Abstract for Hailong Dao:
Let R be a local ring and X the punctured spectrum of R. One can use tensor product and direct sum to turn the category of locally free coherent sheaves on X into a semi-ring. Motivated by constructions and questions over projective varieties, we aim to study two questions on this semi-ring: when can one cancel from products, and when there exist Nori-finite vector bundles? Some partial answers will be discussed when R is regular.