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# Math for Info Age MATH 107

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Date Created: 10/29/15

THIRTEEN OPEN QUESTIONS IN COMMUTATIVE ALGEBRA Speaker Mel Hochster This is an edited version of the slides used at a talk at a conference in honor of Joseph Lipman on the occasion of his sixty fth birthday at a conference at Purdue University in July 2004 Each question is followed by some very terse comments on its status and some references which are sometimes woven into the comments and sometimes follow them The references given are in most cases only a small sample of the relevant literature 1 The Zariski Lipman conjecture Let R be a finitely generated CC algebra and P a prime of R If DerCR Rp is Rp free then Rp is regular Comments Rp is normal Lipman The hypersurface case is known Scheja Storch The graded case is known Hochster The dimension 2 case implies the general case Flenner but is open even for complete intersections in 4 J Lipman Ree derivation modules on algebraic varieties Amer J Math 87 1965 8747898 G Scheja and U StorchUber differentielle Abhangigkeit bei idealen analytischer Algebren Math Z 114 1970 1017112 7 Differentielle Eigenschaften des Lokaliserungen analytischer Algebren Math Ann 197 1972 1377170 M Hochster The Zariski Lipman conjecture in the graded case J of Alg 47 1977 4117424 H Flenner Extendability of differential forms on nonisolated singularities lnvent Math 94 1988 3177326 2 Resolution of singularities Given a reduced and irreducible excellent scheme or simply a domain finitely generated over a eld over Z or over an excellent discrete valuation ring one wants to find a proper birational map of a scheme or variety onto it whose local rings are regular Better yet if R is a domain one wants f1 f E R such that Vf1 f is the singular locus and for all i Rf1fZ fgfi fnfi is regular This is a local algebraic version of blowing up lterations of this process have been the main tool in resolving singularities Comments The literature is HUGE There are a handful of references below The recent exposition by H Hauser listed below contains many more Hauser calls Zariski the grandfather of resolution of singularities Much notable work was done by S S Abhyankar Hironaka solved the problem in equal characteristic zero Lipman gave an optimal treatment for surfaces Recent work of De Jong solves the problem in characteristic 1 and mixed characteristic if the birationality condition is relaxed and one allows a nite field extension of function fields S S Abhyankar Local uniformization of algebraic surfaces over ground elds of characteristic 1 74 0 Ann Math 63 1956 4917526 Resolution of singularities of embedded algebraic surfaces Academic Press New York 1966 H Hironaka Resolution of singularities of an algebraic variety over a eld of characteristic zero 1 H Ann of Math 2 79 1964 1097203 2057326 J Lipman Desingularization of 2 dimensional schemes Ann Math 107 1978 1517207 E Bierstone and P D Milman Uniformization of analytic spaces J Amer Math Soc 2 1989 8017836 A J de Jong Smoothness semi stability and alterations Publ 1HES Math No 83 1996 51793 A J de Jong Families of curves and alterations Ann Inst Fourier Grenoble 47 1997 5997621 H Hauser The Hironaka theorem on resolution of singularities Or A proof we always wanted to understand Bull AMS 40 2003 3237403 N 3 Cancellation of indeterminates Zariski problem If CH1 xnm Ry1 ym is R 2 Chm xn Comments The answer is yes for n 1 or n 2 Fujita Miyanishi Sugie cf T Fujita On Zariski problem Proc Japan Acad Ser A Math Sci 55 1979 1067110 M Miyanishi and T Sugie Af ne surfaces containing cylinderlike 1 open sets J Math Kyoto Univ 20 1980 11742 and T Sugie Characterization of the af ne plane and the af ne threespace in Topological Methods in Algebraic Transformation Groups editors H Kraft T Petrie and G W Schwarz Progress in Math 80 Birkhauser Boston Basel Berlin 1989 1777190 Be wary about cancellation Let R S be Cyzy 7 17 22 ijyz2y 7 1 7 R 9 S but RR 2 Sjt Danielewski surfaces Cf Danielewski On the cancellation problem and automorphism groups of af ne