Reducing systems of parameters and the Cohen-Macaulay property Joint work with J. Stückrad Situation Let R be a local Ring with the maximal ideal m.

Situation Let R be a local Ring with the maximal ideal m. M ≠ 0 be a finitely generated R-module with dim M = d.

Situation Let R be a local Ring with the maximal ideal m. M ≠ 0 be a finitely generated R-module with dim M = d. Definition A system of parameters is called a \underline{reducing system of parameters} if ∀i = 1, …, d − 1 we have x_{i} ∉ P ∀P ∈ Ass M/(x_{1}, …, x_{i−1})M with dim R/P = d − i.

Situation Let R be a local Ring with the maximal ideal m. M ≠ 0 be a finitely generated R-module with dim M = d. Definition A system of parameters is called a \underline{reducing system of parameters} if ∀i = 1, …, d − 1 we have x_{i} ∉ P ∀P ∈ Ass M/(x_{1}, …, x_{i−1})M with dim R/P = d − i.

Proposition For every system of parameters (x_{1}, …, x_{d}) there is a reducing system of parameters (y_{1}, …, y_{d}), such that (x_{1}, …, x_{d})M = (y_{1}, …, y_{d})M Definition and Equivalence A sequence x_{1}, …, x_{r} of elements of the maximal ideal m is called \underline{part of a system of parameters} of M ⟺ There exists x_{r+1}, …, x_{d} ∈ m, such that (x_{1}, …, x_{r}, x_{r+1}, …, x_{d}) is a system of parameters. ⟺ dim M/(x_{1}, …, x_{r})M = d − r Definition and Equivalence A sequence x_{1}, …, x_{r} of elements of the maximal ideal m is called \underline{part of a system of parameters} of M ⟺ There exists x_{r+1}, …, x_{d} ∈ m, such that (x_{1}, …, x_{r}, x_{r+1}, …, x_{d}) is a system of parameters. ⟺ dim M/(x_{1}, …, x_{r})M = d − r

The same holds for reducing systems of parameters.

A sequence y_{1}, …, y_{r} of elements of the maximal ideal m (r < d) is called \underline{part of a reducing system of parameters} of M ⟺ There exists y_{r+1}, …, y_{d} ∈ m, such that (y_{1}, …, y_{r}, y_{r+1}, …, y_{d}) is a reducing system of parameters. ⟺ For all i = 1, …, r holds y_{i} ∉ P if P ∈ Ass M/(y_{1}, …, y_{i−1})M and dim R/P ≥ d − i Remark If M is Cohen-Macaulay TFAE:

(i) x_{1}, …, x_{r} is a regular sequence (ii) x_{1}, …, x_{r} is a part of a reducing system of parameters. (iii) x_{1}, …, x_{r} is a part of a system of parameters.

Remark If M is Cohen-Macaulay TFAE:

(i) x_{1}, …, x_{r} is a regular sequence (ii) x_{1}, …, x_{r} is a part of a reducing system of parameters. (iii) x_{1}, …, x_{r} is a part of a system of parameters. Lemma If x is a zerodevisor on M then for all minimal primes P ∈ Ass M ∩ V(x) holds P ∈ Ass M/xM.

Remark If M is Cohen-Macaulay TFAE:

(i) x_{1}, …, x_{r} is a regular sequence

(ii) x_{1}, …, x_{r} is a part of a reducing system of parameters.

(iii) x_{1}, …, x_{r} is a part of a system of parameters.

Lemma If x is a zerodevisor on M then for all minimal primes P ∈ Ass M ∩ V(x) holds P ∈ Ass M/xM.

Theorem Let (y_{1}, …, y_{d}) be a reducing system of parameters of M. M is Cohen-Macaulay iff y_{d} is a non zerodevisor on M/(y_{1}, …, y_{d−1})M.

Theorem Let (x_{1}, …, x_{r}) be a part of a system of parameters of M where 0 ≤ r < d. TFAE:

(i) (x_{1}, …, x_{r}) is reducing.

(ii) M_{P} is an r-dimensional Cohen-Macaulay-module over R_{P} for all P ∈ Supp M ∩ V(x_{1}, …, x_{r}) satisfying dim R/P = dim M − r.

(iii) There is a part of a reducing system of parameters (y_{1}, …, y_{r}), such that (x_{1}, …, x_{r})M = (y_{1}, …, y_{r})M.

(iv) There is a part of a reducing system of parameters (y_{1}, …, y_{r}), such that Supp M ∩ V(x_{1}, …, x_{r}) ⊆ V(y_{1}, …, y_{r}).

Corollary Let (x_{1}, …, x_{r}) be a part of a reducing system of parameters. If r < d then (x_{π(1)}, …, x_{π(r)}) is a part of a reducing system of parameters for any permutation π of {1, …, r}.

Corollary Let (x_{1}, …, x_{r}) be a part of a reducing system of parameters. If r < d then (x_{π(1)}, …, x_{π(r)}) is a part of a reducing system of parameters for any permutation π of {1, …, r}.

This is not true if r = d as the following example shows: Example R := K[[X, Y, Z]], K field M := R/(XY, XZ)R and x_{1} := Y, x_{2} := X + Y + Z Corollary Let (x_{1}, …, x_{r}) be a part of a reducing system of parameters. If r < d then (x_{π(1)}, …, x_{π(r)}) is a part of a reducing system of parameters for any permutation π of {1, …, r}.

This is not true if r = d as the following example shows: Example R := K[[X, Y, Z]], K field M := R/(XY, XZ)R and x_{1} := Y, x_{2} := X + Y + Z

(x_{1}, x_{2}) is a system of parameters but not reducing. (x_{2}, x_{1}) is a reducing system of parameters.

References

M. Auslander and D.A. Buchsbaum: Codimension and multiplicity, Ann. of Math. 68 (1958) 625-657 D. Eisenbud: Commutative algebra with a view toward algebraic geometry, Springer-Verlag, 1994 D.P. Patil, U. Storch and S. Stückrad: A criterion for regular sequences, Proc. Indian Acad. Sci. (Math. Sci.) 114 (2004) 103-106