-module structure on M, we use M[φ] to denote the left (resp. right) A-module so deﬁned. If L is an A module, a homomorphism u : L → M[φ] is therefore a homomorphism of commutative groups such that u(a.x) = φ(a).u(x) for a ∈ A, x ∈ L; one also calls this a φ-homomorphism L → M, and the pair (u,φ) (or, by abuse of language, u) is a di-homomorphism from (A, L) to (B, M). The pair (A,L) made up of a ring A and an A module L therefore form the objects of a category where the morphisms are di-homomorphisms.

(1.0.3) Under the hypotheses of (1.0.2) if = is a left (resp. right) ideal in A, we use B= (resp. =B) to denote the left (resp right) ideal Bφ(=) (resp φ(=)B) of B generated by φ(=); this is also the image of the canonical homomorphism B ⊗A I → B (resp. = ⊗A B → B) of left (resp right) B modules.

(1.0.4) If A is a (commutative) ring, and B is a not necessarily commutative ring, the data deﬁning the structure of an A-algebra on B is equivalent to the data describing a homomorphism of rings φ : A → B such that φ(A) is contained in the center of B. For each ideal = of A, =B = B= is a doublesided ideal of B, and for each B module M, =M is also a B-module via (B=)M.

1 (1.0.5) We do not review the notions of module of finite type and (commutative) algebra of finite type; to say an A module M is of finite type signifies the existence of an exact sequence Ap → M → 0. One says an A module M admits a finite presentation if it is isomorphic to the cokernel of a homomorphism Ap → Aq; equivalently, there exists an exact sequence Ap → Aq → M → 0. Note that over a noetherian ring, every A module admits a finite presentation. We recall that an A algebra B is said to be integral over A if every element of B is the root in B of a monic polynomial with coefficients in A; recall that this is the same as saying that every element of B is contained in a subalgebra of B that is of finite type as an A module. If this is the case, and B is commutative, the subalgebra of B generated by a finite subset is an A module of finite type; in order that the commutative algebra B be integral and of finite type over A, it suffices to show that B is is of finite type as an A module; one then says that B is an integral finite A-algebra (or simply a finite A-algebra if no confusion will result.) One observes that in making these definitions, one does not require that the homomorphism A → B defining the structure of the A algebra be injective.

(1.0.6) A ring is an integral domain if the product of a ﬁnite family of elements 6= 0 is 6= 0; it is equivalent to say that 0 6= 1 and the product of two elements 6= 0 is not zero. A prime ideal in a ring A is an ideal ℘ such that A/℘ is an integral domain; this requires ℘ 6= A. For a ring A to contain a prime ideal, it is necessary and suﬃcient that A 6= ∅.

(1.0.7) A ring A is local if there exists only one maximal ideal; which is then the complement of invertible elements and contains every ideal 6= A. If A and B are local rings, m and n are their respective maximal ideals, then one says a homomorphism φ : A → B is local if φ(m) ⊂ n (or equivalently, φ−1(n) = m). On passing to quotients, such a homomorphism defines a monomorphism from the residual field A/m to the residual field B/n. The composition of two local homomorphisms is a local homomorphism.

1.1. Radical of an ideal. Nilradical and radical of a ring (1.1.1) Let a be an ideal in a ring A; the radical of A, denoted <(a), is the set of all x ∈ A such that xn ∈ a for some integer n > 0; this

2 is an ideal containing a. One has <(<(a)) = <(a); the relation a ⊂ b implies <(a) ⊂ <(b); and the radical of a finite intersection of ideals is the intersection of their radicals. If φ is a homomorphism from a ring A0 to A, one has <(φ−1(a)) = φ−1(<(a)) for every ideal a ⊂ A. For an ideal to be the radical of an ideal, it is necessary and sufficient that it be the intersection of prime ideals. The radical of an ideal a is the intersection of the prime ideals which are minimal among those which contain a; if A is noetherian, there are a finite number of such ideals. (1.1.2) Recall that the radical <(A) of a (not necessarily commutative) ring A is the intersection of the maximal left ideals of A (and also the intersection of the maximal right ideals) The radical of A/<(A) is 0.

1.2. Modules and rings of fractions (1.2.1) One says a subset S of a ring A is multiplicative if 1 ∈ S and if the product of two elements of S are in S. The most important examples are: 1) n the set Sf of powers f (n ≥ 0) of an element f ∈ A; and 2) the complement A − ℘ of a prime ideal ℘ in A.

