A Property of Finite Free Resolutions Related to Auslander's delta-invariant and Hochster Canonical Element Conjecture
by
Anne-Marie Simon
Universite Libre de Bruxelles

Let N be a finitely generated module of finite projective dimension over a commutative noetherian local ring A and let 0 --> F_n --> ... --> F₀ be its minimal free resolution. The boundary map d_j : F_j --> F_j-1, 1 <= j <= n, is represented by a matrix of size rk F_j-1 ×rk F_j and the entries in an arbitrary column of such a matrix generate an ideal called a column ideal of d_j. (A column ideal of d_j is thus the order ideal in F_j-1 of the image d_j(v) of some minimal generator v of F_j, which is contained in the order ideal of d_j(v) in the j^th syzygy  d_j(F_j).)

When the local ring A is equicharacteristic, we prove that the column ideals of d_j have grade at least j, 1 <= j <= n.

This extends a result of Evans and Griffith, which stated that, for every minimal generator w of a finitely generated j^th syzygy  N_j of finite projective dimension over an equicharacteristic noetherian local ring, the order ideal of w in N_j has height at least j.

Our proof makes use of balanced big Cohen-Macaulay modules, available in equal characteristic, and it is natural to conjecture that the same conclusion holds for any noetherian local ring, possibly of mixed characteristic.

We show that Hochster's Canonical Element Conjecture, which is a conjecture about the whole class of noetherian local rings, which need only be proved for homomorphic images of Gorenstein local ring and which is true when restricted to the class of equicharacteristic noetherian local rings, is equivalent to each of the following ones.

Conjecture A. For every Gorenstein local ring, for every boundary map d_j appearing in a minimal finite free resolution as above, the column ideals of d_j have grade at least j.

Conjecture B. For every Gorenstein local ring, for every finitely generated first syzygy  Y of projective dimension, the annihilator of every minimal generator of Y is null.

Conjecture C. For every Gorenstein local ring R, for any nonull ideal b of R with Ass (R / b) subset Ass (R), the Auslander's delta-invariant \delta(R / b) is null, which means that there is a maximal Cohen-Macaulay module without free direct summands surjecting onto R / b.

Again these conjectures are true when restricted to the class of equicharacteristic Gorenstein local rings.

This is joint work with J. R. Strooker.

Simon Anne-Marie, CP211

ULB, Campus Plaine

Boulevard du Triomphe

B-1050 Brussel

Belgium

Anne-Marie Simon's homepage

Date received: February 1, 2001