For any sheaf of sets F on Sm/k, it is well known that the universal A¹-invariant quotient of F is given as the colimit of sheaves Sⁿ(F) where S(F) is the sheaf of naive A¹-connected components of F. We show that these infinite iterations of naive A¹-connected components in the construction of universal A¹-invariant quotient for a scheme are certainly required. For every n, we construct an A¹-connected variety X_n such that Sⁿ(X_n) ≠ Sⁿ⁺¹(X_n) and Sⁿ⁺²(X_n)=*.

The author is supported by the Prime Minister's Research Fellowship from the Ministry of Human Resource Development, Government of India.