(1.2.2) Let S be a multiplicative subset of a ring A, and M an A-module; in the set M × S, the relation between pairs (m1, s1), (m2, s2):

<< there exists s ∈ S such that s(s1m1 − s2m2) = 0 >> is an equivalence relation. One denotes by S−1M the quotient of M × X by this relation, and m/s denotes the canonical image in S−1M of the pair (m,s); −1 S S one calls the mapping M to S M given by iM : m → m/1 (also denoted i ) the canonical mapping. This mapping is not in general surjective, and the kernel is the set of m ∈ M such that there exist s ∈ S such that sm = 0. On S−1M one deﬁnes the structure of an additive group by

(m1/s1) + (m2/s2) = (s2m1 + s1m2)/s1s2)

. (One verifies that this is independent of the chosen expression of elements −1 −1 of S M). On S A one also defines a multiplication by (a1/s1)(a2/s2) = −1 −1 0 (a1a2)/(s1s2), and finally one defines an action of S A on S M by (a/s)(m/s ) = (am)/(ss0). One verifies that S−1A has the structure of a ring (called the ring of fractions of A with denominators in S) and S−1M has the structure of an S−1A-module (called the ring of fractions of M with denominators in S);

3 for every s ∈ S, s/1 is invertible in S−1A; its inverse is 1/s. The canonical S S mapping iA (resp iM ) is a homomorphism of rings (resp. a homomorphism of A modules, S−1M being considered an A module via the homomorphism S −1 iA : A → S A).

n −1 (1.2.3) If Sf = f n≥0 for each f ∈ A, we write Af and Mf instead of Sf A −1 and Sf M; when Af is considered an algebra over A, we write Af = A[1/f]. Af is isomorphic to the quotient algebra A[T ]/(fT − 1)A[T ]. When f = 1, Af and Mf are canonically identical to A and M; if f is nilpotent, Af and Mf reduce to 0.

−1 S (1.2.4) the ring of fractions S A and the canonical homomorphism iA are the solution to a universal problem: for every homomorphism u from A to a ring B such that u(S) is made up of invertible elements in B, there is a factorization of the form

/ −1 / u : A S S A u∗ B iA where u∗ is a homomorphism of rings. Under the same hypotheses, let M be a A-module, N a B-module, v : m → N a homomorphism of A modules (with the structure of an A module on N deﬁned by u : A → B); then v has a factorization of the form

/ −1 / v : M S S M v∗ N iM where v∗ is a homomorphism of S−1A modules ( with the structure of an S−1A module on N given by u∗)

−1 N −1 (1.2.5) One deﬁnes a canonical isomorphism S A A M ' S M of S−1A modules, which takes an element (a/s) ⊗ m to the element (am)/s, with inverse mapping taking m/s to (1/s) ⊗ m.

0 −1 S −1 0 0 (1.2.6) For each ideal a in S A, a = (iA) (a ) is an ideal in A, and a −1 S −1 is the ideal in S A generated by iA(a), which is denoted S a (1.3.2). The 0 S −1 0 mapping p → (iA) (p ) is an isomorphism of ordered sets between the set of prime ideals in S−1A and the set of prime ideals in A such that p ∩ S = ∅. −1 Furthermore, the local rings Ap and (S A)S−1p are canonically isomorphic (1.5.1).

4 (1.2.7) When A is an integral domain, and K is its ﬁeld of fractions, the S −1 canonical mapping iA : A → S A is injective for all multiplicative subsets S which do not contain 0, and S−1A is canonically identiﬁed with a subring of K containing A. In particular, for every prime ideal p of A, Ap is a local ring containing A, with maximal ideal pAp, and one has pAp ∩ A = p.

(1.2.8) If A is reduced (1.1.1) then the same is true of S−1A: in fact, if (x/s)n = 0 for x ∈ A, s ∈ S, there exists s0 ∈ S such than s0xn = 0, or (s0x)n = 0, which under the hypothesis implies s0x = 0, so x/s = 0.

