Universal localization at semiprime Goldie ideals

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1 at semiprime Goldie ideals Northern Illinois University UNAM 23/05/2017

2 Background If R is a commutative Noetherian ring and P R is a prime ideal, then a ring R P and ring homomorphism λ : R R P can be constructed for which every element of the complement C(P) of P is inverted by λ. The construction uses ordered pairs (c, a) (think c 1 a) where c C(P) and a R, subject to the equivalence relation (c, a) (d, b) if there exists c C(P) with c (ad bc) = 0. The ideal PR P generated by P is the unique maximal ideal of R P.

3 Motivation David Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, p. 57: A local ring is a ring with just one maximal ideal. Every since Krull s paper (1938) local rings have occupied a central position in commutative algebra. The technique of localization reduces many problems in commutative algebra to problems about local rings. This often turns out to be extremely useful. Most of the problems with which commutative algebra has been successful are those that can be reduced to the local case.

4 Motivation Properties of R P : (1). λ : R R P is universal with respect to the property that if c C(P) then λ(c) has an inverse in R P. That is, if φ : R T inverts C(P), then there exists a unique ring homomorphism φ such that the following diagram commutes. R λ φ R P φ T

5 Motivation Properties of R P : (2). The ideal PR P is the unique maximal ideal of R, and R P /PR P is isomorphic to Q(R/P), the quotient field of R/P. (3). The functor R P R : R Mod R P Mod takes short exact sequences to short exact sequences. (4). For any R-module M, the kernel of the mapping M R P R M is the C(P)-torsion submodule {m M cm = 0 for some c C(P)}.

6 The noncommutative case In a noncommutative ring R an ideal P is prime if AB P implies A P or B P, for all ideals A, B of R. [ ] [ ] Z Z 2Z 2Z Example 1. Let R = and P =. Then P is Z Z 2Z 2Z prime since the ideals of R are in one-to-one correspondence with the ideals of the Z. [ ] Note that R/P Z/2Z Z/2Z =. This factor ring has divisors Z/2Z Z/2Z of zero, but at least it is a full matrix ring over a field. The [ logical candidate ] for a localization of R at P is Z(2) Z (2), which can be constructed by inverting all scalar Z (2) Z (2) matrices with an odd entry.

7 The analog of a field of quotients If R is a subring of Q, then R is a left order in Q if (i) each c R that is not a divisor of zero has an inverse in Q, and (ii) each q Q can be written in the form c 1 a, for a, c R, where c is not a divisor of zero. To put the product ac 1 into standard form we need to be able to find a 1 and c 1 with ac 1 = c1 1 a 1, or c 1 a = a 1 c. This is the left Ore condition. Then c 1 a d 1 b can be put into standard form by finding a 1 and d 1 with d 1 a = a 1 d, so that ad 1 = d1 1 a 1 and (c 1 a)(d 1 b) = c 1 (ad 1 )b = c 1 (d1 1 a 1)b = (c 1 d1 1 )(a 1b) = (d 1 c) 1 (a 1 b).

8 Goldie s theorem Goldie s theorem (1958) shows that R is a left order in a full ring of n n matrices over a skew field if and only if R is a prime ring with ACC on left annihilators and finite uniform dimension. (These finiteness conditions always hold when R is left Noetherian.) This ring of quotients is called the classical ring of left quotients of R and is denoted by Q cl (R). We are now ready to look at noncommutative localization. We focus on prime ideals of R for which Q cl (R/P) exists. We would like to invert C(P), which we must now define as the set of elements that are regular modulo P (not as the complement of P). Equivalently, these are the elements inverted by the canonical homomorphism R R/P Q cl (R/P). In Example 1, where P is the set of 2 2 matrices with even entries, C(P) is the set of matrices whose determinant is odd.

