Unramified cohomology and Witt groups of anisotropic Pfister quadrics
Unramified cohomology and Witt groups of anisotropic Pfister quadrics
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各向异性 Pfister 二次曲面的无枝上同调和 Witt 群
DOI：
10.1090/s0002-9947-97-01940-5
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发表时间：
1997
影响因子：
1.3
通讯作者：
R. Sujatha
中科院分区：
文献类型：
--
作者：
R. Sujatha
域k上的各向异性二次曲线的未分歧Witt群，其特征为k 0 2，定义为（1，-a，-B），它是k的Witt群W（k）的商，同构于W（k）/（1，-a，-B，ab）W（k）.我们计算了非分歧上同调群H 43 rk（C），其中C是由k上的二次型（1，-a，-B，ab，-c）定义的三维各向异性二次曲面.我们利用这些计算来研究C的非分歧Witt群。设k是特征不为2的域，K/k是一个非线性生成的域扩张.系数为Z/2的k上K的n阶非分歧上同调群（参见[CT])表示为Hnnr（K/k）（或者当考虑的地面场是清晰的时仅为Hnnr（K））。它是伽罗瓦上同调群Hn（K，Z/2）（简称Hn（K））的一个子群，定义为Hnnr（K/k）= nH 6 nt（Spec Ov，u2）。vEV（K）这里V（K）是K上所有秩为1的离散赋值的集合，这些赋值在k上是平凡的。对于v E V（K），Ov表示相应的离散赋值环;众所周知，群Hnt（Spec Ov，j2）注入到Hn（K）中。如果X/k是光滑射影几何积分簇，k（X）是它的函数域，那么我们称群Hnrn（k（X））为X的非分歧上同调群。对于域F，设W（F）是F [Sc]上二次型的Witt群。对于上面的k和K/k，我们可以类似地定义K/k的非分歧维特群，记为Wtr（K/k），作为W（K）的子群，定义为Wnr（K/k）= n W（ov）vEV（K）这里W（Ov）是离散赋值环Ov的维特群，它是W（K）的子群[Sc，第六章]。如果X/k是光滑射影的，则称Wnr（k（X））为X的非分歧Witt群.回想一下，一个簇X被称为k-有理的，如果X对于射影空间是k-双有理的。然后，自然映射Hn（k）-* Hn（k（X））（分别W（k）-* W（k（X）诱导同构Htn（k）_ Hnrr（k（X））（resp. W（k）_ Wnr（k（X）。因此，k-有理簇的非分歧上同调群（分别为）。未分化的维特1995年11月7日由编辑收到。1991年数学学科分类。小学IIE 70;中学13 K 05，12 G 05。
The unramified Witt group of an anisotropic conic over a field k, with char k 0 2, defined by the form (1, -a, -b) is known to be a quotient of the Witt group W(k) of k and isomorphic to W(k)/(1, -a, -b, ab)W(k). We compute the unramified cohomology group H43rk(C), where C is the three dimensional anisotropic quadric defined by the quadratic form (1, -a, -b, ab, -c) over k. We use these computations to study the unramified Witt group of C. Let k be a field of characteristic not two and K/k be a finitely generated field extension. The n-th unramified cohomology group of K over k with coefficients in Z/2 (cf. [CT]) is denoted Hnnr(K/k) (or just Hnnr(K) when the ground field being considered is clear). It is a subgroup of the Galois cohomology group Hn (K, Z/2) (abbreviated to Hn(K)) and is defined as Hnnr(K/k)= n H6nt (Spec Ov ,u2). vEV(K) Here V(K) is the set of all rank one discrete valuations on K that are trivial on k. For v E V(K), Ov denotes the corresponding discrete valuation ring; it is well-known that the group Hnt(Spec Ov, j2) injects into Hn (K). If X/k is a smooth projective geometrically integral variety and k(X) is its function field, then by abuse of terminology we refer to the groups Hnrn(k(X)) as the unramified cohomology groups of X. For a field F, let W(F) be the Witt group of quadratic forms over F [Sc]. For k and K/k as above, we can similarly define the unramified Witt group of K/k, denoted Wtr(K/k), as the subgroup of W(K) defined by Wnr (K/k) = n W(ov) vEV(K) Here W(Ov) is the Witt group of the discrete valuation ring Ov, which is a subgroup of W(K) [Sc, Chapter VI]. If X/k is smooth projective, then we refer to Wnr(k(X)) as the unramified Witt group of X. Recall that a variety X is said to be k-rational if X is k-birational to the projective space. Then the natural maps Hn(k) -* Hn(k(X)) (resp. W(k) -* W(k(X))) induce isomorphisms Htn(k) _ Hnrr(k(X)) (resp. W(k) _ Wnr(k(X))). Thus the unramified cohomology groups of a k-rational variety (resp. the unramified Witt Received by the editors November 7, 1995. 1991 Mathematics Subject Classification. Primary IIE70; Secondary 13K05, 12G05.