0.1 1.3. Functorial properties (1.3.1) Let M, N be two A-modules, u an A-homomorphism M → N. If S is a multiplicative subset of A, one defines a S−1A homomorphism S−1M → S−1N, denoted S−1u, by (S−1u)(m/s) = u(m)/s; if S−1M and S−1N are −1 N −1 N −1 identified with S A A M and S A A N (1.2.5), S u is identified with 1⊗u. If P is a third A module, and v is an A-homomorphism N → P , one has S−1(v ◦u) = (S−1v)◦(S−1u); so we say that S−1M is a contravariant functor in M, from the category of A modules to that of S−1A modules (where A and S are fixed).

(1.3.2) The functor S−1M is exact; in other words, if the sequence

M u / N v / P is exact, the same is true of the sequence

S−1M / S−1N / S−1P S−1u S−1v

. In particular, if u : m → N is injective (resp. surjective), the same is true of S−1u; if N and P are two submodules of M, S−1N and S−1P are canonically identiﬁed as submodules of S−1M, and one has

S−1(N + P ) = S−1N + S−1P and S−1(N ∩ P ) = (S−1N) ∩ (S−1P )

5 −1 −1 (1.3.3) Let (Mα, φβα) be an inductive system of A modules; then (S Mα,S φβα) −1 −1 −1 is a inductive system of S A modules. Expressing S M and S φβα as tensor products (1.2.5 and 1.3.1) the result on the commutativity of the op- erations of tensor product and inductive limit gives a canonical isomorphism

−1 −1 S lim(Mα) ' lim(S Mα) which we express by saying that the functor S−1M (in M) commutes with inductive limits.

(1.3.4) Let M, N be two A-modules; there exists a canonical functorial isomorphism O O S−1M (S−1N) ' S−1(M N) S−1A A which takes (m/s) ⊗ (n/t) to (m ⊗ n)/st.

(1.3.5) One similarly has a natural transformation (in M and N)

−1 −1 −1 S HomA(M,N) → HomS−1A(S M,S N) which, given u/s, maps it to to m/t → u(m)/st. If M is ﬁnitely presented, this transformation is an equivalence; this is immediate if M is of the form Ar, and one proves it in general for a sequence Ap → Aq → M → 0 by using −1 the exactness of the functor S M and the left exactness of HomA(M,N) in M. One notes that this is always the case when A is Notherian, and the A module M is of ﬁnite type.

1.4. Change of multiplicative subsets (1.4.1) Let S and T be two multiplicative subsets of a ring A such that T,S T,S S ⊂ T ; there exists a canonical homomorphism ρA (or simply ρ ) from S−1A to T −1A, which takes an element denoted a/s in S−1A to the element −1 T T,S S denoted a/s in T A; one has iA = ρA ◦ iA. For each A module M, there similarly exists an S−1A linear mapping from S−1M to T −1M (where the −1 T,S latter is considered a S A module via the homomorphism ρA ), which takes the element m/s in S−1M to the element m/s in T −1M; one denotes this T,S T,S T T,S S mapping ρM , or simply rho , and one again has iM = ρM ◦ iM ; under T,S T,S the canonical identiﬁcation (1.2.5), ρA is identiﬁed with ρA ⊗ 1. The

6 T,S −1 homomorphism ρM is a natural transformation from the functor S M to the functor T −1M; in other words, the diagram

S−1u S−1M / S−1N

T,S T,S ρM ρN T −1u T −1M / T −1N is commutative for each homomorphism u : M → N; one further notes that T −1u is entirely determined by S−1u, because if m ∈ M and t ∈ T , one has

(T −1u)(m/t) = (t/1)−1ρT,S((S−1u)(m/1))

(1.4.2) With the same notation, for two A modules M,N the diagrams (cf 1.3.4 and 1.3.5)

−1 N −1 ' −1 N (S M) S−1A(S N) / S (M A N)

−1 N −1 ' −1 N (T M) T −1A(T N) / T (M A N) and −1 −1 −1 S HomA(M,N) / HomS−1A(S M,S N)

−1 −1 −1 T HomA(M,N) / HomT −1A(T M,T N) are commutative.

(1.4.3) In the important case in which when ρT,S is bijective, one knows that every element of T is a divisor of an element in S; one thus identiﬁes via ρT,S the modules S−1M and T −1M. One says that S is saturated if every divisor in A of an element of S is in S; one can replace S by a set T which contains all the divisors of elements of S (a set which is multiplicative and saturated), and one can limit consideration to modules of fractions S−1M, where S is saturated.

