Projective Resolutions of Coherent Sheaves and Descent Relations between Branes

We notice that, for branes wrapped on complex analytic subvarieties, the algebraic-geometric version of K-theory makes the identification between brane-antibrane pairs and lower-dimensional branes automatic. This is because coherent sheaves on the ambient variety represent gauge bundles on subvarieties, and they can be put in exact sequences (pro-jective resolutions) with sheaves corresponding to vector bundles on the pair; this automatically gives a D(p − 2) as a formal difference of bundles on the Dp − D ¯ p pair, both belonging to the Grothendieck group of coherent sheaves of the ambient.