Hodge-Witt cohomology of complete intersections
In this note, we prove the following assertions. Let \(k\) be a perfect field of characteristic \(p>0\) and \(X\) a smooth complete intersection of dimension \(n\) in a projective space over \(k\). (a) If \(i \neq j\) and \(i + j \neq n\), \(n + 1\), \(H^ j (X,W \Omega^ i_ X) = 0\). (b) If \(2i \neq n\), \(n + 1\) and \(0 \leq i \leq n\), \(H^ i (X,W \Omega^ i_ X) = W\) and \(F\) is bijective on \(H^ i (X,W \Omega^ i_ X)\). (c) \(H^{n-i} (X,W \Omega^ i_ X)\) is a Cartier module [in the sense \textit{L. Illusie} and \textit{M. Raynaud}, Publ. Math., Inst. Hautes Étud. Sci. 57, 73-212 (1983; Zbl 0538.14012)]. (d) If \(2i \neq n + 1\), \(H^{n - i + 1} (X,W \Omega^ i_ X)/F^ \infty B = 0\). (e) If \(2i = n + 1\), \(H^{n - i + 1} (X,W \Omega^ i_ X)/F^ \infty B = W\) and \(F\) is bijective on \(H^{n - i + 1} (X,W \Omega^ i_ X)/F^ \infty B\). Here \(W\) is the ring of Witt vectors. \(H^*(X/W)\) (resp. \(H^ j(X,\Omega^ i_ X))\) denotes the crystalline cohomology group (resp. the Hodge-Witt cohomology group) of \(X\). \(F\) (resp. \(V)\) stands for the Frobenius morphism.
scientific article; zbMATH DE number 469062
scientific article