标题： 格罗滕迪克范畴中的K平坦性：拟凝聚层的应用 显示英文标题

标题： K-flatness in Grothendieck categories: Application to quasi-coherent sheaves

摘要： 设$(\mathcal{G},\otimes)$为任何闭的对称单oidal Grothendieck范畴。 我们证明在链复形范畴中K-平展覆盖普遍存在，并且$K(\mathcal{G})$对K-平展复形的Verdier商总是良好的生成三角范畴。 在进一步假设$\mathcal{G}$拥有一组$\otimes$-平展生成元的情况下，我们可以证明更多内容：(i) 该范畴与$\otimes$-纯导出范畴和通常的导出范畴处于recollement关系，(ii) 通常的导出范畴是其余纤化生成且单oidal模型结构的同伦范畴，其余纤化对象正是K-平展复形。 我们还给出一个条件，保证K-平展的右正交恰好是$\otimes$-纯内射的循环复形。 我们证明该条件对于准凝聚层在拟紧且半分离的概形上成立。 显示英文摘要

摘要： Let $(\mathcal{G},\otimes)$ be any closed symmetric monoidal Grothendieck category. We show that K-flat covers exist universally in the category of chain complexes and that the Verdier quotient of $K(\mathcal{G})$ by the K-flat complexes is always a well generated triangulated category. Under the further assumption that $\mathcal{G}$ has a set of $\otimes$-flat generators we can show more: (i) The category is in recollement with the $\otimes$-pure derived category and the usual derived category, and (ii) The usual derived category is the homotopy category of a cofibrantly generated and monoidal model structure whose cofibrant objects are precisely the K-flat complexes. We also give a condition guaranteeing that the right orthogonal to K-flat is precisely the acyclic complexes of $\otimes$-pure injectives. We show this condition holds for quasi-coherent sheaves over a quasi-compact and semiseparated scheme.