algebraic varieties preprint Warsaw 1989 K H Fieseler On complex af ne surfaces with C action Comment Math Helv 69 1994 5727 4 The Jacobian conjecture If R ijl ac and f R 7 R with fi for 1 S i g n is such that det8fiBacj E C 7 0 then f is an automorphism Comments There have been many incorrect proofs several have been published Cf Bass E Connell and D Wright The Jacobian Conjecture reduction of degree and formal expansion of the inverse Bull AMS 7 1982 2877330 The question remains open even if n 2 A survey is given in A van den Essen Polynomial automorphisms and the Jacobian conjecture Algebre non commutative groupes quantiques et invariants Reims 1995 55781 S min Congr 2 Soc Math Erance Paris 1997 5 Settheoretic de nition Is there a method for determining the least number of generators of an ideal up to radicals how many equations are needed to define an algebraic set This seems incredibly dif cult It is not known whether every irreducible curve in Ag is even locally a set theoretic complete intersection It is not known whether for example the parametric Moh curve at t6t31y t8z 210 is a set theoretic complete intersection at the origin Cf G Lyubeznik A survey of problems and results on the number of de ning equations in Commutative Algebra MSR1 Publ 15 Springer Verlag New York 1989 3757390 6 The strong intersection conjecture Let R m be a local Noetherian ring and let M N 51 0 be finitely generated R modules such that deM lt 00 and M R N has finite length Let I AnnRM Then dim N S depth IR Comments Ci 0 Peskine and L Szpiro Dimension projective nie et cohomologie locale Publ 1HES 42 1973Many other questions considered there have been settled including conjectures of H Bass and M Auslan der 1n equal characteristic several follow from the existence of big Cohen Macaulay modules In mixed characteristic others were settled by P Roberts using intersection theory M Auslander7s rigidity conjecture was settled negatively by R Heitmann See also M Hochster Topics in the homological theory of modules over commutative rings CBMS Regional Conf Ser in Math No 24 Amer Math Soc Providence R1 1975 P Roberts Le theoreme d intersection CR Acad Sc Paris S r 1 304 1987 1777180 P Roberts Multiplicities and Chern classes in local algebra Cambridge Tracts in Mathematics 133 Cambridge Univ Press Cambridge England 1998 R Heitmann A counterexample to the rigidity conjecture for rings Bull AMS New Series 29 1993 94797 7 The Buchsbaum Eisenbud Horrocks problem If R is regular local dim R n and M lt 00 then the jth Betti number of M is 2 Comments See D Buchsbaum and D Eisenbud Algebra structures for nite free resolutions and some structure theorems for ideals of codimension 3 Amer J Math 99 1977 4477485 and R Hartshorne Algebraic vector bundles on projective spaces a problem list Topology 18 1979 1177128 This is known if dim R S 4 Weaker form the sum of the Betti numbers is at least 2 Stronger form the torsion free rank of a minimal jth module of syzygies is 2 1 S j S n Closely related if M lt 00 over Rm a local ring and g 1 ac E m then HM M Z 3 Another somewhat related question if I J Q R with R m regular local dim R n and l J m primary then Torll RIRJ needs 2 n 7 dim Rl 7 dim RJ generators This is open for Kyzjj even if IJ are m primary one expects 1 JIJ to need at least three generators 2 The original problem asks for a lower bound for the dimension of TorfM K The second asks for a lower bound for the dimension of Tor a TorlllRI RJ One may ask more generally for lower bounds for the dimensions of more complicated iterated Tor modules under various conditions on the input modules Why make it harder Sometimes the harder question is easier Another closely related question Let A m be an Artin local ring and let 1 acn E m ls the number of generators of H1x1 A at least 11 This is true if n 1 and n 2 S Dutta 8 Positivity 0f Serre multiplicities Let Rm be regular local dim R 11 Let M N be Noetherian R modules with M R N lt 00 1f xMNZ1 TorfMN then 1 dimM dimN S dimR 2 xM N 2 0 and 3 xMN gt 0 iff equality holds in Comments This was proved by Serre if R contains a field or if 13 is formal power