AI 中文总结
该研究构造两类弗罗贝尼乌斯环作为光滑完全簇$K$-环和周环的模型，证明其满足黎曼-罗赫定理的组合类比，给出相关同构及应用。
AI 中文摘要
我们构造两类交换弗罗贝尼乌斯环：离散弗罗贝尼乌斯商$K_f$和连续弗罗贝尼乌斯商$A_g$，它们分别定义为平移算子代数和微分算子代数模某个多项式零化子的商。这些构造是光滑完全簇的数值$K$-环和周环的模型。我们证明，当多项式满足希策布鲁赫-黎曼-罗赫定理的组合类比时，存在一个自然定义的同构$\boldsymbol{\text{Q}} K_f \boldsymbol{\text{Q}} A_g$，该同构扮演陈特征的角色；具体而言，此时存在可逆元素$\text{td} \boldsymbol{\text{Q}} A_g^\times$（称为托德类）使得$f = \text{td} \boldsymbol{\text{·}} g$。在这种情况下，我们证明$K_f$具备所有必要结构，可作为完全光滑簇$K$-环的合适模型：$K_f$是$\boldsymbol{\text{λ}}$-环，有良定义的陈类、行列式，且满足塞尔对偶的组合类比。我们进一步证明该构造是函子性的，并得到格罗滕迪克-黎曼-罗赫定理的组合类比。最后，我们研究由幂级数在生成集上的作用定义的弗罗贝尼乌斯商之间更大族同构的存在性，例如截断陈特征会产生积分同构$K_f \boldsymbol{\text{≅}} A_g$。我们提供大量例子和应用：给出计算拟阵斯纳珀多项式的新公式，证明$K_f$的对偶类与拟阵及多面体线性族的对偶类一致，将环簇丛的$K$-环实现为弗罗贝尼乌斯商，还在此框架下研究埃哈特扇的$K$-环及例外同构。
英文摘要
We construct two families of commutative Frobenius rings: discrete and continuous Frobenius quotients $K_f$, resp. $A_g$, which are defined as the quotients of shift, resp. differential operator algebras by the annihilator of a polynomial. These constructions model the numerical $K$-rings and Chow rings of smooth complete varieties. We show that there is a naturally defined isomorphism $\mathbb{Q} K_f \cong \mathbb{Q} A_g$ playing the role of the Chern character, precisely when the polynomials satisfy a combinatorial analogue of the Hirzebruch-Riemann-Roch theorem, which states that there exists an invertible element $\mathrm{td} \in \mathbb{Q} A_g^\times$, called the Todd class, such that $f = \mathrm{td} \cdot g$. In this case we show that $K_f$ carries all the structure needed to make it a suitable model for $K$-rings of complete smooth varieties: $K_f$ is a $λ$-ring, has well-defined Chern classes, determinants, and satisfies a combinatorial analogue of Serre duality. We further show that the construction is functorial and obtain a combinatorial analogue of the Grothendieck-Riemann-Roch theorem. Lastly, we investigate the existence of larger families of isomorphisms between Frobenius quotients defined by the action of a power series on a generating set, such as the truncated Chern character which yields an integral isomorphism $K_f \cong A_g$. We provide numerous examples and applications: we give a new formula for computing Snapper polynomials of matroids, show that the dualizing class of $K_f$ coincides with dualizing classes of matroids and linear families of polytopes, realize $K$-rings of toric variety bundles as Frobenius quotients, and study $K$-rings of Ehrhart fans as well as exceptional isomorphisms in this setting.
Comments41 pages
