Portal de Periódicos da CAPES

David Rees, FRS 1918–2013

2016; Wiley; Volume: 48; Issue: 3 Linguagem: Inglês

10.1112/blms/bdw010

1469-2120

Rodney Y. Sharp,

Rings, Modules, and Algebras

David Rees completed his Cambridge undergraduate studies in mathematics in summer 1939; in his first three months of postgraduate work in autumn 1939, he produced a characterization of completely 0-simple semigroups. War then intervened: he worked until the end of the war at Bletchley Park, the British codebreaking centre in Buckinghamshire, where he was part of a team that broke the Enigma code regularly for some critical months during 1940. After the war, he first worked at Manchester University, but moved to Cambridge University in 1948. In the immediate post-war period, he continued with research into semigroups and non-commutative algebra. His first paper was very influential, and he is considered by semigroup theorists to be one of the founding fathers of their subject. At Cambridge, after attending a seminar by Douglas Northcott, Rees changed the focus of his research to commutative Noetherian rings. During an extraordinarily productive period between 1954 and 1961, he produced a string of far-reaching, foundational and deep ideas and results of lasting significance. Highlights include reductions of ideals, his Valuation Theorem, the theory of grade, the graded rings that are nowadays known as ‘Rees rings’, the Artin–Rees Lemma and his characterization of local rings whose completions have zero nilradical. Rees was appointed to the Chair of Pure Mathematics at the University of Exeter in 1958 and elected FRS in 1968. He was awarded the Polya Prize of the London Mathematical Society, and an Honorary DSc by the University of Exeter, in 1993. David Rees was born and brought up in Abergavenny; he was the fourth of five children of Gertrude (née Powell) and (another) David Rees, a corn merchant. The family lived above David's father's corn shop. There is history of both longevity and mathematical ability in David Rees's father's line: his father died at the age of 88, three of his siblings had 90th birthdays and one of his great-great-grandfathers was the Reverend Thomas Rees (1774–1858), a well-known non-conformist minister, who, according to one obituarist, was considered to be the best mathematician in Wales in 1802. David Rees was educated at King Henry VIII Grammar School in Abergavenny. At the time, the school had an excellent headmaster, Wyndham Newcombe, who was also a very good teacher of mathematics. Rees's early teenage years were affected by ill health, and he was absent from school for several terms. During those periods of illness, he studied at home independently, and his mother, armed with lists from the young David, became one of the best customers of the Abergavenny public library. This diligence stood him in good stead when he was able to return to normal schooling: under the guidance of mathematics master L. F. Porter, he was able to catch up quickly with his mathematics. He did rather well in School Certificate examinations in 1934 and 1936, and was awarded a State Scholarship and admission to Sidney Sussex College, Cambridge, where his studies were supervised by Gordon Welchman. Rees started as a Commoner, but was made an Exhibitioner after one year and, after he had come top in the Preliminary Examination for Part II at the end of his second year, he was made a Scholar. Rees was persuaded to take Parts IIB and III together in 1939, and another candidate, Hermann Bondi, with whom he had a friendly rivalry and who only had to take Part IIB at that time, managed to just beat him into second place. Rees began postgraduate work in September 1939, without a proper supervisor, but inspired by ‘wonderful lectures’ by Philip Hall. In the autumn of 1939, he had a rather successful three months, during which he produced a characterization of completely 0-simple semigroups. Here are the relevant definitions. Let S be a semigroup, with operation written multiplicatively. A (two-sided) ideal of S is a non-empty subset A of S such that a s ∈ A and s