MGL的商及其切片和几何部分
Quotients of MGL, their slices and their geometric parts
摘要：
设$x_1, x_2,\ldots$是Lazard环$\mathbb{L}^*=MU^{2*}$的齐次多项式生成元系，$MGL_S$表示基模式$S$上的运动稳定同伦范畴中的Voevodsky代数配边谱，$S$在域$k$上本质光滑.根据Voevodsky切片塔的第0个切片$s_0MGL_S$与$MGL_S/(x_1, x_2,\ldots)$的Hopkins-Morel-Hoyois同构（在反转$k$的特性之后），Spitzweck将$MGL_S$的剩余切片计算为$s_nMGL_S=Σ^n_TH\mathbb{Z}\otimes \mathbb{L}^{-n}$（同样，在反转$k$的特性之后）。我们应用Spitzweck的方法来计算切片的商谱$MGL_S/(\{x_i:i\in I\})$ $I$的任意子集的$\mathbb{N}$，以及模$p$版本$MGL_S/(\{p, x_i:i\in I\})$和本地化相对于一个系统的均匀元素在$\mathbb{Z}[\{x_j:j\not\in I\}]$。在情况下$S=\text{Spec}\, k$，$k$领域的特征零，我们应用这表明，$\mathcal{E}$的本地化的商$MGL$如上所述，有一个自然的同构的理论与支持\[ Ω_*(X)\otimes_{\mathbb{L}^{-*}}\mathcal{E}^{-2*,-*}(k)\to \mathcal{E}^{2m-2*, m-*}_X(M) \]为$X$一个封闭的子计划的一个光滑的准投射$k$计划$M$，$m=$调${}_kM$。
英文摘要：
Let $x_1, x_2,\ldots$ be a system of homogeneous polynomial generators for the Lazard ring $\mathbb{L}^*=MU^{2*}$ and let $MGL_S$ denote Voevodsky's algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme $S$.Take $S$ essentially smooth over a field $k$. Relying on Hopkins-Morel-Hoyois isomorphism of the 0th slice $s_0MGL_S$ for Voevodsky's slice tower with $MGL_S/(x_1, x_2,\ldots)$ (after inverting the characteristic of $k$), Spitzweck computes the remaining slices of $MGL_S$ as $s_nMGL_S=Σ^n_TH\mathbb{Z}\otimes \mathbb{L}^{-n}$ (again, after inverting the characteristic of $k$). We apply Spitzweck's method to compute the slices of a quotient spectrum $MGL_S/(\{x_i:i\in I\})$ for $I$ an arbitrary subset of $\mathbb{N}$, as well as the mod $p$ version $MGL_S/(\{p, x_i:i\in I\})$ and localizations with respect to a system of homogeneous elements in $\mathbb{Z}[\{x_j:j\not\in I\}]$. In case $S=\text{Spec}\, k$, $k$ a field of characteristic zero, we apply this to show that for $\mathcal{E}$ a localization of a quotient of $MGL$ as above, there is a natural isomorphism for the theory with support \[ Ω_*(X)\otimes_{\mathbb{L}^{-*}}\mathcal{E}^{-2*,-*}(k)\to \mathcal{E}^{2m-2*, m-*}_X(M) \] for $X$ a closed subscheme of a smooth quasi-projective $k$-scheme $M$, $m=$dim${}_kM$.
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