series over a DVR Serre proved 1 in general Cf J P Serre Algebre local Multiplicites Springer Verlag Lecture Notes in Math 11 Springer Verlag New York 1961 That X vanishes when it should was proved in R Roberts The vanishing ofintersection multiplicities ofperfect complexes Bull AMS 13 1985 1277130 and Gillet and C Soule K the orie et nullite des multiplicites d intersection C R Acad Sci Paris S rie 1 no 3 t 300 1985 71774 2 was proved by O Gabber an exposition is given in R Berthelot Alterations de varietes algebriques d apres A J de Jong S minaire BOURBAKI 48eme annee n0 815 815 01 7 815 39 using De Jong7s alterations Positivity of multiplicities remains a frustrating open question It is implied by the existence of small Cohen Macaulay modules considered in 12 below 9 Direct summands of regular rings are CohenMacaulay If R is a direct summand as an R module of a regular ring S then must R be Cohen Macaulay This is easy if S is module finite over R It is known in equal char cf Hochster and J L Roberts Rings of invariants of reductive groups acting on regular rings are Cohen Macaulay Advances in Math 13 1974 1157 175 Hochster and C Huneke Applications of the existence of big Cohen Macaulay algebras Advances in Math 113 1995 457117 and 7 Tight closure in equal characteristic zero Springer Verlag to appear The result for finitely generated CC algebras is sharpened in J F Boutot Singularites rationelles et quotients par les groupes r ductz fs lnvent Math 88 1987 658 if S has rational singularities and R is a direct summand then R has rational singularities The question is open in mixed characteristic It would follow from a solution of the fourth form of the problem considered in 12 or from a solution of 13 10 The direct summand and canonical element conjectures understanding superheight Let R be regular and R Q S module nite Is R a direct summand of S This is trivial if R Q Q and then R need only be normal and was proved in char 1 gt 0 in Hochster Contracted ideals from integral extensions of regular rings Nagoya Math J 51 1973 25743 It is easy in dimension 3 2 It reduces to the case of formal power series over a DVR in which 19 generates the maximal ideal In R C Heitmann The direct summand conjecture in dimension three Annals of Math 2 156 2002 6957712 the result was finally proved in mixed characteristic in dimension 3 The direct summand conjecture remains open if dim R 2 4 An equivalent conjecture although it appears stronger is that if R m K is local and g 1 xn is a system of parameters if one maps the Koszul complex IC IC R to a free minimal resolution G of K beginning with 1R then the induced map 1C7 R 7 syz K is such that the image of 1 is not in syz K Letting the system of parameters vary one sees that one has a conjecturally nonzero canonical element in HZsyZ 71K or as pointed out to me by Joe Lipman a conjecturally nonzero canonical element in Torf K Using that the direct summand conjecture implies the canonical element conjecture one sees that it implies many of the early local homological conjectures known from Roberts7 work Cf Hochster Canonical elements in local cohomology modules and the direct summand conjecture J of Algebra 84 1983 5037553 The direct summand conjecture is also equivalent to the statement that if 1 ac is a system of parameters of a local ring Rm then 1 Z 3151 f1R the monomz39al conjecture Another equivalent statement is this if V ath is a complete DVR and T V2xny1 y and f 1acnt 7 21 yaw then if R TfT for any map R 7 S ht 1 S if finite is g n 7 1 More generally one may ask can one tell what the largest height a proper expansion of an ideal I of a Noetherian ring R to a Noetherian R algebra S can achieve le what is the superhez ght of I The problem remains unsolved even in simple special cases in low dimension 3 11 Localization of tight closure tight closure vs plus closure lf R is an excellent Noetherian domain char 1 gt 0 does the tight closure 1 of I commute with localization ls it the same as the elements in IR H I where R is the integral closure of R in an algebraic closure of its fraction field Comments Recall that u E 1 if for