the λ coalescent speed of coming down from infinity consider a coalescent that comes down from infinity meani ng that it starts from a configuration containing infinitely many blocks at time 0 yet it has a finite number nt of blocks at any positive time t 0 we exhibit a deterministic function v 0 0 such that nt v t 1 almost surely and in l p for any p 1 as t 0 our approach relies on a novel martingale technique

localization in a morita context with applications to fixed rings in the study of a noetherian ring r there are two related questions that have played a major role in recent years one is finding prime ideals of r which satisfy the second layer condition the other is finding the second layer link closed sets of prime ideals of r in particular these properties are necessary to determine the classical sets of prime ideals of r which allows one to obtain a well behaved ore localization see for example 24 8 91 one aspect of this problem is the transfer of these properties from the prime spectrum of one ring r to the prime spectrum of another related ring a jategaonkar s has shown that if r is a commutative noetherian ring then the group ring rg satisfies the second layer condition whenever g is polycyclic by finite this result was then generalized by bell 2 to strongly group graded rings more recently letzter lo has studied the relationship between the prime spectrums of the noetherian rings r and s when s is module finite over r showing how the second layer condition and links between prime ideals transfer from one ring to the other let t be a right noetherian ring and let e be an idempotent element of t then it is well known that ete is a right noetherian ring in sections 1 and 2 we examine the transfer of the second layer condition between the prime ideals of t and ete as well as the transfer of classical sets of prime ideals between these rings this enables us to study how these properties of spec ete translate to spec 1 e t 1 e in particular under certain hypotheses we prove that whenever ete satisfies the second layer condition then so does 1 e t 1 e corollary 2 6 in addition we are able to show that under the same hypotheses if every prime ideal of ete belongs to a finite classical set then so does every prime ideal of 1 e t l e another way to describe the relationship between t and ete is via a morita context see 11 let r and s be rings let s w and r v be bimodules and let 6 vos w r wor v s be bimodule 373 oozi 8693 91 3 00

quotient rings and localization for noetherian rings abstract criteria are obtained for the localizability of a noetherian ring at a semiprime ideal s particularly when s is the nil radical of r the results are applied to the existence of quotient rings a decomposition theorem is obtained for certain noetherian rings which admit artinian quotient rings

finite dimensional duality on the generalized lie algebra 2 q we show a local duality between the left and right finite dimensional representations on the enveloping algebra of 2 q the generalized lie algebra introduced by lyubashenko and sudbery this duality works in a general context of algebras satisfying certain algebraic properties our goal in this paper is to prove that the enveloping algebra of 2 q satisfies these abstract properties

stochastic flows associated to coalescent processes we study a class of stochastic flows connected to the coalescent processes that have been studied recently by mohle pitman sagitov and schweinsberg in connection with asymptotic models for the genealogy of populations with a large fixed size we define a bridge to be a right continuous process b r r 0 1 with nondecreasing paths and exchangeable increments such that b 0 0 and b 1 1 we show that flows of bridges are in one to one correspondence with the so called exchangeable coalescents this yields an infinite dimensional version of the classical kingman representation for exchangeable partitions of we then propose a poissonian construction of a general class of flows of bridges and identify the associated coalescents we also discuss an important auxiliary measure valued process which is closely related to the genealogical structure coded by the coalescent and can be viewed as a generalized fleming viot process

localization and ideal theory in iterated differential operator rings abstract in this paper we study primality hypercentrality simplicity and localization and the second layer condition in skew enveloping algebras and iterated differential operator rings we give sufficient conditions for the skew enveloping algebra of a nilpotent lie algebra with coefficient ring containing the rational numbers to be a simple ring and we give necessary and sufficient conditions in the case that the lie algebra is abelian our main results show that if l is a finite dimensional solvable lie algebra over a field k of characteristic zero and r is an artinian or a commutative noetherian algebra over k then the skew enveloping algebra r u l satisfies the second layer condition we discuss consequences of this for localization and use the localization theory to state a classical krull dimension versus global dimension inequality when k is uncountable

on skew brownian motion we consider the stochastic equation x t w t beta l x 0 t where w is a standard wiener process and l x 0 cdot is the local time at zero of the unknown process x there is a unique solution x and it is adapted to the fields of w if beta leq 1 but no solutions exist if beta 1 in the former case setting alpha beta 1 2 the unique solution x is distributed as a skew brownian motion with parameter alpha this is a diffusion obtained from standard wiener process by independently altering the signs of the excursions away from zero each excursion being positive with probability alpha and negative with probability 1 alpha finally we show that skew brownian motion is the weak limit as n rightarrow infty of n 1 2 s lbrack nt rbrack where s n is a random walk with exceptional behavior at the origin

localization and ideal theory in noetherian strongly group graded rings abstract we study primality hypercentrality simplicity and localization and the second layer condition in skew groups rings and group graded rings we give necessary and sufficient conditions for the skew group ring of a torsion free nilpotent group to be a simple ring and if the coefficient ring is commutative we give necessary and sufficient conditions for the skew group ring of an abelian group to be simple our method involves showing certain group graded rings are hypercentral our main results show that if g is a polycyclic by finite group and r is an artinian ring or a commutative noetherian ring then a strongly g graded ring with base ring r satisfies the second layer condition we discuss consequences of this for localization in such rings

