The fundamental group-scheme
The fundamental group-scheme
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DOI：
10.1007/bf02967978
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发表时间：
1982-07
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影响因子：
--
通讯作者：
M. Nori
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文献类型：
--
作者：
M. Nori
设M是紧Riem~nn曲面，JSA有限Oalois非分支覆盖，有Galois群O，设If是具有G作用的向量空间。O在)~·上的d作用是自由的，其商是向量丛IF在X上。A井证明了存在两个非负整系数多项式sf和g，其中f#g和f(If)与g(It)同构。M上向量丛的同构类S关于直和和与张量积形成一个半环，因此表达式f(L 0和g(L 0)有意义为X上的向量丛。满足这一性质的X上的向量丛B称为有限的。我们证明了反之亦然：有限向量丛~产生于适当的转移覆盖~-的Galois群的表示。X.很容易看出线丛L是有限的当且仅当L是M的雅可比矩阵中的时间阶点。对于这样的线丛，~的函数域是Y的函数域的简单Kumrn：R扩张。因此，我们关于线丛的定理简单地断言了的基本群的特征。它是MARE的拓扑基本群与有限阶线丛一一对应的奢侈完备化。T*。当然，这是一个事实，也是一个非常有用的事实，因为所有这些线丛的阿贝尔群的结构很容易由雅可比的拓扑确定。由于不清楚如何从~‘上稳定丛的变化确定Flaite丛，因此我们的定理没有实用价值。它是域k上有限向量块上的一个完全连通约化格式。死还是马来的意思。本质有限丛仅是I#的一个子商，属于半稳定范畴。如果G是有限群方案，P是M上的主O-丛，则O的表示在上产生本质有限的btmdles。实际上，所有本质上有限的丛都是以这种方式获得的。在特征零中，有限=本质上有限。这是第一章的内容。
Let M be a compact Riem~ nn surface, JSa finite Oalois unramifled covering, with Galois group O. Let IF be a vector space with G-action. The d~ agonal action of O on)~• Vis free and the quotient is a vector bundle IF on X. It was shown by A Well that there are two polynomialsf and g with nononegative integer coefficients with f# g andf (IF) isomorphic to g (It). Isomorphism clasr~ s of vector bundles on M form a semioring with respect to direct sums and tensor products, so the expressions f (l 0 and g (l 0 make sense as vector bundles on X. A vector bundle B, on Xsatisfying this property is called finite. We prove the converse: a finite vector bundle~ arises from a representation of the Galois group for a suitable tmramified covering~--. X. It is easy to see that a line bundle L is finite if and only if L is a point of timte order in the Jacobian of M. For such a line bundle, the function field of~ is junta simple Kumrn: r extension of the function field of, Y. Thus our theorem for line bundles simply asserts that the characters of the etale fundamental group of. It (waich is the proflnite completion of the topological fundamental group of Mare in one-to-one correspondence with line bundles of finite order on. t*. This is, of course, a weU-i~ nown fact, and a very useful one because the structure of the abelian group of all such line bundles is determined very easily by the topology of the Jacobian. Whereas it is not clear how to go about determining the flaite bundles from the variety of stable bundles on~'; consequently our theorem has met with no utility.It" Mis a complete connected reduced scheme over a field k, finite vector bur. dies still malae sense. An essentially finite bundle is justa sub-quotient of I#, amaining in the semi-stable category. If G is a finite group-scheme and P is a principal O-bundle on M, the representations of O give rise to essentially finite btmdles on. Y and in fact all essentially finite bundles are obtained in this manner. In characteristic zero, finite= essentiaLly finite. This is the content of Chapter I.