7 (1.4.4) If S, T, U are three multiplicative subsets of A, such that S ⊂ T ⊂ U, one has ρU,S = ρU,T ◦ ρT,S.

(1.4.5) Consider an finite filtered family of multiplicative subsets (Sα) of A (one writes α ≤ β for Sα ⊂ Sβ), and let S be the multiplicative subset ∪αSα; Sβ ,Sα we put ρβα = ρA for α ≤ β; by virtue of (1.4.4) the homomorphism ρβα 0 −1 defines a ring A as the inductive limit of the system of rings (Sα A, ρβα). Let −1 0 S,Sα ρα be the canonical mapping S A → A , and set φα = ρA ; as φα = φβ ◦ρβα for α ≤ β by (1.4.4) one defines a unique homomorphism φ : A0 → S−1A by the commutative diagram

−1 Sα A 55 ρ 5 βα55 55 ρα S−1A 55φα β 55 z HH 55 zz HH 5 zρ HH 5 zz β φβ HH 5 Ö }zz H# 5 0 / −1 A φ S A

In fact, φ is an isomorphism. It is immediate from the construction that φ is 0 surjective. For injectivity, if ρα(a/sα) ∈ A is such that φ(ρα(a/sα)) = 0, this −1 implies that a/sα = 0 in S A, and so there exists s ∈ S such that sa=0; but there is the β ≥ α such that s ∈ Sβ, and it follows that ρα(a/sα) = ρβ(sa/ssα) = 0, and φ is injective. One treats similarly the cas of an A module M, and one deﬁnes canonical isomorphisms

−1 −1 −1 −1 lim Sα A ' (lim Sα) A, lim Sα M ' (lim Sα) M the second being functorial in M.

(1.4.6) Let S1 and S2 be two multiplicative subsets of A, then S1S2 is also 0 a multiplicative subset of A. Designate by S2 the canonical image of S2 in the −1 ring S1 A, which is a multiplicative subset of this ring. For each A module M, there exists a functorial isomorphism

0−1 −1 −1 (S2 )S1 M) ' (S1S2) M which takes (m/s1)/(s2/1) to the element m/(s1s2).

8 1.5. Change of Rings (1.5.1) let A, A0 be two rings, φ a homomorphism A0 → A, S (resp S0) a multiplicative subset of A(resp A) such that φ(S0) ⊂ S; the composition A0 → A → S−1A factorizes as A0 → S0−1A0 → S−1A by virtue of (1.2.4); one has φS0 (a0/s0) = φ(a0)/φ(s0). if A = φ(A0) and S = φ(S0), φS0 is surjective. If 0 S0 S,S0 A = A and φ is the identity, φ is none other than the homomorphism ρA deﬁned in (1.4.1)

(1.5.2) Under the hypotheses of (1.5.1) let M be an A module. There 0−1 −1 exists a canonical natural transformation σ : S (M[φ]) → (S M)[φS0 ] of 0−1 0 0 0−1 S A modules, that corresponds each element m/s of S (M[φ]), to the 0 −1 element m/φ(s ) of (S M)[φS0 ]; one verifies immediately from the definitions that this does not depend on the expression m/s0 of the element considered. When S = φ(S0), the homomorphism σ is bijective. When A0 = A and S,S0 φ is the identity, σ, is none other than the homomorphism ρM defined in 1.4.1. When in particular M = A, the homomorphism φ defines in A 0−1 the structure of an A’ algebra; S (A[φ]) has then the structure of a ring, 0 −1 0−1 −1 identified with (φ(S )) A, and the homomorphism σ : S (A[φ]) → S A is a homomorphism of S0−1A0 algebras.

(1.5.3) Let M and N be two A modules; one composes the homomorphisms deﬁned in (1.3.4) and (1.5.2) to get a homomorphism

−1 O −1 0−1 O (S M S N)[φS0 ] ← S ((M N)[φ]) S−1A A which is an isomorphism when φ(S0) = S. Similarly, one composes the homomorphisms (1.3.5) and (1.5.2) to get the homomorphism

0−1 −1 −1 S ((HomA(M,N))[φ]) → (HomS−1A(S M,S N))[φS0 ] which is an isomorphism when φ(S0) = S and M admits a ﬁnite presentation.