a ∈ A for all a ∈ A and s ∈ S . A zero element of S is a (necessarily uniquely determined) element 0 ∈ S such that 0 s = 0 = s 0 for all s ∈ S . The semigroup S with zero is called 0-simple if { 0 } and S are its only ideals and there exist s , t ∈ S such that s t ≠ 0 . The semigroup S is said to be completely 0-simple if it is 0-simple and has a non-zero idempotent element e such that the only idempotent f ∈ S for which e f = f e = f ≠ 0 is e itself. Paper [1] was submitted in early May 1940, and represents a very successful start by Rees to postgraduate research. Given Rees's intensive work at Bletchley Park from December 1939 (see §3), most of the work for [1] must have been completed in Rees's first three months of research. In that paper, Rees does thank ‘Mr. P. Hall, both for his encouragement, while this paper was being written, and his very considerable assistance in preparing the paper for publication’. It should be noted that paper [1] is explicitly mentioned in the citation that accompanied David Rees's election as FRS. The phrases ‘Rees matrix semigroup’ and ‘Rees Theorem’ ensure that his name will live on among the semigroup community. By summer 1939, Gordon Welchman had been appointed to work at Bletchley Park, the British codebreaking centre in Buckinghamshire. In December 1939, Welchman knocked on the door of Rees's college rooms to tell him that he had a job for him to do. Rees naturally wanted to know details, but Welchman refused to elaborate, and only after prompting did he tell Rees to meet him a few days later at Bletchley railway station. Rees did so, and in this way was recruited to a team of codebreakers in Hut 6 at Bletchley Park. Welchman recruited several other young mathematicians he knew from Cambridge, including some he had taught at Sidney Sussex College. Even in later life after the veil of secrecy that covered the war-time exploits of Bletchley Park had been lifted, David Rees did not like to talk about his time there. However, it is now clear that he was part of a team that broke the Enigma code regularly for some critical months during the summer and autumn of 1940. The German operators of the Enigma machines were told which three of the five available rotors and which settings to use each day, but they had to choose the initial positions of the rotors and indicate their choices by means of the first three letters of their initial messages. John Herivel, who had also been recruited to Bletchley Park from Sidney Sussex College by Welchman, predicted in February 1940 that some German operators, when tired or stressed, might use short cuts that could be exploited by the Bletchley Park codebreakers. For three months, this lateral thinking by Herivel produced no result; but in May 1940 some of the German operators began to make the predicted mistakes, and David Rees and his fellow codebreakers were able to use the technique known as the ‘Herivel tip’ to break Enigma ciphers for some critical months from May 1940. Herivel has written an account 〈7〉 of the Herivel tip and related matters, in which he attributes the first successful use of the tip to David Rees: see 〈7, pp. 118–119〉. Interestingly, the same book contains a reproduction of a statement by David Rees about the Herivel tip in which he declared that he did not recollect being the person responsible for the first successful use of it, although he conceded that ‘it is possible that my memory is at fault’; see 〈7, p. 122〉. What is not in doubt is that the first successful use of the Herivel tip resulted in much rejoicing, shouting and standing on chairs. Rees thought very highly of Herivel's idea: he described it as ‘brilliant’ in the above-mentioned statement 〈7, p. 122〉; and he is quoted in 〈7, p. 11〉 as having said, in 2000, that ‘of course, the Herivel tip was one of the seminal discoveries of the Second World War’. Rees told the present author in 2007 that, in his opinion, Herivel did not receive the recognition that he deserved. In late 1941, David Rees was seconded to the Enigma Research Section at Bletchley Park, run by Dillwyn (‘Dilly’) Knox, and where