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

beta coalescents and continuous stable random trees coalescents with multiple collisions also known as a coalescents were introduced by pitman and sagitov in 1999 these processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block in the case that the measure a is the beta 2 a a distribution they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring here we use a recent result of birkner et al to prove that beta coalescents can be embedded in continuous stable random trees about which much is known due to the recent progress of duquesne and le gall our proof is based on a construction of the donnelly kurtz lookdown process using continuous random trees which is of independent interest this produces a number of results concerning the small time behavior of beta coalescents most notably we recover an almost sure limit theorem of the present authors for the number of blocks at small times and give the multifractal spectrum corresponding to the emergence of blocks with atypical size also we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and the allele frequency spectrum associated with mutations in the context of population genetics

algebre homologique dans la categorie mod r u g abstract letkbe an algebraically closed field of characteristic zero ga finite dimensional lie algebra overk u g the enveloping algebra ofg ra noetheriank algebra on whichgacts by derivations via a lie algebra morphism andr u g the differential operator rings generated byrandu g this paper is concerned with the homological algebra forr u g modules which areg locally finite especially with injective and projective modules minimal injective resolutions and cohomology soientkun corps algebriquement clos de caracteristique nulle gunek algebre de lie de dimension finie u g l algebre enveloppante deg runek algebre noetherienne sur laquellegopere par derivations via un morphisme d algebres de lie etr u g le produit croise derparu g ce papier traite de l algebre homologique dans la categorie desr u g modulesg localement finis plus particulierement des modules injectifs et projectifs des resolutions injectives minimales et de la cohomologie

the influence of ideal structure on representation theory the purpose of this paper is to explore the representation theoretic consequences of the two sided ideal structure of a noetherian ring with special emphasis on an additional combinatorial structure on the prime spectrum of the ring the graph of links we obtain existence theorems for extensions and for rings satisfying the strong second layer condition we obtain extensive information about the indecomposable injective modules particularly detailed results are obtained when the ring is polynormal

actions of linearly reductive groups on pi algebras let g be a linearly reductive group acting rationally on a pi algebra r we study the relationship between r and the fixed ring r g generalizing earlier results obtained under the additional hypothesis that r is affine

connections between optimal stopping and singular stochastic control ii reflected follower problems the stochastic control problem of following a brownian path by a process of bounded variation and subject to a reflecting barrier at the origin reflected follower is reduced to a question of optimal stopping with absorption direct probabilistic arguments are used to establish the equivalence of the two problems under suitable conditions on the cost functions

a note on two sided stochastic control problems we consider a class of two sided stochastic control problems for each continuous process t t t with bounded variation the state process xt is defined by xt bt f0t i xs a d s f0t i xs a d s where a is a positive constant and bt is a standard brownian motion we show the existence of an optimal policy so as to minimize the cost function j e f0 e sxs2 ds with discount rate 0 associated with

a taxonomy of 2 primal rings abstract various conditions on a noncommutative ring imply that it is 2 primal i e the ring s prime radical coincides with the set of nilpotent elements of the ring we will examine several such conditions and show that their known interdependencies are their only ones of particular interest will be the ps i condition on a ring i e every factor ring modulo the right annihilator of a principal right ideal is 2 primal we will see that even within a fairly narrow class of rings ps i is a strictly stronger condition than 2 primal we will show that the ps i condition is left right asymmetric we will also study the interplay between various types of semilocal rings and various types of 2 primal rings the kothe conjecture will make a cameo appearance in section 6 we will examine subideals of prime ideals of commutative rings that are invariant under derivations

affine noetherian pi rings have enough clans abstract a noetherian pi ring which is affine over the centre can be localized at every finite or infinite link connectivity component p in the sense that the multiplicative set b p satisfies the ore condition and determines p