9 Ore localization If P is a prime Goldie ideal for which C(P) satisfies the left Ore condition and is left reversible (if ac = 0 for c C(P), then c a = 0 for some c C(P)) then the construction of a localization R P goes through much as in the commutative case, and all four of the properties listed above still hold. Even in very nice cases the left Ore condition may not hold. Example [ 2. Z 0 R = Z Z ] [ 2Z 0, P 1 = Z Z ] [ Z 0, P 2 = Z 2Z Then R/P i = Z/2Z and so Qcl (R/P i ) = R/P i is a field, making P i as nice a prime Goldie ideal as possible. ].

10 Checking the Ore condition The ideal P[ 1 satisfies the ] left Ore condition: [ ] a11 0 c11 0 given a = R and c = C(P a 21 a 22 c 21 c 1 ) we 22 [ need to solve ] [ c a = a c ] with [ c C(P). ] [ ] c11 0 a11 0 a11 0 c11 0 =. 0 0 a 21 a c 21 c 22 The ideal [ P 2 does ] not satisfy [ the] left Ore condition: given R and C(P ) the equation [ ] [ ] [ ] [ ] c a = c 21 c a 21 a has no solution with c 22 odd.

11 Category theoretic localization In the absence of the left Ore condition we could focus on the last two conditions that hold in the commutative case. (3). The functor R P R : R Mod R P Mod takes short exact sequences to short exact sequences. (4). For any R-module M, the kernel of the mapping M R P R M is the C(P)-torsion submodule {m M cm = 0 for some c C(P)}. The C(P)-torsion submodule is changed to {m M r R, crm = 0 for some c C(P)}. This determines a closure operator, a module of quotients, and an exact functor into an abelian category consisting of these quotient modules.

12 Category theoretic localization This construction can be done more generally relative to any torsion theory, and there is a large body of work studying it. The best answer to its origin probably lies in Gabriel s thesis Des catégories abéliennes, Bull. Soc. Math. France , published in In the case of a prime Goldie ideal, this quotient functor does produce a ring R C(P), but properties (1) and (2) from the commutative case may be lost. For the prime ideal P 2 in Example [ 2 the torsion-theoretic ] Z(2) Z construction produces R C(P2 ) = (2), not a bad answer, Z (2) Z (2) but its unique maximal ideal is not closely connected to P 2.

13 We now turn to properties (1) and (2) of the commutative case: (1). λ : R R P is universal with respect to the property that if c C(P) then λ(c) has an inverse in R P. (2). The ideal PR P is the unique maximal ideal of R, and R P /PR P is isomorphic to Q(R/P). A ring satisfying (1) can be defined, but it may be the zero ring. A new approach inverting matrices rather than elements was introduced by Cohn in Free Rings and Their Relations (1971) and Inversive localization in Noetherian rings, Commun. Pure Appl. Math. 26 (1973), It s convenient to generalize to a semiprime ideal S for which Q cl (R/S) exists and is semisimple Artinian, i.e. for which R/S is a semiprime left Goldie ring. In this case we say that S is a semiprime Goldie ideal.

14 Definition of the universal localization Let S be a semiprime Goldie ideal, and let Γ(S) be the set of all square matrices inverted by the canonical mapping R R/S Q cl (R/S). Definition (Cohn, 1973, Noetherian case) The universal localization R Γ(S) of R at a semiprime Goldie ideal S is the ring universal with respect to inverting all matrices in Γ(S). That is, if φ : R T inverts all matrices in Γ(S), then there exists a unique ring homomorphism φ such that the following diagram commutes. λ R R Γ(S) φ φ T

15 Elementary properties Note: If S is left localizable, then R Γ(S) coincides with the Ore localization R S defined via elements. Theorem Let S be a semiprime Goldie ideal of R. (a) (Cohn, 1971) The universal localization of R at S exists. (b) (Cohn, 1971) The canonical mapping λ : R R Γ(S) is an epimorphism in the category of rings. (c) (1981) The ring R Γ(S) is flat as a right module over R if and only if S is a left localizable ideal.

