A grade five Gorenstein algebra with no minimal algebra resolutions.

*(English)*Zbl 0856.13011A free resolution of a cyclic module \(R/I\) over a noetherian local ring \(R\) is called an algebra resolution if it admits a multiplication which makes it an associative, commutative differential graded algebra. It is known that if \(I\) is a Gorenstein ideal of grade less than five, then \(R/I\) has an algebra resolution, and there are examples of Gorenstein ideals of grade six with no minimal algebra resolutions.

Let \(R = k[[X_i,Y_{ij},\;1\leq i<j \leq 5]]\) with \(k\) a field of characteristic \(\neq 2\). Put \(Y=(Y_{ij})\) with \(Y_{ij}= -Y_{ji}\), \(X=(X_1, \dots,X_5)\) and let \(I\) be the ideal generated by the \(4\times 4\) Pfaffians of \(Y\) and the five entries of \(XY\). Then the author shows that \(I\) is a grade five Gorenstein ideal of \(R\) and constructs a minimal free resolution \(F_\bullet\) of \(R/I\) which does not admit an algebra structure. – Moreover, there is an ideal \(J\) generated by a regular sequence contained in \(I\) such that \(F_\bullet\) does not admit a structure of a differential graded module over the Koszul resolution of \(R/J\).

Let \(R = k[[X_i,Y_{ij},\;1\leq i<j \leq 5]]\) with \(k\) a field of characteristic \(\neq 2\). Put \(Y=(Y_{ij})\) with \(Y_{ij}= -Y_{ji}\), \(X=(X_1, \dots,X_5)\) and let \(I\) be the ideal generated by the \(4\times 4\) Pfaffians of \(Y\) and the five entries of \(XY\). Then the author shows that \(I\) is a grade five Gorenstein ideal of \(R\) and constructs a minimal free resolution \(F_\bullet\) of \(R/I\) which does not admit an algebra structure. – Moreover, there is an ideal \(J\) generated by a regular sequence contained in \(I\) such that \(F_\bullet\) does not admit a structure of a differential graded module over the Koszul resolution of \(R/J\).

Reviewer: A.Ooishi (Yokohama)