The singular Riemann-Roch theorem and Hilbert-Kunz functions

Toric Hirzebruch-Riemann-Roch via Ishida's theorem on the Todd genus

We give a simple proof of the Hirzebruch-Riemann-Roch theorem for smooth complete toric varieties, based on Ishida's result that the Todd genus of a smooth complete toric variety is one.

奇异曲线的Riemann-Roch空间快速计算

Fast computation of Riemann-Roch spaces for singular curves

本文提出了一种适用于任意分歧曲线的快速算法，用于计算Riemann-Roch空间的基础，该算法在任意完美域上均有效。

Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations

We study the category of matrix factorizations for an isolated hypersurface singularity. We compute the canonical bilinear form on the Hochschild homology of this category. We find explicit expressions for the Chern character and the boundary-bulk maps and derive an analog of the Hirzebruch-Riemann-Roch formula for the Euler characteristic of the Hom-space between a pair of matrix factorizations. We also establish G-equivariant versions of these results.

格罗滕迪克对希策布鲁赫-黎曼-罗赫定理的证明

Grothendieck's proof of Hirzebruch-Riemann-Roch theorem

本文针对代数闭域上光滑射影簇，自洽介绍并完成格罗滕迪克框架下希策布鲁赫-黎曼-罗赫定理的完整证明，验证其可还原经典情形。

Riemann-Roch for tensor powers

Mapping a locally free module to its l-th tensor power gives rise to a natural map from the Grothendieck group of all locally free modules to the Grothendieck group of all locally free representations of the l-th symmetric group. In this paper, we prove some formulas of Riemann-Roch type for the behaviour of this tensor power operation with respect to the push-forward homomorphism associated with a projective morphism between schemes. We furthermore establish analogous formulas for higher K-groups.

Monomials, Binomials, and Riemann-Roch

The Riemann-Roch theorem on a graph G is related to Alexander duality in combinatorial commutive algebra. We study the lattice ideal given by chip firing on G and the initial ideal whose standard monomials are the G-parking functions. When G is a saturated graph, these ideals are generic and the Scarf complex is a minimal free resolution. Otherwise, syzygies are obtained by degeneration. We also develop a self-contained Riemann-Roch theory for artinian monomial ideals.