1.Faithfully flat modules. 2.Properties of the ring of fractions. 3.Properties of the module of fractions. 4.Algebras of fractions. 5.Modules over algebras of fractions.
6.Special algebras of fractions.
In this section we study the forming of quotient modules and quotient algebras with respect to multiplicative subsets of R. Since this is closely related to tensoring with quotient rings ofR, we recall some properties of tensor functors for the category of R-modules.
Recall that an R-moduleU isfaithfully flatif UR is flat and, for anyR-module N, U ⊗RN = 0 implies N = 0. The following is shown in [40, 12.17]:
16.1 Faithfully flat modules.
For an R-module U, the following assertions are equivalent:
(a) U is faithfully flat;
(b) U is flat and U ⊗RR/I 6= 0, for every proper (maximal) ideal I ⊂R;
(c) U is flat and U I 6=U, for every proper (maximal) ideal I ⊂R;
(d) the functor U ⊗R−:R-Mod→ZZ-Mod (i) is exact and reflects zero morphisms, or (ii) preserves and reflects exact sequences.
An important class of flat R-modules is obtained by forming the ring of fractions with respect to multiplicative subsets S ⊂R containing 1:
On the set R×S an equivalence relation is given by
(a, s)∼(b, t) if (at−bs)s0 = 0, for some s0 ∈S.
For the equivalence classes of (a, s)∈R×S, denoted by [a, s], we define addition and multiplication by
[a, s] + [b, t] = [at+bs, st], [a, s][b, t] = [ab, st].
This yields a ring denoted byRS−1, called the ring of fractions of Rwith respect to S. There is a ring morphism
R →RS−1, r7→[r,1], with kernel {r ∈R|rs= 0, for some s∈S}.
16.2 Properties of the ring of fractions. We use the notation above.
(1) RS−1 is a flat R-module.
(2) Every ideal of RS−1 is of the form IRS−1, for some ideal I ⊂R.
(3) IRS−1 is a prime ideal in RS−1 if and only if I is a prime ideal in R with I∩S=∅.
(4) If R is artinian or noetherian, then RS−1 has the same property.
(5) For any ideal I ⊂R and π :R→R/I, RS−1/IRS−1 '(R/I)π(S)−1.
Proof. (1) It is to show that I⊗RS−1 → IRS−1, a⊗[1, s] 7→ [a, s], is injective for any idealI ⊂R (e.g., [40, 12.16]). This is a special case of an isomorphism which will be obtained in 16.3(1).
(2) For any ideal L⊂RS−1, consider I ={a∈R|[a,1]∈L}.
Then obviouslyIRS−1 ⊂L. For [a, s]∈L, [1, s][a,1]∈IRS−1 and so IRS−1 =L.
(3) For a prime idealP ⊂RS−1,p={a∈R|[a,1]∈P}is a prime ideal inR and P =pRS−1. Since P contains no invertible elements of RS−1, p∩S =∅.
Now let p be a prime ideal of R with p∩S = ∅ and [a, s][b, t] ∈ pRS−1. Then [ab,1][1, st][st,1] ∈ pRS−1 and ab ∈ p, implying a ∈ p or b ∈ p. Hence [a, s] or [b, t]∈pRS−1 and pRS−1 is a prime ideal.
(4) This is an obvious consequence of (2).
(5) The isomorphism is given by
RS−1/IRS−1 →(R/I)π(S)−1, [a, s] +IRS−1 7→[a+I, s+I]. 2 For any R-module M and a multiplicative subset S ⊂R containing 1, we define a module of fractions M S−1 in the following way:
On the set M ×S consider the equivalence relation
(m, s)∼(n, t) if (mt−ns)s0 = 0, for somes0 ∈S.
On the set of equivalence classes, denoted by [m, s], addition and scalar multipli-cation are defined by
[m, s] + [n, t] = [mt+ns, st], [a, s][m, t] = [am, st].
This makes M S−1 an RS−1-module, and there is a canonical R-linear map M →M S−1, m7→[m,1], with kernel {m∈M |ms= 0 for some s ∈S}.
16.3 Properties of the module of fractions.
Let M, N be R-modules. We use the above notation.
(1) M⊗RRS−1 'M S−1.
(2) If M is a flat R-module, then M S−1 is a flat RS−1-module.
(3) Every RS−1-submodule of M S−1 is of the form KS−1, for some R-submodule K ⊂M.
(4) If M is a finitely generated R-module, then M S−1 is a finitely generated RS−1-module.
(5) If M is a noetherian (artinian)R-module, thenM S−1 is a noetherian (artinian) RS−1-module.
(6) A finite subset {m1, . . . , mk} ⊂M is in the kernel of M →M S−1 if and only if there exists s ∈S with smi = 0, for i= 1, . . . , k.
(7) M S−1⊗RN 'M S−1⊗RS−1 N S−1 '(M ⊗RN)S−1.
(8) For an RS−1-module L, the map L→ LS−1, l 7→[l,1] is an isomorphism and HomR(M, L)'HomRS−1(M S−1, L).
(9) For RS−1-modules L, L0, we have L⊗RL0 'L⊗RS−1 L0 and HomR(L, L0)'HomRS−1(L, L0).
(10) For any submodule N ⊂M, M S−1/N S−1 '(M/N)S−1.
Proof. (1) The map M ×RS−1 → M S−1,(m,[r, s]) 7→ [rm, s], is R-bilinear and hence yields anR-linear mapM ⊗RRS−1 →M S−1, which is in factRS−1-linear.