classical localizability in solvable enveloping algebras and poincare birkhoff witt extensions abstract in several classes of noncommutative noetherian rings jategaonkar s intersection conditions are verified for finite unions of cliques of prime ideals and it follows that these finite unions of cliques are classically localizable if they satisfy incomparability among the rings in question are the enveloping algebra of any finite dimensional solvable lie algebra g over any field k of characteristic zero any twisted smash product r u g where r is a commutative noetherian k algebra and g acts on r via k linear derivations and certain iterated differential operator rings over r in contrast to previously verified cases k is allowed to be countable one tool used here a lying over theorem for cliques has some independent interest namely if r s is a flat finite centralizing extension of noetherian rings satisfying the second layer condition and if x is a clique in spec r there exists a clique y in spec s such that p r p y x

representational repleteness of differential operator rings abstract in this paper we use results of goodearl describing the link graph of a differential operator ring over a commutative noetherian3 algebra to derive sufficient conditions for a prime ideal in such a ring to be representationally replete

prime ideals in differential operator rings and crossed products of infinite groups abstract we use a version of the martindale ring of quotients to study prime ideals in extensions of a ringr corresponding to two cases ifg is a group of automorphisms ofr we form the crossed product r g if g is a lie algebra of derivations of r we have the twisted differential operator ring denoted byr g sometimes known as the twisted smash product r t u g we obtain analogues of incomparability for crossed products of nilpotent groups and differential operator rings of solvable lie algebras in the case of crossed products the incomparability result of d s passman and m lorenz for the infinite cyclic group also proved by g bergman is generalized

integrality for pi rings abstract in this paper we study schelter integral extensions of pi rings we prove in particular lying over going up and incomparability for prime ideals a major result is transitivity of integrality if r s b are pi rings such that b is integral over s and s is integral over r then b is integral over r next we obtain a powerful criterion for integrality if s is a prime pi ring such that its center is integral over a noetherian subring r of s then s is integral over r this allows interesting applications to the theory of finite group actions further topics concern eakin nagata type results and embeddings of quotient rings for integral extensions finally we analyze the relationship between module finite extensions and finitely generated integral extensions obtaining positive results for affine noetherian pi algebras and algebras satisfying certain restrictions on pi degrees e g algebras of low pi degree

links between prime ideals in differential operator rings in recent years much work has been done on a localization theory for noetherian rings links between prime ideals play a central role in this development when a ring r is not commutative it is rarely possible to localize r at a prime ideal p having a link between p and another prime ideal is an obstruction to localization in this paper we will investigate the links between prime ideals in differential operator rings showing that it is possible to localize these rings in a certain sense recall that a multiplicatively closed subset s of a ring r is a right ore set if for all r in r and all s in s we have rs n sr qr the set s is an ore set if it is both a right and a left ore set it is always possible to construct a ring rf where the elements of s become units however if we insist on writing the elements of rf as right fractions rs with r in r and s in s then the set s must be a right ore set 9 sect 12 11 let p be a prime ideal of r and let 7 p r e r 1 r p is regular in r p we say that p is a localizable prime ideal if the set u p is an ore set we will shortly give an example of a prime ideal which is not localizable but first we introduce the differential operator ring let r be a ring and let 6 be an additive map from r to itself we call 6 a derivation if it satisfies the product rule 6 ab a b ad b the differential operatqr ring over r denoted r 0 s is a free left r module with basis 1 8 13 thus every element of r 0 s is a polynomial in 0 with coefficients from r the addition in r l s is defined as usual for polynomials but multiplication is extended from r by the rule br rti 6 r the differential operator ring is an associative ring and is determined up to isomorphism by an obvious universal property 9 sect 12 21

localization in ore extensions of commutative noetherian rings there is extensive literature on the subject of localization at prime ideals in noetherian rings in the non commutative setting where it is seldom possible to localize at a prime ideal one central concept which has emerged is the notion of a link between prime ideals as an obstruction to localization in recent years much work has been devoted to describing linked prime ideals in several important classes of non commutative noetherian rings notably group rings k a brown in 4 8 universal enveloping algebras k a brown in s 6 and differential operator rings g sigurdsson in 173 in this paper we analyze the localizability of prime ideals in an ore extension of a ring r if r is a ring with automorphism g the ore extension or skew polynomial ring r x a is defined to be the set of polynomials of the form cy r xi where rie r with termwise addition and with multiplication determined by distributivity and the relation xr r x r x a is an associative ring if r is right noetherian the hilbert basis theorem shows that r x a is also right noetherian in this paper we shall assume that the coefficient ring r is a commutative noetherian ring all rings are associative with an identity element a ring is noetherian if it is both right and left noetherian the same convention applies to other ambidextrous properties for an r bimodule m i m and r m are respectively the left and right annihilators of m in r for an ideal a of r v ja denotes the collection of elements of r which are regular modulo a earlier versions of the results in this paper appeared in the author s ph d thesis where krull dimension techniques were used extensively the author would like to thank his thesis advisor at mcmaster university

