## Discriminants, resultants and a conjecture of S. Halperin.(English)Zbl 1018.55008

The Halperin conjecture states that for every orientable fibration $$X\to Y\to B$$ with fibre $$X$$ a space of type $$F_0$$ (i.e. $$\dim \pi_* (X)\otimes Q<\infty$$, $$H^{\text{odd}}(X,\mathbb{Q})=0$$, and $$\dim H^*(X,\mathbb{Q}) <\infty)$$ the Serre spectral sequence degenerates at the term $$E_2$$. The conjecture has been proved for a large family of spaces of type $$F_0$$, for example for homogeneous spaces $$G/H$$ of rank zero by Shiga and Tezuka, and in the case of $$H^*(X,\mathbb{Q})$$ being generated by $$\geq 3$$ elements by Hao Chen. The paper under review contains a very nice survey on the subject. The author also introduces differential calculus and algebraic geometry around the conjecture. Consider all the algebras of the form $$\mathbb{Q}[a_1, \dots,a_n]/(f_1, \dots,f_n)$$ where $$n$$, the degrees of the $$a_i$$ and the degrees of the homogeneous polynomials $$f_i$$ are fixed. Each sequence $$(f_1,\dots,f_n)$$ corresponds to a point in some $$\mathbb{Q}^N$$. The sequence is regular if some resultant does not vanish. Therefore regular sequences form a Zariski open set $$V$$. The author proves that the sequences $$(f_1,\dots,f_n)$$ that satisfy the Halperin conjecture form an open subset of $$W$$.

### MSC:

 55P62 Rational homotopy theory 55T10 Serre spectral sequences 13D10 Deformations and infinitesimal methods in commutative ring theory 13E10 Commutative Artinian rings and modules, finite-dimensional algebras 14B07 Deformations of singularities 14B12 Local deformation theory, Artin approximation, etc. 14M10 Complete intersections