the Abwehr Enigma used by the German Secret Service was broken. The so-called ‘Double Cross Committee’ used captured German agents to persuade Hitler that the D-Day landings would be south of Calais rather than in Normandy. It is said that, without the break into the Abwehr Enigma, British intelligence officers could not have known that the deception was working. David Rees subsequently moved to the ‘Newmanry’, the department at Bletchley Park led, for the second half of the war, by M. H. (Max) Newman, for which the first Colossus computer was constructed to assist with codebreaking. The list of subsequently famous mathematicians whom David Rees encountered during his service at Bletchley Park includes A. O. L. Atkin, I. J (Jack) Good, J. A. (Sandy) Green (who worked at Bletchley Park as a teenager), Peter Hilton, Max Newman, G. B. Preston and Shaun Wylie. Sandy Green and Peter Hilton were later to become coauthors of mathematical papers with David Rees, and Rees's third paper [3] (written after the war) was about a paper by Jack Good. There are now available in print numerous articles detailing aspects of the war-time exploits at Bletchley Park; two recent ones are The Guardian's obituary of Peter Hilton 〈23〉 and the Royal Society's Biographical Memoir about William Tutte 〈31〉. Following the end of the war, David Rees resumed his academic studies, and soon found himself working under Max Newman in a different context: he was appointed in 1945 to an Assistant Lectureship in the Department of Mathematics at Manchester University, and Newman was the head of that department. Rees remained at Manchester until 1948, when he was appointed to a University Lectureship at Cambridge; in 1949 he was appointed to a Fellowship at Downing College. He worked in semigroup theory and non-commutative algebra while at Manchester, and continued with these themes for his first years as a Cambridge don. He was very pleased with his joint paper [8] (with Sandy Green) from this time; in it they considered, for positive integers n and r with r ⩾ 2 , the semigroup S n , r (again written multiplicatively) generated by n elements in which each element x satisfies x r = x , but which is otherwise free, and they showed that the question as to whether S n , r is finite is intimately related to Burnside's Conjecture in group theory. Recall that the latter conjecture for r is the statement that, for all n > 0 , the group B n r generated by n elements in which each element x satisfies x r = e , but which is otherwise free, is finite. A striking result from the Green–Rees paper [8] is that the Burnside conjecture for r is true if and only if S n , r + 1 is finite for all n > 0 . David Rees wrote just five papers on semigroup theory, but their influence on the development of that subject has been very substantial. Interested readers might like to consult the tribute 〈11〉 to Rees in Semigroup Forum, where he is described as ‘one of the pioneers of semigroup theory’, as ‘one of the subject's founding fathers’, and as having ‘laid the foundations for a number of important avenues of future research’. However, as David Rees published about forty papers in commutative algebra, it is appropriate that the majority of this obituary be devoted to his contributions to that field. Another addition to the Mathematics faculty at Cambridge in 1948 was Douglas G. Northcott, who had spent 21 post-war months in Princeton, where he had been greatly stimulated by a seminar with the title ‘Valuation theory’ run by Emil Artin and Claude Chevalley, and by much informal guidance from Artin. Northcott returned to Cambridge having become a dedicated algebraist (his PhD work concerned a theory of integration for functions with values in a Banach space). In Princeton, Northcott had, at Artin's suggestion, studied the famous paper 〈28〉 by Weil, and, as a consequence, began to work in the algebra underlying what some refer to as the ‘pre-Grothendieck’ era of algebraic geometry. Thus Northcott became a commutative algebraist. Back in Cambridge, Northcott organized a very successful working seminar on Weil's book 