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BULLETIN(New Series)OF THE
有锁和无锁的区别
AMERICAN MATHEMATICAL SOCIETY
Volume33,Number3,July1996
Commutative algebra with a view toward algebraic geometry,by David Einbud, Graduate Texts in Math.,vol.150,Springer-Verlag,Berlin and New York, 1995,xvi+785pp.,$69.50,ISBN0-387-94268-8
In the title of a story[To]written in1886,Tolstoy asks,“How Much Land Does a Man Need?”and answers the question:just enough to be buried in.One may ask equally well,“How much algebra does an algebraic geometer—man or woman—need?”and prospective algebraic geometers have been known to worry that Tolstoy’s answer remains accurate.Authors of textbooks have at times given vastly different answers,albeit with a general tendency over time to be monotonely increasing in what they expect will be uful.In crudely quantitative terms,one has at one extreme the elegant minimalist introduction of Atiyah and Macdonald [AM]at128pages,then Nagata[Na]at234pages,Matsumura[Ma]at316pages, Zariski and Samuel[ZS]at743pages,and the prent work at785pages.吃水果有什么好处
I suspect that most of my fellow algebraic geometers,as a practical matter, would answer my question by saying,“Enough to read Hartshorne.”Hartshorne’s Algebraic geometry[Ha],appearing in1977,rapidly became the central text from which recent generations of algebraic geometers have learned the esntial tools of their subject in the aftermath of the“French revolution”inspired by the work of Grothendieck and Serre(Principles of algebraic geometry by Griffiths and Har-ris[GH]plays a comparable role on the geometric side for the infusion of com-plex analytic techniques entering algebraic geometry at about the same period). Hartshorne’s book is peppered with references to the then-existing texts in com-mutative algebra,no one of which contained everything he needed.Indeed,one of the goals of Einbud’s book was to provide a single work containing all of the algebraic results needed in Hartshorne’s book.
It is sobering to consult Hodge and Pedoe’s Methods of algebraic geometry[HP] and e how little commutative algebra they got away with—for example,there is no entry for“Noetherian”in the index.The student reader(I was one)was t loo in algebraic geometry armed with not much more than the fundamental theorem of algebra,resultants,the Hilbert basis theorem,the Nullstellensatz,and the Pl¨u cker equations for the Grassmannian(which sneaks in a bit of reprenta-tion theory),plus some splendid geometric insights.It is heartening to feel that algebraic geometry has attained the poi
nt where we have at our disposal the power of scheme-theoretic techniques without losing the inexhaustible wellspring of inspi-ration supplied by geometric insights(the insights were kept alive most vividly in the work of Griffiths and Mumford).
It is interesting to obrve that about the time that Hartshorne’s book was being published,a cond wave of algebraic input into algebraic geometry,admittedly less powerful than thefirst,was gathering strength.Rather than being insights having universal application throughout algebraic geometry,such as Serre’s FAC [Se],they tend to be more specific and to involve more specialized algebra.The discovery by Kempf[Ke]and Kleiman-Laksov[KL]that the singularities of the theta-divisor of the Jacobian variety of an algebraic curve may be studied using 1991Mathematics Subject Classification.Primary13-01,14-01,13A50,13C15.
c 1996American Mathematical Society
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368BOOK REVIEWS
determinantal varieties(varieties defined by the minors of matrices who entries are polynomials)mi生煎馒头
ght rve as an opening of this pha of the influence of algebra on algebraic geometry.About the same time,a theorem of Macaulay about what we would now call Gorenstein rings appears as the crucial step in Griffiths’proof [Gr]that the derivative of the period map for projective hypersurfaces is injective (infinitesimal Torelli).The beautiful u of the Eagon-Northcott resolution in the bound by Gruson-Lazarsfeld-Peskine[GLP]for the regularity of ideals of projective curves is another example.Einbud’s book has the great virtue of incorporating the commutative algebra that lies behind this cond wave of algebraic influence on algebraic geometry as well.
疯狂出租车4There are certainly topics in algebra that lie outside the scope of Einbud’s book which belong in the arnal of many algebraic geometers.Reprentation theory in many guis appears in algebraic geometry,for example,in the geometry of the period domains which appear in Hodge theory,in geometric invariant theory,or the various more specialized us of the reprentation theory of the general linear group,such as Kempf’s derivation of the Eagon-Northcott complex.A student planning to work in arithmetic algebraic geometry might wish for a more abstractly oriented package of tools—simplicial objects,the derived category,algebraic K-theory.It should be noted that Tolstoy’s protagonist collaps in the attempt to include just one additional tract of ground,and Einbud has managed to distill the gist of some of the topics into a ries of highly concentrated appendices.
背影的作者The subject of commutative algebra itlf has undergone considerable changes in the period since Hartshorne’s book was written—although probably not,with one exception,a revolution.There is one change which has overtaken commuta-tive algebra that is in my view revolutionary in character—the advent of symbolic computation.This is as yet an unfinished revolution.At prent,many rearchers routinely u Macaulay,Maple,Mathematica,and CoCoA to perform computer experiments,and as more people become adept at doing this,the list of theorems that have grown out of such experiments will enlarge.The next pha of this de-velopment,in which the questions that are considered interesting are influenced by computation and where the questions make contact with the real world,is just beginning to unfold.I suspect that ultimately there will be a sizable applied wing to commutative algebra,which now exists in embryonic form.Einbud has been very much involved in computational developments;he has,for example,authored many of the basic scripts in u with Macaulay.He has included a highly uful chapter on Gr¨o bner bas containing most of the basic theorems and with a ries of suggested computational projects.I am in agreement with him that this is an area that most young algebraic geometers ought to learn.
