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).
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5$\begingroup$ Could you please remind us what "type 3" and "type 2" mean for perfect ideals (or provide a reference)? $\endgroup$– Jason StarrCommented Jan 14, 2016 at 13:11

$\begingroup$ 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 $\endgroup$– NanCommented Jan 16, 2016 at 22:19
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1 Answer
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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 HunekeUlrich). 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.

$\begingroup$ Why is an almost complete intersection directly linked to a Gorenstein ideal? $\endgroup$ Commented Aug 13, 2019 at 1:36

$\begingroup$ @user2154420 Because if $I$ and $J$ are linked, the type of one is bounded above by the deviation of the other. This appears in HunekeUlrich's "Structure of Linkage" paper. $\endgroup$ Commented Dec 28, 2019 at 21:07