〈29〉. David Rees was a member of the audience, and he was so inspired by the seminar that he too became a commutative algebraist. (Another aspect of Northcott's seminar that was life-changing for Rees was the presence in the audience of Joan Cushen: David and Joan were married in 1952.) David Rees's transition from semigroup theory was gradual and his first paper in commutative algebra ([9], written jointly with Northcott) only appeared in 1954. That paper is central to the next section. Paper [9], written jointly with Douglas Northcott, is, by a long way, David Rees's most-cited research paper: Mathematical Reviews records more than 200 citations of it. It introduced the notion of reductions of ideals. This concept and the related concept of integral closure have had a major influence on research in commutative algebra in the more than 60 years since they were introduced; indeed, even in the present century, hardly a top-level international conference in commutative algebra passes without there being several mentions of reductions. Let b and a be proper ideals of R. The ideal b is said to be a reduction of a if b ⊆ a and there exists s ∈ N 0 (the set of non-negative integers) such that b a s = a s + 1 . One can view such a b as an approximation to a that nevertheless retains some of the properties of a: for example, a prime ideal p of R is a minimal prime ideal of b if and only if it is a minimal prime ideal of a, and when that is the case, the multiplicity of b corresponding to p is equal to the multiplicity of a corresponding to p. (The multiplicity of a corresponding to its minimal prime ideal p is the multiplicity e ( a R p ) of the ideal a R p of the localization R p .) The inspiration for the definition of reduction came to David Rees while he was thinking about so-called irrelevant ideals in a (commutative Noetherian) positively graded ring S = ⨁ n ∈ N 0 S n that is generated, as an algebra over S 0 , by homogeneous elements of degree 1. Set S + : = ⨁ n ∈ N S n (where N denotes the set of positive integers); let A = ⨁ n ∈ N A n be a graded ideal of R generated by homogeneous elements of degree 1; Rees noted that A n = S n for all sufficiently large n (that is, A is irrelevant) if and only if there exists v ∈ N 0 such that A ( S + ) v = ( S + ) v + 1 . This observation led to the birth of the concept of reduction. The fundamental connections between reductions and integral closures can be summarized as follows. Let b ⊆ a be ideals of R. Then b is a reduction of a if and only if each element of a is integrally dependent on b. Furthermore, the set J of all ideals of R that have b as a reduction has a unique maximal member, b ¯ say: b ¯ is the union of the members of J, and this ideal b ¯ is precisely the set of all elements of R that are integrally dependent on b. The ideal b ¯ is called the integral closure of b; it has the property that the ideals of R that have b as a reduction are precisely those between b and b ¯ . We say that b is integrally closed if b = b ¯ . The ideal b is said to be a minimal reduction of a if b is a reduction of a and there is no reduction c of a with c ⊂ b (the symbol ‘ ⊂’ denotes strict inclusion). Most of [9] is written under the hypothesis that R is a local ring Q with infinite residue field, and so that hypothesis will be in force until further notice; also m will denote the maximal ideal of Q. Rees and Northcott defined the analytic spread ℓ ( a ) of a; this turns out to be equal to the dimension of G ( a ) / m G ( a ) , where G ( a ) denotes the associated graded ring ⨁ i ∈ N 0 a i / a i + 1 of a. They proved that every reduction of a requires at least ℓ ( a ) generators, that a reduction of a is a minimal reduction of a if and only if it can be generated by ℓ ( a ) elements, and that each reduction of a contains a minimal reduction of a. Thus all minimal generating sets of all minimal reductions of a have exactly ℓ ( a ) elements. They went on to show that ℓ ( a ) can be interpreted as follows. Elements u 1 , … , u t ∈ a are said to be analytically independent in a if, whenever h ∈ N and f ∈ R [ X 1 , … , X