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In a Noetherian local ring $R$, an ideal $I$ is called an almost complete intersection ideal if $\mu(I)=\text{ht}(I)+1$.

Q) Is it true that $I$ is generated by a $d$-sequence?

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Due to the comments to the previous answer (see the next answer) I give the following counterexample for the case where $d$-sequences are defined without considering permutations:

Let $R=K[x,y]$ and consider the ideal $(x^2,y^2,xy)$. It can be seen that no permutation of $x^2,y^2,xy$ is a d-sequence, e.g. $(xy):x^2y^2\neq (xy):x^2$. But in order to prove that the ideal is not generated by any $d$-sequence it is better to use theorems. In view of the theorems of the cited papers below, the ideal generated by a $d$-sequence is an ideal of linear type i.e. its Rees Algebra and Symmetric Algebra are equal. Now, by, e.g., using Macaulay2, one can see that the ideal $(x^2,y^2,xy)$ is not of linear type.

[Theorem, page 472, Craig Huneke, MR 608547 Powers of ideals generated by weak $d$-sequences, J. Algebra 68 (1981), no. 2, 471--509.]

[Page 5, J. Herzog, A. Simis, and W. V. Vasconcelos, MR 686942 Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) 79--169.]

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Not, necessarily. For example, in $K[x,y]$, $x^3,xy^2$ is not a $d$-sequence, but it generates an almost complete intersection. However, in some cases, the answer is positive. For example, see (5) of the cited paper below.

Craig Huneke, MR 563225 On the symmetric and Rees algebra of an ideal generated by a $d$-sequence, J. Algebra 62 (1980), no. 2, 268--275.

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  • $\begingroup$ Is not xy2,x3 a d-sequence? I think it is. (xy2:x3) and (xy2:x6) both are generated by (y2). $\endgroup$
    – tessellation
    Commented Feb 9, 2017 at 16:06
  • $\begingroup$ $(x^3):xy^2=(x^2)$, but $(x^3):x^2y^4=(x)$, therefore $x^3,xy^2$ is not a $d$-sequence. $\endgroup$
    – Aurora
    Commented Feb 9, 2017 at 16:10
  • $\begingroup$ But the ideal J=(xy2,x3) is almost intersection and generated by d-sequence. $\endgroup$
    – tessellation
    Commented Feb 9, 2017 at 16:27
  • $\begingroup$ It depends on how you are going to define a sequence to be a $d$-sequence. For instance, in the Huneke's paper which I cited above, to define $d$-sequences, colon ideals are considered for any permutation of the sequence. $\endgroup$
    – Aurora
    Commented Feb 9, 2017 at 16:57

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