The complete intersection locus of certain ideals. (English) Zbl 0574.13013
For a local Noetherian ring R and an ideal I, this paper approaches the comparison between the height, ht(I), and the minimum number of generators, \(\nu\) (I), of I through the properties of the set of primes P where the localization \(I_ P\) is a complete intersection. This defines an open set, CIL(I), of Spec(R); the codimension of CIL(I) is denoted by c(I).
Broadly speaking the aim is to estimate c(I) for ideals where the conormal module, \(I/I^ 2\), is sufficiently well known. An extreme case of this formulation is the following conjecture: Let I be an ideal of the regular local ring R. If \(I/I^ 2\) has finite projective dimension over R/I then I is a complete intersection. From the cases settled thus far the emerging evidence for it and for similar conjectures on the other (co-)normal modules is quite strong. The backdrop for the especial estimates of the paper are general - but surprisingly sharp - estimates of Bruns, Faltings and Huneke relating c(I) to ht(I), the number of generators and relations of I, and its analytic spread \(\ell (I).\)
Section 2 discusses an enrichment by Gulliksen of the Tate resolution of R/I. A rephrasing of this proof leads to the following assertion: Let R be a local ring and let I be an ideal of finite projective dimension; the first homology module, \(H_ 1\), of the Koszul complex built on a minimal generating set of generators of I does not admit a nonzero (R/I)-free summand. As a consequence one can write bounds for c(I) that may, under certain conditions, be very tight. - The conjectural homological rigidity for \(I/I^ 2\)- i.e. whether for an ideal I of a regular ring R, 0 and \(\infty\) are the only possible values for \(pd_{R/I}(I/I^ 2)\)- is then studied. Along with analogs for the other (co-)normal modules one asks whether local complete intersections are characterized by the finiteness of the projective dimension of these modules. The results of § 2 could then be stated as asserting that \(pd_{R/I}(I/I^ 2)\neq 1.\)
In the next section it is shown that the canonical module of R/I, for the ideals in the conjecture, has the expected form: it is cyclic. This says that the Cohen-Macaulay ideals in the conjecture are in fact Gorenstein ideals, and settles the question in several cases - e.g. arbitrary ideals of height two in rings containing a field. Finally it discusses the form the estimates of c(I) assume when applied to ideals satisfying \(pd_{R/I}(I/I^ 2)\leq 3\). It allows for extending the catalog of solved cases of the conjecture up to various Cohen-Macaulay ideals of height four.
Broadly speaking the aim is to estimate c(I) for ideals where the conormal module, \(I/I^ 2\), is sufficiently well known. An extreme case of this formulation is the following conjecture: Let I be an ideal of the regular local ring R. If \(I/I^ 2\) has finite projective dimension over R/I then I is a complete intersection. From the cases settled thus far the emerging evidence for it and for similar conjectures on the other (co-)normal modules is quite strong. The backdrop for the especial estimates of the paper are general - but surprisingly sharp - estimates of Bruns, Faltings and Huneke relating c(I) to ht(I), the number of generators and relations of I, and its analytic spread \(\ell (I).\)
Section 2 discusses an enrichment by Gulliksen of the Tate resolution of R/I. A rephrasing of this proof leads to the following assertion: Let R be a local ring and let I be an ideal of finite projective dimension; the first homology module, \(H_ 1\), of the Koszul complex built on a minimal generating set of generators of I does not admit a nonzero (R/I)-free summand. As a consequence one can write bounds for c(I) that may, under certain conditions, be very tight. - The conjectural homological rigidity for \(I/I^ 2\)- i.e. whether for an ideal I of a regular ring R, 0 and \(\infty\) are the only possible values for \(pd_{R/I}(I/I^ 2)\)- is then studied. Along with analogs for the other (co-)normal modules one asks whether local complete intersections are characterized by the finiteness of the projective dimension of these modules. The results of § 2 could then be stated as asserting that \(pd_{R/I}(I/I^ 2)\neq 1.\)
In the next section it is shown that the canonical module of R/I, for the ideals in the conjecture, has the expected form: it is cyclic. This says that the Cohen-Macaulay ideals in the conjecture are in fact Gorenstein ideals, and settles the question in several cases - e.g. arbitrary ideals of height two in rings containing a field. Finally it discusses the form the estimates of c(I) assume when applied to ideals satisfying \(pd_{R/I}(I/I^ 2)\leq 3\). It allows for extending the catalog of solved cases of the conjecture up to various Cohen-Macaulay ideals of height four.
MSC:
| 13H10 | Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) |
| 13D05 | Homological dimension and commutative rings |
| 14M10 | Complete intersections |
| 13E05 | Commutative Noetherian rings and modules |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13A99 | General commutative ring theory |
Keywords:
local Noetherian ring; ideal; height; minimum number of generators; complete intersection; conormal module; analytic spread; projective dimension; canonical module; Cohen-Macaulay ideals; Gorenstein idealsReferences:
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