t ] (the ring of polynomials over R in t indeterminates) is a homogeneous polynomial of degree h such that f ( u 1 , … , u t ) ∈ a h m , then all the coefficients of f lie in m. Then, if b is a reduction of a, dim Q / m ( b / m b ) = : t and { u 1 , … , u t } is a minimal generating set for b, it turns out that b is a minimal reduction of a if and only if u 1 , … , u t are analytically independent in a. Consequently, ℓ ( a ) is equal to the largest number of elements of a that are analytically independent in a, and ht a ⩽ ℓ ( a ) ⩽ dim Q / m ( a / m a ) . As mentioned above, the appearances in the literature of the concepts of reduction and integral closure in the 60 years since Rees and Northcott published [9] are very numerous; far-reaching extensions, generalizations and related concepts have been studied in depth. The reader can glean some idea of the enormous influence that these ideas of Rees and Northcott have had, and continue to have, in commutative algebra by studying the book 〈25〉 by Swanson and Huneke on integral closures. That book (which, incidentally, is dedicated to Joseph Lipman and David Rees) contains, inter alia, a wealth of information and detail about many of Rees's contributions to commutative algebra. In this section, in which we revert to consideration of the general commutative Noetherian ring R, we recall some graded rings used by Rees to good effect. Nowadays, these rings are referred to as ‘Rees rings’ and ‘extended Rees rings’. A homogeneous isomorphism between graded rings is an isomorphism that preserves degrees. There is an obvious homogeneous isomorphism between R [ a T ] and R ( a ) . Notice that the graded ring R ( a ) / a R ( a ) is homogeneously isomorphic to the associated graded ring G ( a ) : = ⨁ i ∈ N 0 a i / a i + 1 of a. The ring R ( a ) is also called the blowing up ring of a; this terminology has its roots in the fact that the projective spectrum of R ( a ) is the topological space underlying the scheme obtained by blowing up Spec ( R ) with respect to a. The 0th component of R [ a T , T − 1 ] is R, and Rees used to very good effect the observation that, for an i ∈ N 0 , the 0th component of the graded ideal R [ a T , T − 1 ] U i of R [ a T , T − 1 ] is just a i . In other words, R [ a T , T − 1 ] U i ∩ R = a i . By means of this observation, Rees was able to reduce some questions about powers of an ideal in a Noetherian ring to the case where the ideal is principal and generated by a non-zerodivisor. In that special case, simplifications are often available. The following proof of Krull's Intersection Theorem, based on the proof in Rees [14], illustrates his use of the above device. Theorem 7.1 (W. Krull's Intersection Theorem 〈10〉)..(Recall that R is Noetherian.) If r ∈ ⋂ i = 1 ∞ a i , then there exists a ∈ a such that r = a r . Proof..We deal first with the case where a is the principal ideal R u generated by a non-zerodivisor u. Since r ∈ ⋂ i = 1 ∞ a i , we can, for each i ∈ N , write r = u i s i for some s i ∈ R . Then s i = u s i + 1 for all i ∈ N , since u is a non-zerodivisor in R. Therefore, R s 1 ⊆ R s 2 ⊆ ⋯ ⊆ R s i ⊆ ⋯ , and so there exists j ∈ N such that R s j = R s j + 1 . Thus s j + 1 = s j b for some b ∈ R , from which we see that s j = u s j + 1 = s j ( b u ) , with b u ∈ R u . Therefore, r = u j s j = ( b u ) u j s j = ( b u ) r . In the general case, consider the (Noetherian) extended Rees ring S : = R [ a T , T − 1 ] , and set U : = T − 1 , a non-zerodivisor of that ring. Let r ∈ ⋂ i = 1 ∞ a i . Then r ∈ ⋂ i = 1 ∞ S U i , and, by the first paragraph of this proof, we can write r = f U r for some f ∈ S . Write f = ∑ i = − v w b i T i , where b i ∈ a i for all i = − v , … , w . Compare components of degree 0 to see that r = b 1 r , and note that b 1 ∈ a . In the same paper [14], Rees also gave a proof of what is now known as ‘the Artin–Rees Lemma’. That proof also uses the extended Rees ring. Lemma 7.2 (The Artin–Rees Lemma [14, Lemma 1])..