Einbud’s book is clearly intended to rve both as an introduction for students and as a reference work.It is difficult to harmonize the two goals,and indeed many reference works,stating theorems i
n maximal generality,are virtually un-readable.Einbud’s strategy for surmounting this difficulty is quite interesting and successful.A typical chapter begins with an informal discussion,in which he attempts to explain to the reader what is really going on and why the topic is important and interesting.The discussions are almost invariably illuminated by Einbud’s remarkable gift for producing the telling example.He then wipes the slate clean and begins again,giving formal definitions and proofs.He frequently takes the unusual step of explaining what is not true and why the theory is not
BOOK REVIEWS369 simpler than it is.This is then followed by a profusion of exercis,drawn from the heartland of commutative algebra and from algebraic geometry.
The book is informed by Einbud’s broad knowledge of algebraic geometry and his encyclopaedic knowledge of commutative algebra;he works actively in both fields.The algebra is illuminated whenever possible by its geometric interpretation, a feature that I found extraordinarily uful and one which I imagine some readers from the commutative algebra side willfind enlightening.He brings the reader as clo as he can to late-breaking developments in commutative algebra,with numerous references to many of the important things currently going on.
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My friend Harsh Pittie once gave the following description of the styles of math-ematical exposition of three of the leading mathematicians of the day:A paper or lecture by X was like a walk in a beautiful garden.Y would take you up in an airplane and show you the rervoir from which the garden ultimately got its water,while with Z you got into a jeep and went careening through the shrubbery. The expository style of this book is mostly of the“walk-in-the-garden”variety,al-though Einbud does take the reader up in a plane when necessary,and there are rare but occasional crashes through the shrubbery.The style is delightfully old-fashioned,with digressions,interesting stories,apostrophes to the reader(including one exhorting him or her to generalize a conjecture of mine),puns,historical excur-sions,and advice.The book is infud with an evident affection for both subjects, commutative algebra and algebraic geometry,and Einbud displays equal relish in showing the reader the Hilbert-Burch Theorem and the geometry of a trigonal canonical curve.
The existence of this book rais some interesting questions about how students in algebraic geometry ought to be trained.Traditionalfirst-year graduate cours in algebra often have a rather perfunctory treatment of rings,ideals,and modules, emphasizing instead group theory andfield theory.For a future algebraic geome-ter,field theory is esntial,an introduction to reprentation theory would be more uful than the Sylow theorems,and a solid introduction to commutative algebr
a is vital.The increasing sophistication and diversity of the algebraic tools now in u in algebraic geometry require students to be quite lective if they are going to get a Ph.D.in a reasonable amount of time and make it really important to acquire the habit of lifelong learning and persistently expanding one’s repertoire of mathematical techniques.My advice to a student would be to read the portions of Einbud’s book relevant to Hartshorne,skimming where appropriate and per-haps shifting back and forth between the two books,and then to nibble further at Einbud’s book over the succeeding years.
This volume is a major and highly welcome addition to the mathematical litera-ture,providing a unified,elegant,and exhaustive survey of tho topics in commu-tative algebra likeliest to be of u to algebraic geometers.The rigorous treatment is supplemented by substantial heuristic,historical,and motivational ctions and a wide range of exercis.There is an exceptionally thorough bibliography and nu-merous links to recent developments.I anticipate that it will soon be found on the bookshelf of virtually any practicing algebraic geometer or commutative algebraist.
370BOOK REVIEWS
References
[AM]M.Atiyah and I.Macdonald,Introduction to commutative algebra,Addison-Wesley,Read-ing,MA,1969.MR39:4129
[GH]P.Griffiths and J.Harris,Principles of algebraic geometry,Wiley,New York,1978.MR 80b:14001
[GLP]L.Gruson,R.Lazarsfeld,and C.Peskine,On a theorem of Castelnuovo,and the equations defining space curves,Invent.Math.72(1983),491–506.MR85g:14033
[Gr]P.Griffiths,On the periods of certain rational integrals:I and II,Ann.Math.90(1969), 460–495,498–541.MR41:5357
[Ha]R.Hartshorne,Algebraic geometry,Springer-Verlag,New York,1977.MR57:3116 [HP]W.Hodge and D.Pedoe,Methods of algebraic geometry,Cambridge Univ.Press,Cam-bridge,1947.MR10:396b
[Ke]G.Kempf,On the geometry of a theorem of Riemann,Ann.of Math.98(1973),178–185.
MR50:2180
[KL]S.Kleiman and D.Laksov,On the existence of special divisors,Amer.J.Math.94(1972), 431–436.MR48:2148
[Ma]H.Matsumura,Commutative ring theory,Cambridge Univ.Press,Cambridge,1986.MR 88h:13001
[Na]M.Nagata,Local rings,Wiley,New York,1962.MR27:5790
[Se]J.-P.Serre,Faisceaux alg´e briques coh´e rents,Ann.Math.61(1955),197–278.MR16:953c [To]L.Tolstoy,How much land does a man need?,Short Stories,vol.2,Modern Library,New York,1965,pp.183–199.
[ZS]O.Zariski and P.Samuel,Commutative algebra,Vol.I,Springer-Verlag,New York,1975.
MR52:5641,19:833e
Mark Green
University of California at Los Angeles
E-mail address:mlg@math.ucla.edu
