zbMATH — the first resource for mathematics

Modules of finite projective dimension with negative intersection multiplicities. (English) Zbl 0588.13020
For finitely generated modules M, N over an unramified regular local ring R a theorem of Serre asserts that \((i)\quad \chi (M,N)=0\) if \(\dim M+\dim N<\dim R,\) and \((ii)\quad \chi (M,N)>0\) if \(\dim M+\dim N=\dim R,\) provided, of course, that M, N has finite length. (\(\chi\) (M,N) denotes the intersection multiplicity of M and N.) It has been an open problem whether Serre’s theorem is still true if one only assumes that R is an arbitrary local ring and one of the modules has finite projective dimension. Partial positive results have been achieved by various authors. The main objective of the article under review is to disprove the generalizations of (i) and (ii) just stated. Let K be a field, \(R=K[X_ 1,...,X_ 4]/(X_ 1X_ 4-X_ 2X_ 3)\) localized at the irrelevant maximal ideal, and \(P=(x_ 1,x_ 2)\subset R.\) The authors study a class of modules of finite length and finite projective dimension over R, and in particular produce such modules M with \(\chi (M,R/P)=\pm 1\) and, in case \(\chi (M,R/P)=1,\) \(\chi (M,P^ t)<0\) for \(t>>0\). We cannot comment the details of this elaborate construction here. The last section of the article discusses the consequences for the Grothendieck group of modules of finite length and finite projective dimension and related questions, such as the positivity of partial Euler characteristics.
Meanwhile it has been proved by P. Roberts [Bull. Am. Math. Soc., New Ser. 13, 127-130 (1985; Zbl 0585.13004)] and, independently, by H. Gillet and C. Soulé [C. R. Acad. Sci., Paris, Sér. I 300, 71-74 (1985; Zbl 0587.13007)] that (i) holds over complete intersections (including ramified regular local rings) if both M and N have finite projective dimension.
Reviewer: W.Bruns

13H15 Multiplicity theory and related topics
13D05 Homological dimension and commutative rings
13D15 Grothendieck groups, \(K\)-theory and commutative rings
13C12 Torsion modules and ideals in commutative rings
Full Text: DOI EuDML
[1] [A1] Auslander, M.: Modules over unramified regular local rings. Illinois J. Math.5, 631-645 (1961) · Zbl 0104.26202
[2] [A2] Auslander, M.: Modules over unramified regular local rings. Proc. Intern. Congress of Math. 1962, pp. 230-233
[3] [D1] Dutta, S.P.: Generalized intersection multiplicities of modules. Trans. Amer. Math. Soc.276, 657-669 (1983) · Zbl 0531.13008 · doi:10.1090/S0002-9947-1983-0688968-4
[4] [D2] Dutta, S.P.: Weak linking and multiplicities. J. Pure and Applied Algebra27, 111-130 (1983) · Zbl 0523.13009 · doi:10.1016/0022-4049(83)90010-5
[5] [D3] Dutta, S.P.: Symbolic powers, intersection multiplielty, and asymptotic behavior of Tor. J. London Math. Soc. (2)28, 261-281 (1983) · Zbl 0512.13016 · doi:10.1112/jlms/s2-28.2.261
[6] [D4] Dutta, S.P.: Frobenius and multiplicities. J. of Algebra85, 424-448 (1983) · Zbl 0527.13014 · doi:10.1016/0021-8693(83)90106-0
[7] [D5] Dutta, S.P.: Generalized intersection multiplicities of modules II. Proc. A.M.S. (in press) (1984)
[8] [DHM] Dutta, S.B., Hochster, M., McLaughlin, I.E.: Appendix to ?Modules of Finite Projective Dimension with Negative Intersection Multiplicities,? manuscript available from the authors
[9] [E] Eisenbud, D.: Homological algebra on a local complete intersection, with an application to group representations. Trans. Amer. Math. Soc.260, 35-64 (1980) · Zbl 0444.13006 · doi:10.1090/S0002-9947-1980-0570778-7
[10] [F1] Fossum, R.:æ-theory. Preprint Series, Politecnico di Torino, 1977
[11] [F2] Fossum. R.: Correspondence, 1983
[12] [FFI] Fossum, R., Foxby, H.-B., Iversen, B.: Characteristic classes of complexes. Preliminary preprint
[13] [H1] Hochster, M.: Cohen-Macaulay modules. Conference on Commutative Algebra. Lecture Notes in Math. Vol. 311, pp. 120-152. Berlin-Heidelberg-New York Springer 1973
[14] [H2] Hochster, M.: Topics in the homological theory of modules over commutative rings, C.B.M.S. Regional Conference Series in Math., Vol. 24. Amer. Math. Soc., Providence, R.I., 1975 · Zbl 0302.13003
[15] [H3] Hochster, M.: The local homological conjectures. In: Commutative Algebra: Durham 1981. Cambridge Univ. Press, London Math. Soc. Lecture Note Series. Vol. 72, pp. 32-54 (1982)
[16] [H4] Hochster, M.: Euler characteristics over unramified regular local rings. Illinois J. Math. in press (1984) · Zbl 0562.13019
[17] [L] Lichtenbaum, S.: On the vanishing of Tor in regular local rings. Illinois J. Math.10, 220-226 (1966) · Zbl 0139.26601
[18] [M] Malliavin-Brameret, M.-P.: Une remarque sur les anneaux locaux réguliers, Sém. Dubreil-Pisot (Algèbre et Théorie des Nombres), 24 année, 1970/71, no 13
[19] [PS1] Peskine, C., Szpiro, L.: Dimension projective finie et cohomologie locale. Inst. Hautes Etudes Sci. Publ. Math.42, 323-395 (1973) · Zbl 0268.13008 · doi:10.1007/BF02685877
[20] [PS2] Peskine, C., Szpiro, L.: Syzygies et multiplicités. C.R. Acad. Sci. Paris Ser. A-B278, 1421-1424 (1974) · Zbl 0281.13004
[21] [S] Serre, J.P.: Algèbre locale. Multiplicités. Lecture Notes in Math. Vol. 11, 3éme édition. Berlin-Heidelberg-New York: Springer 1975
[22] [W] Weil, A.: Foundations of algebraic geometry. Amer. Math. Soc. Colloq. Publ. Vol. 29, Amer. Math. Soc., Providence, R.I., Revised ed., 1962 · Zbl 0168.18701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.