We prove that the$\infty $-category of$\mathrm{MGL} $-modules over any scheme is equivalent to the$\infty $-category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite$\mathbf{P} ^1$-loop spaces, we deduce that very effective$\mathrm{MGL} $-modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers.Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that$\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $is the$\mathbf{A} ^1$-homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for$n>0$,$\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $is the$\mathbf{A} ^1$-homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension$-n$.

中文翻译：

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代数协边的模块

我们证明了$\infty $-类别$\mathrm{MGL} $-modules 在任何方案上都等效于$\infty $-具有有限符号转移的运动光谱类别。使用无限的识别原理$\mathbf{P} ^1$-loop 空间，我们推断非常有效$\mathrm{MGL} $- 完美域上的模等价于具有有限符号传递的类群动机空间。在此过程中，我们根据有限准光滑派生方案的模堆栈描述了由非负秩的虚拟向量丛构建的任何动机 Thom 谱相应的切向结构。特别是，在一个规则的等特征基上，我们证明了$\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $是个$\mathbf{A} ^1$-虚拟有限平面局部完全交点的模堆栈的同伦类型，对于$n>0$,$\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $是个$\mathbf{A} ^1$-虚维有限拟光滑派生格式模栈的同伦类型$-n$.