Consider the assignment M S−1 →M⊗RRS−1,[m, s]7→m⊗[1, s].
If [m, s] = [m0, s0], then ts0m =tsm0 for some t∈S and
m⊗[1, s] =m⊗[ts0, tss0] =ts0m⊗[1, tss0] =tsm0 ⊗[1, tss0] =m0⊗[1, s0].
Hence our assignment defines in fact an inverse to the above mapping.
(2) Since RS−1 is a flat R-module by 16.2, this follows from (1).
(3) For any RS−1-submodule L⊂M S−1, consider K ={m ∈M|[m,1]∈L} ⊂M.
K is an R-submodule and obviously KS−1 ⊂ L. For [l, s] ∈ L we observe [l, s] = [1, s][l,1]∈KS−1 and hence KS−1 =L.
(4) and (5) are immediate consequences of (3).
(6) Under the given conditions, there existsi ∈S with simi = 0, ands :=s1· · ·sk has the property demanded.
(7) This follows from (1) and basic properties of the tensor product.
(8) Assume [l,1] = 0. Then there exists s ∈ S with sl = 0 which implies l = [1, s][s,1]l = 0. Hence the map is injective.
Consider [l, s] ∈ LS−1. The image in LS−1 of [1, s]l ∈ L is in fact [l, s] and the map is surjective.
Referring to (1), the second isomorphism is obtained from 15.6.
(9) These isomorphisms are derived from (7) and (8).
(10) Since − ⊗RRS−1 is an exact functor this follows from (1). 2 Applied to R-algebras, the above techniques again yield algebras and we want to consider some basic relations between an algebra and its algebra of fractions. Recall that an algebra is said to be reduced if it contains no non-zero nilpotent elements.
16.4 Algebras of fractions.
Assume A is an R-algebra and S ⊂R is a multiplicative subset. Then:
(1) AS−1 'A⊗RRS−1 is an RS−1-algebra.
(2) Every (left) ideal of AS−1 is of the form KS−1, for some (left) ideal K ⊂A.
(3) The canonical map A→AS−1, a7→[a,1], is an R-algebra morphism.
(4) If A is commutative, associative, alternative, power-associative or a Jordan al-gebra, then AS−1 is of the same type.
(5) Assume A is a power-associative algebra. Then:
(i) For any nil (left) ideal N ⊂A, N S−1 is a nil (left) ideal in AS−1.
(ii) Every nilpotent element [a, s] ∈ AS−1 is of the form [a, s] = [ta, ts], for somet ∈S and ta∈A nilpotent.
(iii) If A is reduced then AS−1 is reduced.
Proof. (1) AS−1 is a scalar extension of A byRS−1 (see 15.4).
(2) This can be seen with the proof of (3) in 16.3.
(3),(4) and (5)(i) are obvious.
(5)(ii) Assume for [a, s]∈ AS−1 and n ∈IN, we have [a, s]n= [an, sn] = 0. Then tan = 0, for some t∈S, and (ta)n= 0. Obviously [a, s] = [ta, ts].
(5)(iii) This is immediate from (ii). 2
If A is an associative R-algebra the above constructions can also be extended to unitalA-modules and with the proofs of 16.3 we obtain:
16.5 Modules over algebras of fractions.
Let A be an associative R-algebra with unit, S a multiplicative subset of R, and M a unital left A-module. Considering M as an R-module we form the module of fractions M S−1. Then:
(1) M S−1 'M ⊗RRS−1 'M⊗AAS−1, and for any right A-module N, N ⊗AM S−1 'N S−1⊗AS−1M S−1.
(2) If M is a flat A-module, then M S−1 is a flat AS−1-module.
(3) Every AS−1-submodule of M S−1 is of the form KS−1, for some A-submodule K ⊂M.
(4) For any submodules U, V ⊂M, (U∩V)S−1 =U S−1∩V S−1.
(5) If M is a finitely generated A-module, then M S−1 is a finitely generated AS−1-module.
(6) If M is a noetherian (artinian)A-module, thenM S−1 is a noetherian (artinian) AS−1-module.
(7) If U is a direct summand in M, then U S−1 is a direct summand in M S−1. (8) If every (finitely generated) A-submodule of M is a direct summand, then every
(finitely generated) AS−1-submodule of M S−1 is a direct summand.
As an application of the above observations we state:
16.6 Special algebras of fractions. Let A be an R-algebra.
(1) If A is alternative and (strongly) regular, then AS−1 is (strongly) regular.
(2) If every (finitely generated) left ideal of A is a direct summand, then every (finitely generated) left ideal of AS−1 is a direct summand.
(3) If every (finitely generated) ideal of A is a direct summand, then every (finitely generated) ideal ofAS−1 is a direct summand.
Proof. (1) Assume A is regular. For any [a, s] ∈ AS−1, there exists b ∈ A with a=aba, and we have [a, s][bs,1][a, s] = [a, s], i.e., AS−1 is regular.
If A is reduced, then AS−1 is reduced by 16.4.
(2) Left ideals of A are L(A)-submodules of A, where L(A) denotes the left mul-tiplication algebra of A (see 2.1). Similar to the construction in 15.11, we obtain a surjective ring morphism L(A)⊗RRS−1 → L(AS−1) and we see that left ideals of AS−1 are exactly its L(A)⊗RRS−1-submodules.
Now the assertion follows from 16.5(7).
(3) This is obtained by the same proof as (2). 2
References: Bourbaki [7], Matsumura [25].