(Recall that R is Noetherian.) Let a , b be two ideals of R. Then there exists k ∈ N such that a n ∩ b = a n − k ( a k ∩ b ) for all n ⩾ k . Proof..Set S : = R [ a T , T − 1 ] , the extended Rees ring of a, and let B = b R [ T , T − 1 ] ∩ S , an ideal of S. Thus an element ∑ i = − v w r i T i of R [ T , T − 1 ] belongs to B if and only if r i ∈ a i ∩ b for all i = − v , … , w . Hence B is a graded ideal of the Noetherian ring S, and so has a finite generating set of homogeneous elements, { b i 1 T i 1 , … , b i q T i q } say, where b i j ∈ a i j ∩ b for all j = 1 , … , q . Let k = max { i 1 , … , i q } . Then a n ∩ b = a n − k ( a k ∩ b ) for all n ⩾ k . (The reader might find it helpful to note that ( a i ∩ b ) a n − i ⊆ ( a i + 1 ∩ b ) a n − ( i + 1 ) for integers n , i with 0 ⩽ i < n .) David Rees explained the name of the lemma as follows: David had his proof of the lemma in 1954, but he did not submit it for publication until May 1955; paper [14] appeared in 1956, in the very month in which Emil Artin lectured, at a conference in Japan, about his discovery of the same argument and result; M. Nagata was asked to adjudicate as to who should receive the credit, and responded that ‘it is obviously the Artin–Rees Lemma’. There is a version of the lemma for modules, which states that if N is a submodule of a finitely generated R-module M, then there exists k ∈ N such that a n M ∩ N = a n − k ( a k M ∩ N ) for all n ⩾ k . This result means that the topology induced on N by the a-adic topology on M is the a-adic topology on N. The interested reader might like to consult 〈13, Theorem 8.5〉. As well as being well suited to the study of powers of a fixed ideal a of R, the extended Rees ring R [ a T , T − 1 ] can be used to explore the integral closures of the powers of a, because it turns out that R [ a T , T − 1 ] U i ¯ ∩ R = a i ¯ for each i ∈ N 0 (where, once again, U = T − 1 ). However, Rees's Valuation Theorem, which is the subject of the next section, also provides information about the integral closures of powers of a. In a series of papers [11, 12, 15–17] published during an exceptionally productive period from 1955 to 1957, David Rees established what he called his ‘Valuation Theorem’, which can be viewed as describing the integral closures of the powers of an ideal a of R in terms of certain uniquely determined discrete valuation rings (DVRs). These DVRs are nowadays referred to as ‘the Rees valuations rings’, while the associated discrete valuations are called ‘the Rees valuations’. Intimately related to the Rees valuations is the asymptotic Samuel function, defined as follows. Definition 8.1..Let a be a proper ideal of the (Noetherian) ring R. The order function of a is the function w a : R ⟶ N 0 ∪ { ∞ } for which Lemma and Definition 8.2 (Rees [11, Lemma 1.2])..With the notation of Definition 8.1, for each r ∈ R , the limit The name is in recognition of P. Samuel's initiation of the study of the asymptotic theory of ideals in 〈21〉. Samuel's work had a formative influence on Rees. The definition of the order function and the definition of integral closure can be extended in obvious ways to the case where the underlying ring, A say, is not Noetherian, and the analogue of Lemma 8.2 still holds: see McAdam 〈14, Proposition 11.1〉. Indeed, for a k ∈ N 0 , an ideal I of A and a , a ' ∈ A , one can show that, if a ∈ I k ¯ , then w I ¯ ( a ) ⩾ k , while if w I ¯ ( a ' ) > k , then a ' ∈ I k ¯ . See McAdam 〈14, Proposition 11.2〉. For the statement of Rees's Valuation Theorem, we require the concept of discrete integer-valued valuation of R, including in the case where R is not a domain. For basic properties of DVRs and the associated discrete valuations, the reader is referred to 〈13, Chapter 4〉. Definition 8.3..By a discrete integer-valued valuation of R, we shall mean the composition v ∗ of the natural ring homomorphism R ⟶ R / p for some minimal prime ideal p of R and a (conventional) discrete integer-valued valuation v of the quotient field of R / p that is non-negative on R / p . (Strictly speaking, the values of v and v ∗ lie in Z ∪ { ∞ } .) Note that, for an r ∈ R , we have v ∗ ( r ) = ∞ if and only if r ∈ p . Theorem 8.4 (Rees's Valuation Theorem [16])..