this paper is a slightly revised version of an introduction into singularity theory corresponding to a series of lectures given at the ``advanced school and conference on homological and geometrical methods in representation theory'' at the international centre for theoretical physics (ictp), miramare - trieste, italy, 11-29 january 2010. we show how to associate to a triple of positive integer...

We introduce a new category of lf-graded vector bundles on a smooth projective variety X over an algebraically closed field k. This category includes in particular all stable bundles. We then show that the category of strongly lf-graded bundles is a neutral Tannaka category. We study the associated GrothendieckTannaka group scheme. This enables us to prove an analogue of the classical Narasimha...

For a weighted projective line, the stable category of its vector bundles modulo line bundles has a natural triangulated structure. We prove that, for any positive integers p, q, r and r′ with r′ r, there is an explicit recollement of the stable category of vector bundles on a weighted projective line of weight type (p, q, r) relative to the ones on weighted projective lines of weight types (p,...

A tangent category is a equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of bundle functor from classical differential geometry. Cockett and Cruttwell introduced bundles in 2017 as algebraic alternative to vector arbitrary category. In this paper, we prove smooth manifolds are precisely bundles. particular, means can give characterisation exhibits...

Abstract In this article, we establish a motivic analog of an enumeration result James–Thomas [ 28] on non-stable vector bundles in topological setting. Using this, obtain results projective modules rank $d$ over smooth affine $k$-algebra $A$ dimension $d$, recovering particular Suslin and Bhatwadekar cancellation such bundles. Admitting conjecture Asok Fasel, prove $d-1$ if the base field $k$ ...