论文标题
软功能代数的DE RHAM协同学
The de Rham cohomology of soft function algebras
论文作者
论文摘要
我们研究了dg-algebra $ω^\ bullet_ {a | \ mathbb {r}} $的代数de rham的形式的真实软功能代数$ a $，即，$ c_x $的软性子eaf的全球段的代数，$ c_x $，$ c_x $，sheaf sheaf a sheaf of Space $ x $ $ x $ $ x $。我们获得了一个规范分裂$ \ MATHRM H ^n（ω ^\ bullet_ {a | \ Mathbb {r}}））\ cong \ mathrm h ^n（x，x，x，\ mathbb {r}）\ oplus v $，其中$ v $在$ v $中是一定的。特别是，我们考虑$ a = c（x）$ for $ x $紧凑的hausdorff空间和$ a = c^\ infty（x）$ for $ x $ a n compact smoth Smooth歧管。对于代数$ \ mathrm {ppol} _k（| k |）$ $ k $ potiewise多项式函数的$ k $上述分裂减少至规范的同构$ \ \ m mathrm h ^** \ cong \ mathrm h ^*（| k |，\ mathbb {r}）$。我们还证明，代数de rhAM共同体$ \ mathrm h ^n（ω ^\ bullet_ {c（x）| \ mathbb {r}}）$对于每个$ n \ geq 1 $都是不繁琐的，如果$ x $是无限的compact cartact cartact compact cartact hausdorff space。
We study the dg-algebra $Ω^\bullet_{A|\mathbb{R}}$ of algebraic de Rham forms of a real soft function algebra $A$, i.e., the algebra of global sections of a soft subsheaf of $C_X$, the sheaf of continuous functions on a space $X$. We obtain a canonical splitting $\mathrm H ^n (Ω^\bullet_{A|\mathbb{R}}) \cong \mathrm H ^n (X,\mathbb{R})\oplus V$, where $V$ is some vector space. In particular, we consider the cases $A=C(X)$ for $X$ a compact Hausdorff space and $A = C^\infty (X)$ for $X$ a compact smooth manifold. For the algebra $\mathrm{PPol}_K (|K|)$ of piecewise polynomial functions on a polyhedron $K$ the above splitting reduces to a canonical isomorphism $\mathrm H ^* (Ω^\bullet_{\mathrm{PPol}_K (|K|)|\mathbb{R}}) \cong \mathrm H ^* (|K|,\mathbb{R})$. We also prove that the algebraic de Rham cohomology $\mathrm H ^n (Ω^\bullet_{C(X)|\mathbb{R}})$ is nontrivial for each $n\geq 1$ if $X$ is an infinite compact Hausdorff space.