16 Cohn s construction Cohn s construction showing that R Γ(S) exists: For each n and each n n matrix [c ij ] in Γ(S), take a set of n 2 symbols [d ij ], and take a ring presentation of R Γ(S) consisting of all of the elements of R, as well as all of the elements d ij as generators; as defining relations take all of the relations holding in R, together with all of the relations [c ij ][d ij ] = I and [d ij ][c ij ] = I which define all of the inverses of the matrices in Γ(S). Theorem (Cohn, 1971) Each element of R Γ(S) is an entry in a matrix of the form (λ(c)) 1, for some C Γ(S).

17 A characterization of R Γ(P) Theorem Let S be a semiprime Goldie ideal of R. (a) (Cohn, 1973) R Γ(S) modulo its Jacobson radical is naturally isomorphic to Q cl (R/S). (b) (1981) R Γ(S) is universal with respect to the property in (a). Theorem (1981) Let R be left Noetherian, let N be the prime radical of R, and let K = ker(λ), for λ : R R Γ(N). (a) The kernel K is the intersection of all ideals I N such that C(N) C(I ). (b) The ring R/K is a left order in a left Artinian ring, and R Γ(N) is naturally isomorphic to Q cl (R/K).

18 Symbolic powers Definition Let S be a semiprime Goldie ideal of R, with λ : R R Γ(S). The n th symbolic power of S is S (n) = λ 1 (R Γ(S) λ(s n )R Γ(S) ). Theorem If R is left Noetherian, then the following conditions hold for the symbolic powers of the semiprime ideal S. (a) S (n) is the intersection of all ideals I such that S n I S and C(S) C(I ). (b) C(S) is a left Ore set modulo S (n). (c) R Γ(S) λ(s n )R Γ(S) = (J(R Γ(S) ) n, for all n > 0. (d) R/S (n) is an order in the left Artinian ring R Γ(S) /(J(R Γ(S) )) n.

19 Goldie s localization In two papers in 1967 and and 1968, Goldie defined a localization at a prime ideal P of a Noetherian ring R by first factoring out the intersection n=1 P(n) of the symbolic powers. He then took the inverse limit of the Artinian quotient rings Q cl (R/P (n) ), and finally defined an appropriate subring of this inverse limit. Theorem (1984) Let P be a prime ideal of the Noetherian ring R. Then Goldie s localization of R at P is isomorphic to R Γ(P) / n=1 Jn, where J is the Jacobson radical of R Γ(P).

20 Some information about the kernel Theorem (1976) If R K S is finitely generated, with SK = K, then K ker(λ). Proof. Let K = n i=1 Rx i, for x 1,..., x n R. Since K = SK, we have K = n i=1 Sx i. For x = (x 1,..., x n ) we have x t = Ax t, where the n n matrix A has entries in S. Thus (I n A)x t = 0 t. But I n A is invertible modulo S, so it certainly belongs to Γ(S). Therefore the entries of x must belong to ker(λ), and so K ker(λ). Corollary R Γ(P) can be determined for a prime ideal of an hereditary Noetherian prime ring, since in HNP rings each prime ideal is either localizable or idempotent.

21 Example 2 again [ ] [ ] [ ] Z 0 Z 0 Z 0 Example 2: R =, P Z Z 2 =, K =. Z 2Z Z 0 Then K 2 = K, so P 2 K = K, and therefore K kerλ, for λ : R R Γ(P2 ). It follows easily that K = kerλ and R Γ(P2 ) is isomorphic to Z (2). [ ] 2Z 0 An alternate approach: Recalling that P 1 =, we can Z Z invert [ the scalar ] matrices in C(P 1 P 2 ) to obtain [ R P1 P 2 Z(2) 0 = with maximal ideal Z (2) Z P Z(2) 0 2 = (2) Z (2) 2Z (2) Factoring out n i=n P 2 yields RΓ(P2 ) = Z (2). This illustrates a two-step approach: use the Ore localization at a suitable semiprime ideal, followed by its universal localization, which in this case is just a factor ring. ].