some c 74 0 cape 6 We Ve gtgt 0 where We ipg iE IR IR R Q 1 By E SmithTight closure ofparameter ideals Invent Math 115 1994 41760 they are equal if RP is generated by part of a system of parameters in Rp for all P Q I Brenner Brenner Tight closure and plus closure for cones over elliptic curves preprint 7 Tight closure and plus closure in dimension two preprint has done the case of homogeneous ideals in the homogeneous coordinate rings of elliptic curves and of other nonsingular curves if the base field is the algebraic closure of ZpZ Plus closure commutes with localization and so the second question would solve the localization problem for tight closure 12 Existence of big CohenMacaulay CM modules and algebras Let RmK be local We shall say here that M is a big C M module or algebra if mM 74 M and every system of parameters R is a regular sequence on M We say that M is a small C M module if it is finitely generated 1 Does every complete local domain have a small C M module 2 Do there exist big C M modules 3 Do there exist big C M algebras 4 Given a local map of complete local domains R gt S do there exist big C M algebras BR BS and an R algebra map BR a BS weakly functorial big C M algebras 5 Does a version of tight closure theory exist in mixed characteristic Comments both 2 and 3 reduce to the case of complete even normal local domains Small C M modules are known to exist in dimension 2 and for nitely generated N graded domains R of char 1 gt 0 with R0 a perfect field if the ring has an isolated non C M point at the origin this was rst observed by R Hartshorne The argument is given in Hochster Big Cohen Macaulay modulesand algebras and embeddability in rings of Witt vectors in Proceedings of the Queen s University Commutative Algebra Conference Queen7s Papers in Pure and Applied Math No 42 1975 1067195 Big C M modules were shown to exist in equal char using reduction to char 1 in 7 Topics in the Homological Theory of Modules over Commutative Rings Proceedings of the Nebraska Regional CBMS Conference Lincoln Nebraska 1974 AMS Providence 1975 4 was proved in char pL0 in 7and C Huneke In nite integral extensions and big Cohen Macaulay algebras Annals of Math 135 1992 53789 R works and in char 0 in the sequel 7 Applications of the existence of big Cohen Macaulay algebras Advances in Math 113 1995 457117 But 4 is also true for two rings of mixed char in dimension 3 3 This is shown making use of Heitmann7s results in Hochster Big Cohen Macaulay algebras in dimension three via Heitmann s theorem J of Alg 254 2002 3957408 1 have heuristic reasons for believing that the problem of proving 4 in mixed characteristic in general and the problem of developing an analogue of tight closure in mixed characteristic should be nearly equivalent 13 The vanishing conjecture for maps of Tor and the strong direct summand conjecture lf A Q R a S are Noetherian rings with A S regular and R is module finite and torsion free over A the Tor4 M R a Toria M S is 0 for all A modules M and all i 2 1 Comments This can be proved in equal characteristic using either tight closure theory or the weakly functorial existence of big C M algebras This powerful result implies both the direct summand conjecture when S is a field and the conjecture that direct summands of regular rings are Cohen Macaulay when A M K is regular local M K and i 1 Cf Hochster and C Huneke Applications of the existence of big Cohen Macaulay algebras Advances in Math 113 1995 457117 fln mixed characteristic it is a conjecture ln N Ranganathan Splitting in module nite extension rings and the vanishing conjecture for maps of Tor Thesis University of Michigan 2000 the surprising result is obtained that 13 is equivalent to the following strong direct summand conjecture let RmK be a regular local ring in which it E m 7 m2 let S be a module nite extension domain and let Q be a height one prime of S lying over xR Then zR Q Q splits as a map of R modules which implies that R splits from S There is no easy proof known in equal char 0 and the result is open in mixed characteristic

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