Rational homotopy theory is distinguished by its effective algebraic models (e.g. commutative differential graded algebra models). In a certain technical sense, these algebraic models completely describe the rational homotopy theory of spaces. Furthermore, these algebraic models allow a translation from the rational homotopy setting to the local algebra setting, and vice-versa. At least since L. Avramov and S. Halperin [Algebra, algebraic topology and their interactions, Proc. Conf., Stockholm 1983, Lect. Notes Math. 1183, 1–27 (1986; Zbl 0588.13010)], the ability to pass to-and-fro between these settings has proved fruitful. This paper continues the cross-fertilization of commutative algebra and homotopy theory, focussing on the complete intersection (ci) condition on the rational homotopy side of the looking glass.
The authors give four conditions on a simply connected, rational space \(X\), any of which may be called a ci condition and each of which is adapted into the rational homotopy setting from a corresponding condition on a local ring. Two of these conditions, with their acronyms from the paper, are as follows: (1) \(X\) is sci if it may be constructed as an iterated spherical fibration starting from a product of even Eilenberg-Mac Lane spaces; (2) \(X\) is gci if \(H^*(X)\) is Noetherian and \(H_*(\Omega X)\) has polynomial growth. The other two conditions go by the acronyms eci and zci, and are expressed in terms of resolutions of a commutative cochain model for \(X\). One striking aspect of the results here is that these conditions are qualitatively quite different from each other, and yet end up being closely related, if not actually equivalent.
On the local algebra side of the looking glass, results of D. Eisenbud [Trans. Am. Math. Soc. 260, 35–64 (1980; Zbl 0444.13006)] and T. Gulliksen [Math. Scand. 47, 5–20 (1980; Zbl 0458.13010)] show that, for a complete, local Noetherian ring, the algebraic conditions from which these four topological conditions are translated are equivalent. In this paper, the main results establish dependency amongst the topological conditions; with certain extra side conditions, they are shown to be equivalent. Specifically, for a rational space, it is shown that sci \(\Rightarrow\) eci \(\Rightarrow\) gci. If the space is also assumed “strongly Noetherian,” these three are equivalent; with a further side condition, these three are also equivalent to the zci condition. The implication gci \(\Rightarrow\) sci under the “strongly Noetherian” hypothesis translates into rational homotopy the result of Gulliksen ({ibid}.) in local algebra. As the authors note, this implication (in either setting) is significant, as it obtains a structural consequence from a growth condition.
The paper is written in a style that this reviewer found refreshing: the authors appear to have gone out of their way to be helpful to the reader. The overall expository style is rich, with frequent remarks that clarify or contextualize. A number of basic results—used in establishing the main results—of the kind that are well-known to experts, but all too often hidden or missing from the literature, are here carefully stated and proved. A whole section of examples is given, including separating examples for the various ci conditions.
The article [D. J. Benson, J. P. C. Greenlees and S. Shamir, Algebr. Geom. Topol. 13, No. 1, 61–114 (2013; Zbl 1261.13007)] continues this investigation into the complete intersection condition, looking at characteristics away from \(0\).