22 Chain conditions on R Γ(S) [ ] [ Z 0 Z(2) 0 Example 3. If P 3 =, then R Z 3Z Γ(P1 P 3 ) = Q Z (3) This ring is no longer Noetherian. Bill Blair and I had to work much harder to produce such an example for a prime ideal. Recall that in a ring finitely generated as a module over its center, the clique of a prime ideal P is the set of prime ideals with the same intersection down to the center of the ring. Theorem (2016, with Christine Leroux) If R is finitely generated as a module over its Noetherian center, and P is a prime ideal that does not contain the intersection of symbolic powers of any other prime ideal in the clique of P, then R Γ(P) is the homomorphic image of the Ore localization at the clique of P, and therefore it is Noetherian. ].

23 Some questions Question: Are there conditions under which the kernel of R R Γ(P) is the intersection of the symbolic powers of P, as in the commutative case? Fully bounded Noetherian? Question: Is there a natural class of rings for which the universal localization works well? In particular, are there (weak) finiteness conditions that might be preserved? Piecewise Noetherian? Question: The top of R Γ(P) is well-behaved since R/P (n) is a left order in R Γ(P) /(J(R Γ(P) ) n. Is it possible to use universal localization to give new proofs of some results for modules over Noetherian rings? e.g. Stafford s results on generating modules efficiently e.g. Q cl (R/P) R (M/PM) = R Γ(P) R M / J(R Γ(P) R M)

24 Some future directions for research Future directions: of enveloping algebras. McConnell [1968] obtained some results on Goldie s localization for certain enveloping algebras. of group rings. Some topologists have been interested in universal localization at the augmentation ideal. of additive categories.

25 Another construction of the universal localization Let S be a semiprime Goldie ideal of R. Each element in the universal localization R Γ(S) has the form λ(a)λ(c) 1 λ(b) t for some a, b R n and some matrix C Γ n (S). Instead of modeling elements of the form c 1 a where c C(S), via ordered pairs (c, a), we model elements of the form q ij = e i C 1 e t j, where C Γ(S) and e i, e j are unit vectors. Let X be a left R-module. To construct a module of quotients, consider ordered triples (a, C, x t ) where a R n, C Γ n (S), and x X n, for all positive integers n.

26 Addition of congruence classes Model for addition: Suppose C, D are already invertible. [ ] 1 [ ] [a b] C 0 x t 0 D y t = [a b] [ ] [ C 1 0 x t 0 D 1 y t [ac 1 bd 1 [ ] ] x t = ac 1 x t + bd 1 y t y t ] = Definition (a : C : x t ) + (b : D : y t ) = ( [ C 0 [a b] : 0 D ] [ x t : y t ]) This is a commutative, associative binary operation.

27 Scalar multiplication, avoiding the Ore conditiion Model for scalar multiplication: Suppose C, D are invertible. [ [a 0] C r t ] 1 [ ] b 0 0 D y t = [ [a 0] C 1 C 1 r t bd 1 ] [ ] 0 0 D 1 y t = [ac 1 ac 1 r t bd 1 [ ] ] 0 y t = ac 1 r t bd 1 y t Definition (a : C : r t ) (b : D : y t ) = ( [ C r [a 0] : t b 0 D ] : [ 0 y t ])

28 The module of quotients Theorem (1989) (a) The above addition and multiplication define a ring of quotients Γ 1 R and a module of quotients Γ 1 X. (b) Each element of Γ 1 R is an entry in the inverse of a matrix in Γ(S). Theorem (1989) Γ 1 R = R Γ(S) and Γ 1 X = R Γ(S) R X.

Universal Localization of Piecewise Noetherian Rings

Universal Localization of Piecewise Noetherian Rings Northern Illinois University UCCS November 29, 2017 Acknowledgments For commutative rings, results on piecewise Noetherian rings are from two joint