论文标题
在三角形类别上，框架动机$ \ text {dfr} _ { - }^{eff}（k）$
On the triangulated category of framed motives $\text{DFr}_{-}^{eff}(k)$
论文作者
论文摘要
Voevodsky在他的注释中发明了框架通信的类别$ fr _*（k）$，以便为SH（k）提供另一个框架，以使其更适合显式计算。基于该注释及其JAMS Paper Garkusha和作者在最近的论文中引入了一个三角形类别的Bispectra $ \ text {sh} _ {nis}^fr}（k）$。在后一篇文章中显示了$ \ text {sh} _ {nis}^{fr}（k）$恢复Bispectra $ \ text {sh}（k）$的经典Morel-Voevodsky三角形类别。对于任何无限的完美字段$ k $ a $ \ mathbb {f} \ text {r} $的三角形类别 - 动机$ \ text {d} \ mathbb {f} \ text {r} {r} _ { - } _ { - }^{eff}^{eff}（k）$构建了voev odssky Contuntron $ \ text {dm} _-^{eff}（k）$。在我们的方法中，Nisnevich Sheaves的Voevodsky类别用$ \ Mathbb {f} \ text {r} $ - 模块代替。对于每个平滑的$ k $ -variety $ x $ $ \ mathbb {f} \ text {r} $ - Motive $ \ text {m} _ {\ Mathbb {f} \ text {r}} \ text {r}}}（x）$与$ \ text {d} \ mathbb {f} \ text {r} _ { - }^{eff}（k）$。我们标识三角形类别$ \ text {d} \ mathbb {f} \ text {r} _ { - }^{ - }^{aff} {aff}（k）$，带有完整的三角形子类别$ \ text {sh} {sh}^{eff}^{eff} { - { - } _ { - }（k）classical morel-morel-vodian-niran-morel-vo ev。有效动机Bispectra的$ \ text {sh}^{eff}（k）$。此外，三角类别$ \ text {d} \ mathbb {f} \ text {r} _ { - }^{ - }^{eff}（k）$自然是对称的单型。提到的三角类别的识别尊重两侧的对称单体结构。
The category of framed correspondences $Fr_*(k)$ was invented by Voevodsky in his notes in order to give another framework for SH(k) more amenable to explicit calculations. Based on that notes and on their JAMS paper Garkusha and the author introduced in a very recent paper a triangulated category of framed bispectra $\text{SH}_{nis}^{fr}(k)$. It is shown in the latter paper that $\text{SH}_{nis}^{fr}(k)$ recovers classical Morel-Voevodsky triangulated category of bispectra $\text{SH}(k)$. For any infinite perfect field $k$ a triangulated category of $\mathbb{F}\text{r}$-motives $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$ is constructed in the style of the Voevodsky construction of the category $\text{DM}_-^{eff}(k)$. In our approach the Voevodsky category of Nisnevich sheaves with transfers is replaced with the category of $\mathbb{F}\text{r}$-modules. To each smooth $k$-variety $X$ the $\mathbb{F}\text{r}$-motive $\text{M}_{\mathbb{F}\text{r}}(X)$ is associated in the category $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$. We identify the triangulated category $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$ with the full triangulated subcategory $\text{SH}^{eff}_{-}(k)$ of the classical Morel-Voevodsky triangulated category $\text{SH}^{eff}(k)$ of effective motivic bispectra. Moreover, the triangulated category $\text{D}\mathbb{F}\text{r}_{-}^{eff}(k)$ is naturally symmetric monoidal. The mentioned identification of the triangulated categories respects the symmetric monoidal structures on both sides.
