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A grade five Gorenstein algebra with no minimal algebra resolutions. (English) Zbl 0856.13011
A 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 $$R/J$$.

##### MSC:
 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, AndrĂ©-Quillen, cyclic, dihedral, etc.) 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
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