prime ideals in enveloping rings let l be a lie algebra over the field k of characteristic 0 and let u l denote its universal enveloping algebra if r is a k algebra and l acts on r as derivations then there is a natural ring generated by r and u l which is denoted by r u l and called the smash product of r by u l the aim of this paper is to describe the prime ideals of this algebra when it is noetherian specifically we show that there exists a twisted enveloping algebra u x on which l acts and a precisely defined one to one correspondence between the primes p of r u l with p n r 0 and the l stable primes of u x here x is a lie algebra over some field c 2 k let l be a lie algebra over the field k and let u l denote its universal envelop ing algebra if r is a k algebra and l acts on r as derivations then this action determines in a natural manner a ring generated by rand u l which is denoted by r u l and called the smash product of r by u l the aim of this paper is to study the prime ideals p of r u l disjoint from r that is with p n r o the main result here describes these primes when char k 0 and r u l is right noetherian when dealing with smash products it is frequently necessary to study more general objects for example if x is a lie ideal of l then r u l is determined by its subalgebra r u x and the quotient u lj x indeed

simple modules and their essential extensions for skew polynomial rings let r be a commutative noetherian ring and alpha an automorphism of r this paper addresses the question when does the skew polynomial ring s r theta alpha satisfy the property diamond that for every simple s module v the injective hull e s v of v has all its finitely generated submodules artinian the question is largely reduced to the special case where s is primitive for which necessary and sufficient conditions are found which however do not between them cover all possibilities nevertheless a complete characterisation is found when r is an affine algebra over a field k and alpha is a k algebra automorphism in this case diamond holds if and only if all simple s modules are finite dimensional over k this leads to a discussion involving close study of some families of examples of when this latter condition holds for affine k algebras s r theta alpha the paper ends with a number of open questions

prime ideals in differential operator rings catenarity let r be a commutative algebra over the commutative ring k and let a 51 5 be a finite set of commuting k linear derivations from r to r let t r 61i 6 al 5 n be the corresponding ring of differential operators we define and study an isomorphism of left r modules between t and its associated graded ring r xl a polynomial ring over r this isomorphism is used to study the prime ideals of t with emphasis on the question of catenarity we prove that t is catenary when r is a commutative noetherian universally catenary k algebra and one of the following cases occurs a k is a field of characteristic zero and a acts locally finitely b k is a field of positive characteristic c k is the ring of integers r is affine over k and a acts locally finitely

prime ideals in skew polynomial rings and quantized weyl algebras the concern of this paper is to investigate the structure of skew polynomial rings ore extensions of the form t r where and are both nontrivial and in particular to analyze the prime ideals of t the main focus is on the case that r is commutative noetherian in this case the prime ideals of t are classified polynomial identities and artin rees separation in prime factor rings are investigated and cliques of prime ideals are studied the second layer condition is proved as well as boundedness of uniform ranks for the prime factor rings corresponding to any clique futher q skew derivations on noncommutative coefficient rings are introduced and some preliminary results on contractions of prime ideals of t are obtained in this setting finally prime ideals in quantized weyl algebras over fields are analyzed

connections between optimal stopping and singular stochastic control i monotone follower problems the stochastic control problem of tracking a brownian motion by a nondecreasing process monotone follower is related to a question of optimal stopping direct probabilistic arguments are employed to show that the two problems are equivalent and that both admit optimal solutions

coagulation transport equations and the nested coalescents the nested kingman coalescent describes the dynamics of particles called genes contained in larger components called species where pairs of species coalesce at constant rate and pairs of genes coalesce at constant rate provided they lie within the same species we prove that starting from rn species the empirical distribution of species masses numbers of genes n at time t n converges as n rightarrow infty to a solution of the deterministic coagulation transport equation begin aligned partial t d partial x psi d a t left d star d d right end aligned where psi x cx 2 star denotes convolution and a t 1 t delta with delta 2 r the most interesting case when delta 0 corresponds to an infinite initial number of species this equation describes the evolution of the distribution of species of mass x where pairs of species can coalesce and each species mass evolves like dot x psi x we provide two natural probabilistic solutions of the latter ipde and address in detail the case when delta 0 the first solution is expressed in terms of a branching particle system where particles carry masses behaving as independent continuous state branching processes the second one is the law of the solution to the following mckean vlasov equation begin aligned dx t psi x t dt v t delta j t end aligned where j is an inhomogeneous poisson process with rate 1 t delta and v t t ge 0 is a sequence of independent random variables such that mathcal l v t mathcal l x t we show that there is a unique solution to this equation and we construct this solution with the help of a marked brownian coalescent point process when psi x x gamma we show the existence of a self similar solution for the pde which relates when gamma 2 to the speed of coming down from infinity of the nested kingman coalescent