Examples of non-rigid cotangent complexes (Q1125870)

According to F. Planas-Vilanova, for any commutative ring \(C\) there is an augmented commutative \(C\)-algebra \(A\) with the property \(H_2(A,C,\cdot)\equiv 0\not\equiv H_3(A,C,\cdot)\). Let us modify and generalize this example. The following result will be proved [see \textit{F. Planas}-\textit{Vilanova}, ``On the vanishing and non-rigidity of the André-Quillen (co)homology'', J. PUre Appl. Algebra 120, No. 1, 65-75 (1977), too, for the case \(n=2\)]. Theorem. For any commutative ring \(C\) and any integer \(n\geq 2\), there is an augmented commutative \(C\)-algebra \(B\) which is non-rigid in the sense \(H_k(B,C,C)\cong 0\) for \(0\leq k\leq n+3\), with one exception, \(H_{n+1}(B,C,C)\cong C\). In general the next module \((k=n+4)\) is not trivial. When \(C\) is a field of characteristic 0, the result is complete and minimal as follows: All the modules \(H_k(B,C,C)\) are equal to 0 with a unique exception \(H_{n+1}(B,C,C)\cong C\) if \(n\) is odd, with two exceptions \(H_{n+1}(B,C,C)\cong C\cong H_{2n+2}(B,C,C)\) if \(n\) is even.