Primitivity of the \(c_ 2\)-invariant
The authors study the \(c_ 2\)-invariant \(c(\tilde f)\) [see \textit{A. Zabrodsky}, Steenrod Algebra Appl., Lect. Notes Math. 168, 308-317 (1970; Zbl 0205.276)] of the map \(\tilde f\) in the diagram \[ \begin{matrix} &&E \\ &\tilde f \nearrow & \downarrow \\ \Omega X & @>>f> & K & @>>g> & k_1 \end{matrix} \] under the following hypotheses: X is a simply connected finite H-space, K and \(K_ 1\) are generalized Eilenberg-MacLane spaces, g is an infinite loop map, E is its fibZbl 0154.038)] these algebras are characterized by the conditions: \(J/J^ 2\) is a projective B-module and \(fd_ A(B)<\infty\) (where \(fd_ A(B)\) denotes the flat dimension of the A-module B). If B is a flat A-algebra, then [see \textit{A. Grothendieck}, EGA IV, Publ. Math., Inst. Hautes Étud. Sci. 20 (1964; Zbl 0136.159)] B is c.i. over A if and only if the following condition is verified: for each prime ideal q in B and its inverse image \(p=q^ c\) in A, the local ring \(B_ q/pB_ q\) is complete intersection (recall that a local noetherian ring (A,m) is said to be complete intersection if the m-adic completion of A is isomorphic to R/J, where R is a regular local ring and J is generated by a regular sequence). - It is also possible to get a characterization of complete intersections using the André-Quillen homology functors. B is c.i. over A if and only if \(H_ 2(A,B,-)=0.\) In this paper we obtain another characterization based on the vanishing of the functors \(H_ n(A,B,-)\) and the finiteness of the projective dimension of Kähler differentials module \(\Omega_{B| A}\). We also study the significance of the condition \(fd_{B\otimes_ AB}(B)<\infty\). Using some results of L. L. Avramov and D. Quillen, we show that, in the characteristic zero case, this condition is equivalent to the smoothness of B over A.''
projective dimension of Kähler differentials module
scientific article; zbMATH DE number 4048660
scientific article