Homotopy types of algebraic varieties
Homotopy types of algebraic varieties
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代数簇的同伦类型
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通讯作者：
B. Toën
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作者：
B. Toën
设k是一个域。对于k上的任意光滑射影代数簇X，可以对应一定的几何上同调空间H_∞（X），它是系数域K上的有限维向量空间。例如H et（X，Ql）（X的l-adic上同调：= X k k）、HDR（X）（例如k = C时X的代数de Rham上同调）、H * cris（X）（当k为正特征时X的cristalline上同调）。. . .这些几何上同调理论H是所谓的Weil上同调理论。它们对X的几何属性进行编码，并且不应该看到基域k的算术属性。更确切地说，基域k的算术性质并不反映在空间H（X）本身，而是表现为空间H（X）上自然附加结构的存在。例如，H_（？）（X，Q_1）具有Galois群Gal（k/k）的连续作用，H_DR（X）具有纯Hodge结构，H_（？）（X）具有k上的F -等晶结构. . . . Tannakian形式主义进一步告诉我们，这些额外的结构被编码在空间H（X）上的前代数群1 H的作用中。群H当然取决于所选择的上同调理论，在上面的例子中，它是Gal（k/k）的连续有限维l-adic表示的Tannakian范畴、纯Hodge结构的Tannakian范畴、k上的F -等晶的Tannakian范畴的基本群。. . .从这些观察中，我们可以得出以下一般原则。
Let k be a field. To any smooth and projective algebraic variety X over k one can associate certain geometric cohomology spaces H ∗ (X), which are finite dimensional verctor spaces over some coefficients field K. For example H et(X, Ql) (the l-adic cohomology of X := X ⊗k k ), H DR(X) (the algebraic de Rham cohomology of X when e.g. k = C), H ∗ cris(X) (the cristalline cohomology of X when k is of positive characteristic) . . . . These geometric cohomology theories H ∗ are the so-called Weil cohomology theories. They encode geometric properties of X, and are not suppose to see the arithmetic properties of the base field k. More precisely, the arithmetic nature of the base field k is not reflected in the spaces H ∗ (X) themselves but rather appears as the existence of natural additional structures on them. For example, H ∗ et(X, Ql) comes equiped with a continuous action of the Galois group Gal(k/k), H DR(X) is endowed with a pure Hodge structure, H ∗ cris(X) has a structure of an F -isocristal over k . . . . The Tannakian formalism tells us furthermore that these additional structures are encoded in an action of a pro-algebraic group1 H on the space H ∗ (X). The group H of course depends on the cohomology theory one chose, and in the example above is the foundamental group of the Tannakian categories of continuous finite dimension l-adic representations of Gal(k/k), of pure Hodge structures, of F -isocristals over k . . . . From these observations one extracts the following general principle.