0 0 0 N (1.5.4) We consider an A module N , formed as the tensor product N A0 A[φ], which can be considered as an A module via a.(n ⊗ b) = n0 ⊗ (ab). There exist an functorial isomorphism of S−1A modules

0−1 0 O −1 −1 0 O τ :(S N ) (S A)[φS0 ] ' S (N A[φ]) S0−1A0 A0

9 which, for an element (n0/s0) ⊗ (a/s), gives the element (n0 ⊗ a)/(φ(s0)s); one seperately verifies the effect of replacing n0/s0 (resp a/s) by a different expression of the same element (n0 ⊗ a)/(φ(s0)s) is no change; for the other part, one defines an inverse for τ by mapping (n0 ⊗ a)/s to the element 0 −1 0 N (n /1) ⊗ (a/s): use the fact that S (N A0 A[φ]) is canonically isomorphic 0 −1 0 −1 to (N ⊗A0 A[φ]) ⊗A S A (1.2.5) therefore also to N ⊗A0 (S A)[ψ], where ψ is the homomorphism a0 → φ(a0)/1 from A’ to S−1A.

(1.5.5) If M’ and N’ are two A’ modules, one composes the isomorphisms (1.3.4) and (1.5.4) to obtain the isomorphism O O O O S0−1M 0 S0−1N 0 S−1A ' S−1(M 0 N 0 A) S0−1A0 S0−1A0 A0 A0 similarly, if M’ admits a ﬁnite presentation, one composes (1.3.5) and (1.5.4) to get an isomorphism 0−1 0 0−1 0 O −1 −1 0 0 O HomS0−1A0 (S M ,S N ) S A ' S (HomA0 (M ,N ) A). S0−1A0 A0

(1.5.6) Under the hypotheses of (1.5.1) let T (resp T’) be a second multiplicative subset of A (resp A’) such that S ⊂ T (resp S0 ⊂ T 0) and φ(T 0) ⊂ T . Then the diagram 0 φS S0−1A0 / S−1A

0 0 ρT ,S ρT,S 0−1 0 −1 T A 0 / T A φT is commutative. If M is an A module, the diagram

0−1 0 σ −1 S M[φ] / (S M)[φS0 ]

0 0 ρT ,S ρT,S −1 0−1 0 T M[φ] σ / (T M)[φT ] is commutative. Finally, if N’ is an A’ module, the diagram 0−1 0 −1 ' N 0 −1 0 N (S N ) S0−1A0 (S A)[φS ]τ / S (N A0 A[φ])

ρT,S 0−1 0 −1 τ N 0 −1 0 N (T N ) T 0−1A0 (T A)[φT ]' / T (N A0 A[φ])

10 is commutative, the vertical arrow on the left being obtained by applying T 0,S0 0−1 0 T,S −1 ρN 0 to S N and ρA to S A.

(1.5.7) Let A00 be a third ring, φ0 : A00 → A0 a homomorphism of rings, S00 a multiplicative subset of A00 such that φ0(S00) ⊂ S0. Put φ00 = φ ◦ φ0; then one has φ00S00 = φS0 ◦ φ0S00 . 0 00 Let M be an A module; it is evident that M[φ00] = (M[φ])[φ0]; if σ and σ are the homomorphisms deﬁned by φ0 and φ00 as σ is deﬁned by φ as in (1.5.2), one has the formula of transitivity

σ00 = σ ◦ σ0.

00 Finally, let N” be a A”-module; the A module N ⊗A0 A[φ00] is canoni- 00 0 −1 cally identiﬁed with (N ⊗A00 A[φ0]) ⊗A0 A[φ], and similarly, the S A module 00−1 00 N −1 (S N ) S00−1A00 (S A)[φ00S00 ] is canonically identiﬁed with

00−1 00 O 0−1 0 O −1 ((S N ) (S A )[φ0S0 ]) (S A)[φS0 ] S00−1A00 S0−1A0 . With these identifications, if τ 0 and τ 00 are the isomorphisms defined with respect to φ0 and φ00 and τ is defined as in (1.5.4) with respect to φ, one has the formula of transitivity

τ 00 = τ ◦ (τ 0 ⊗ 1)

(1.5.8) Let A be a subring of a ring B; for every minimal prime ideal p of A, there exists a minimal prime ideal q of B such that p = A ∩ q. In fact, 0 Ap is a subring of Bp (1.3.2) and posseses a single prime ideal p (1.2.6); if 0 Bp does not reduce to zero, it too posseses a prime ideal q and necessarily 0 0 0 q ∩Ap = p ; the prime ideal q1 in B is the inverse image of q and is therefore such that q1 ∩ A = p, and a fortiori one has q ∩ A = p for each prime ideal q of B containing q1.

