The Poincaré Duality of a Surface with Rational Singularities

In this article, we proof the Poincaré duality with coefficient Ql on a surface with isolated rational singularities. INTRODUCTION The Poincaré duality theorem in the étale cohomology was established in [1, XVIII, 3.2] on the smooth varieties. But in many times, the duality on singular varieties has to be considered. In this paper, we study the surfaces with at most isolated rational singularities. Roughly speaking, these are isolated singularities of a surface which are “cohomologically trivial” (in the sense of cohomology of coherent sheaves). In [3], J. Lipman prove that a two-dimensional normal local ring R has a rational singularity if and only if R has a finite divisor class group. This makes me think that rational singularities only affect the torsion part of the étale cohomology with coefficient Zl, but leave the free part invariant. In other words, surfaces with rational singularities should share the same good properties with nonsingular surfaces in the étale cohomology in with coefficient Ql. In particular, we obtain the Poincaré duality with coefficient Ql. But the duality with coefficient Zl or finite coefficient is no longer valid, which we shall see in the proof of the main theorem. Notation and Conventions. For a vector space V over a field K , we use V ∨ to denote its dual space. If X is a scheme and P a point on X, we use P to denote associated geometric point on X. An algebraic scheme over a field k is a scheme separated, of finite type over k. A variety over k is a geometric integral algebraic scheme over k. If X is an algebraic scheme over a field k, then we define X := X ⊗k k̄ where k̄ is the algebraic closure of k. For an algebraic scheme X over a field k, we use KX := RpQl to denote the dualizing complex of X, where p : X → Spec k is the structure morphism. If F • is a complex of sheaves on the étale site of a scheme X, we write F •〈r〉 := F •(r)[2r] for each r ∈ Z. 1. THE MAIN THEOREM Let X be a surface over an algebraically closed field, P an isolated singular point. We say that X has rational singularity at P if there exists a desingularization π : Ũ → U of an open neighborhood U of P such that π∗Oe U = OU and R π∗Oe U = 0 for all q > 0. An important fact of rational singularity is that the exceptional divisor of the desingularization π is a tree of nonsingular rational curves with normal crossings.