(Recall that R is Noetherian.) Let a be a proper ideal of R. Then there exist discrete integer-valued valuations v 1 ∗ , … , v h ∗ of R (in the sense of Definition 8.3), and positive integers e 1 , … , e h , such that Where do the Rees valuations come from? Key points in an argument that proves their existence are that, in a DVR, every ideal is integrally closed, and the Mori–Nagata Theorem, the statement of which uses the concept of Krull domain. Definition 8.5..A Krull domain is an integral domain D such that for each prime ideal p of D of height 1, the localization D p is a DVR; D = ⋂ p ∈ Spec ( D ) , ht p = 1 D p ; each non-zero a ∈ D belongs to only finitely many of the prime ideals of D of height 1. Theorem 8.6 (The Mori–Nagata Theorem 〈16, 17〉)..(Recall that R is Noetherian.) Suppose that R is an integral domain. Then its integral closure R ¯ is a Krull domain. This result is due to Y. Mori in the case where R is local and to M. Nagata in the general case. (Note that R ¯ need not be Noetherian.) In the following hints about how the Mori–Nagata Theorem can be used to prove Rees's Valuation Theorem, attention will be concentrated on the case where R is a domain, because in that case it is easier to see where the Rees valuations come from. A key point in the extension of the Valuation Theorem from a Noetherian domain to a general commutative Noetherian ring R is the fact that, for an ideal a of R and r ∈ R , we have r ∈ a ¯ if and only if r + p ∈ a + p / p ¯ for each minimal prime ideal p of R. Remark 8.7..The asymptotic Samuel function w a ¯ of Lemma 8.2 is related to the integral closures of powers of a. We have already noted earlier in the section that if r ∈ a c ¯ for a c ∈ N , then w a ¯ ( r ) ⩾ c ; the Valuation Theorem can be used to prove the converse statement in our Noetherian ring R. Thus, for c ∈ N and r ∈ R , it is the case that r ∈ a c ¯ if and only if w a ¯ ( r ) ⩾ c . This consequence of the Valuation Theorem is useful in applications, such as to questions about whether two ideals a , b of R are projectively equivalent, that is, such that a s ¯ = b t ¯ for some s , t ∈ N . Rees thought that [16] was his best paper, but another of which he was particularly proud, and in which valuations also featured, was [26]. In that, he settled a problem that had been posed by Zariski in 〈32〉 and was related to Hilbert's 14th problem. The latter problem can be stated as follows: if S denotes the ring of polynomials in n indeterminates over a field k, and if F is a subfield of the field of fractions of S that contains k, must the ring S ∩ F be finitely generated over k? In 〈32〉, Zariski asked the following question: if F is a finitely generated field extension of a field k, and S is a finitely generated integrally closed integral domain over k whose field of fractions contains F, must the ring S ∩ F be finitely generated over k? Zariski himself proved in 〈32〉 that if the transcendence degree of F over k is 1 or 2, then S ∩ F is indeed finitely generated over k. In [26], Rees constructed an example that showed that the answer to Zariski's problem is negative. For this, he used delicate and very impressive geometric arguments involving an extended Rees ring, over the homogeneous coordinate ring of a projective complex elliptic curve C, of the ideal defining a point on C. Hilbert's 14th problem was settled, again negatively, by Nagata in 〈18〉. The concept of grade, fundamental to the theory of Cohen–Macaulay rings, is also due to David Rees. Elements u 1 , … , u t in R are said to form a regular sequence if they generate a proper ideal and ( ( u 1 , … , u i − 1 ) : u i ) = ( u 1 , … , u i − 1 ) for all i = 1 , … t . (The particular case of this equation when i = 1 is interpreted as ( 0 : u 1 ) = 0 , that is, u 1 is a non-zerodivisor on R.) In [18, 19], Rees proved that, for a proper ideal a of R, each maximal R-sequence contained in a has length equal to the least integer i such that Ext R i ( R / a , R ) ≠ 0 . Consequently, all maximal R-sequences contained in a have the same length, and Rees defined this length to be the grade of a. His methods enabled him to deduce quickly that