1.6. Identiﬁcation of a module Mf as an inductive limit. (1.6.1) Let M be an A module, f an element of A. Consider the sequence (Mn) of A modules, all identically M, such that for each pair of integers

11 n−m m ≤ n, φnm is the homomorphism z→ f z from Mm to Mn; it is immediate than ((Mn),(φnm)) is an inductive system of A modules; let N = lim Mn be the inductive limit of the system. We define a canonical A-isomorphism n from N to Mf . For this, we remark that, for all n, θn : z → z/f is an A- homomorphism from M = Mi to Mf , and it results from the definitions that one has θn ◦φnm = θm for m ≤ n. There therefore exists an A-homomorphism θ : N → Mf such that, if φn denotes the canonical homomorphism Mn → N, one has θn = θ ◦ φn. With this holding, every element of Mf is of the form z/f n for some n, so it is clear that φ is surjective. For the other part, if n θ(φn(z)) = 0, or in other words, z/f = 0, there exist an integer k > 0 such k that f z = 0, so φn+k,n(z) = 0, which implies φn(z) = 0. One identifies Mf and lim Mn via θ.

f f (1.6.2) We use the notation Mf,n, φnm and φn instead of Mn, φnm and φn. Let g be a second element of A. Since f n divides f ngn, one has a functorial homomorphism ρfg,f : Mf → Mfg (1.4.1 and 1.4.3); if one identifies Mf and Mfg with lim Mf,n and lim Mfg,n respectively, ρfg,f is identified with the n n n inductive limit of maps ρfg,f : Mf , n → Mfg,n defined by ρfg,f (z) = g z. In fact, this result is immediate from the commutativity of the diagram

n ρfg,n Mf,n / Mfg,n

f fg φn φn

ρfg,f Mf / Mfg .

1.7. Support of a module (1.7.1) Given an A module M, we name the support of M and denote by Supp(M) the set of prime ideals p in A such that Mp 6= 0. For M to equal 0, it is necessary and suﬃcient that Supp(M)=∅ for if Mp = 0 for all p, the annihilator of an element x ∈ M is not contained in any prime ideal of A, hence is invertible in A.

(1.7.2) If 0 → N → M → P → 0 is an exact sequence of A modules, one has Supp(M)=Supp(N)∪ Supp(P) since for all prime ideals in A, the sequence 0 → Np → Mp → Pp− > 0 is exact (1.3.2) and Mp = 0 if and only if Np = Pp = 0.

12 (1.7.3) If M is the sum of a family (Mλ) of submodules, Mp is the sum of (Mλ)p for each prime ideal p in A (1.3.3 and 1.3.2) so

Supp(M) = ∪λSupp(Mλ)

(1.7.4) If M is an A module of ﬁnite type, Supp(M) is the set of prime ideals which contain the annihilator of M. In fact, if M is generated by a single element x, then Mp = 0 implies there exist s∈ / p such that s.x = 0, so that p is not contained in the annihilator of x. If M admits a ﬁnite system (xi)1≤i≤n of generators and if ai is the annihilator of xi, by (1.7.3) Supp(M) is the set of p containing the ai, where, by the same, the set p contains a = ∩iai, which is the annihilator of M.

(1.7.5) If M and N are two A modules of finite type, one has O Supp(M N) = Supp(M) ∩ Supp(N) A One can see that if p is a prime ideal of A, the condition M N N 6= p Ap p 0 is equivalent to << Mp 6= 0 and Np 6= 0 >>) (compare (1.3.4)). In other words, one sees that if P,Q are two modules of finite type over a local ring B which does not reduce to zero, then P ⊗B Q 6= 0. Let m be the maximal ideal of B. By Nakayama’s lemma, the vector spaces P/mP and Q/mQ do not reduce to 0, hence the same is true of the tensor products N (P/mP ) ⊗B/mB (Q/mQ) = (P B Q) ⊗B (B/m), thus the conclusion. In particular, if M is an A module of finite type, a an ideal of A, Supp(M/aM) is the set of prime ideals containing a as well as the annihilator n of M (1.7.4), or the set of prime ideals containing a + n.