论文标题
正式的Bott-Thurston Cocycle和正式Riemann-Roch定理的一部分
Formal Bott-Thurston cocycle and part of a formal Riemann-Roch theorem
论文作者
论文摘要
Bott-Thurston Cocycle是该圆圈的定向性差异的$ 2 $ cocycle。我们介绍并研究了Bott-Thurston Cocycle的正式类似物。正式的Bott-Thurston Cocycle是$ 2 $ cocycle的连续$ a $ a $ a $ automormormormormormormormormorphisms $ a（（t））laurent系列的$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a^*$ a^*$ a $ a $的价值。我们证明，正式的Bott-Thurston Cocycle给出的中央扩展相当于$ 12 $ fold baer的确定性中央扩展名，当$ a $是$ \ mathbb q $ -Algebra时。由于这个结果，我们证明了新的正式Riemann-Roch定理的一部分。该Riemann-Roch定理应用于$ \ Mathbb Q $的分离方案$ s $上的环形空间，其中环形空间的结构捆绑在$ s $ s $ sugmorphic上，对捆$ {\ Mathcal o} _s（（t））$ and the Transiention Automorphisms continallisms contrance contincy contince contince contine contrance。在$ s $上，这个环上的空间对应于平滑形态的一部分的刺穿正式邻域，以相对尺寸为$ 1 $的$ u $，其中一个开放子集$ u \ u \ subset s $。
The Bott-Thurston cocycle is a $2$-cocycle on the group of orientation-preserving diffeomorphisms of the circle. We introduce and study a formal analog of Bott-Thurston cocycle. The formal Bott-Thurston cocycle is a $2$-cocycle on the group of continuous $A$-automorphisms of the algebra $A((t))$ of Laurent series over a commutative ring $A$ with values in the group $A^*$ of invertible elements of $A$. We prove that the central extension given by the formal Bott-Thurston cocycle is equivalent to the $12$-fold Baer sum of the determinantal central extension when $A$ is a $\mathbb Q$-algebra. As a consequence of this result we prove a part of new formal Riemann-Roch theorem. This Riemann-Roch theorem is applied to a ringed space on a separated scheme $S$ over $\mathbb Q$, where the structure sheaf of the ringed space is locally on $S$ isomorphic to the sheaf ${\mathcal O}_S((t))$ and the transition automorphisms are continuous. Locally on $S$ this ringed space corresponds to the punctured formal neighbourhood of a section of a smooth morphism to $U$ of relative dimension $1$, where an open subset $U \subset S$.
