# what are mutations of sheaves all about?

Suppose I have a smooth projective variety $X$ and a semi-orthogonal decomposition of its bounded derived category of coherent sheaves $D^b(X)$. Then I can apply right or left mutations to the full asmissible triangulated subcats that make up the semi-orth decomposition. Algorithmically it is clear to me what happens, but what is the spirit of such a transformation? what is the geometry behind it? in particular I would be curious to understand this in the case of full exceptional sequences and when all subcats in the semi-orth decomposition are generated by sheaves, line or vector bundles.

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However, in some cases there is some meaning. For example, if $D(X) = < A, {}^\perp A >$ then the left mutation functor $L_A:{}^\perp A \to A^\perp$ is isomorphic to the composition of the Serre functor of $D(X)$ and the inverse Serre functor of ${}^\perp A$: $$L_A = S_X\circ S_{{}^\perp A}^{-1}.$$ In the particular case, when ${}^\perp A$ is generated by one exceptional object, one has $L_A = S_X$.