论文标题
无限仿射空间和代数K理论的希尔伯特计划
The Hilbert scheme of infinite affine space and algebraic K-theory
论文作者
论文摘要
我们从$ \ mathbb {a}^a}^1 $ - 霍尼伯视图中研究希尔伯特方案$ \ mathrm {hilb} _d（\ mathbb {a}^\ infty）$。我们表明，希尔伯特方案$ \ mathrm {hilb} _d（\ mathbb {a}^\ infty）$ is $ \ mathbb {a}^1 $ - 等于$（d-1）$ - planes $（d-1）$ - planes in $（d-1）$ - $ \ m athbb {a a} a}^\ infty $。然后，我们描述$ \ mathrm {hilb} _d的$ \ mathbb {a}^1 $ -HOMOTOPY类型在一个范围内的$ n $，与$ d $相比。例如，我们在一个范围内计算$ \ mathrm {hilb} _d（\ mathbb {a}^n）$的整体共同体学（\ Mathbb {a}^n）$。我们还推断出健忘的地图$ \ mathrm {fflat} \ to \ mathrm {vect} $从有限的本地免费方案中从有限的本地免费滑轮的模型堆栈中是$ \ mathbb {a} a}^1 $ - 等价。这意味着Moduli stack $ \ mathrm {fflat} $，被视为带有框架传输的预毛，是代表代数K理论的有效动机频谱$ \ mathrm {kgl} $的模型。将我们的技术与Bachmann的最新工作相结合，我们获得了Hilbert方案模型，用于$ \ Mathrm {kgl} $ - 在完美领域的平滑适当方案的同源性。
We study the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ from an $\mathbb{A}^1$-homotopical viewpoint and obtain applications to algebraic K-theory. We show that the Hilbert scheme $\mathrm{Hilb}_d(\mathbb{A}^\infty)$ is $\mathbb{A}^1$-equivalent to the Grassmannian of $(d-1)$-planes in $\mathbb{A}^\infty$. We then describe the $\mathbb{A}^1$-homotopy type of $\mathrm{Hilb}_d(\mathbb{A}^n)$ in a range, for $n$ large compared to $d$. For example, we compute the integral cohomology of $\mathrm{Hilb}_d(\mathbb{A}^n)(\mathbb{C})$ in a range. We also deduce that the forgetful map $\mathrm{FFlat}\to\mathrm{Vect}$ from the moduli stack of finite locally free schemes to that of finite locally free sheaves is an $\mathbb{A}^1$-equivalence after group completion. This implies that the moduli stack $\mathrm{FFlat}$, viewed as a presheaf with framed transfers, is a model for the effective motivic spectrum $\mathrm{kgl}$ representing algebraic K-theory. Combining our techniques with the recent work of Bachmann, we obtain Hilbert scheme models for the $\mathrm{kgl}$-homology of smooth proper schemes over a perfect field.
