# ideals linked to an almost complete intersection

Is a grade $3$ type $3$ perfect ideal in a Noetherian ring linked to a grade $3$ almost complete intersection? I know that grade $3$ type $2$ perfect ideals are (by a work of Anne Brown).

• Could you please remind us what "type 3" and "type 2" mean for perfect ideals (or provide a reference)? – Jason Starr Jan 14 '16 at 13:11
• the grade is the projective dimension of R/I and the type is rank(F3) so a minimal free resolution can be given by $$0 \rightarrow F_3 \rightarrow F_2 \rightarrow F_1 \rightarrow F_0 \rightarrow R/I \rightarrow 0$$ with $F_3= R^2$ for type 2 and $F_3=R^3$ for type 3 – Nancy Abdallah Jan 16 '16 at 22:19

Let $I=(x^3,y^3,z^3,xy^2,y^2z,x^2z^2)$. Then $I$ is grade $3$ and has type $3$. But $I$ is not licci (see "Liason of Monomial Ideals" by Huneke-Ulrich). But any grade 3 almost complete intersection is licci, since almost complete intersections are directly linked to Gorenstein ideals, and grade 3 Gorenstein ideals are licci. Hence $I$ is not linked to an almost complete intersection.
• @user2154420 Because if $I$ and $J$ are linked, the type of one is bounded above by the deviation of the other. This appears in Huneke-Ulrich's "Structure of Linkage" paper. – user194928 Dec 28 '19 at 21:07