Pure sheaves and Kleinian singularities (Q2313933)

Let \(\mathcal E\) be a locally free sheaf on a projective line \(\mathbb P^1\) over a field \(k.\) Grothendieck proved that \(\mathcal E\) decomposes uniquely into a direct sum of line bundles on \(\mathbb P^1.\) This gives a complete classification of locally free sheaves and also of indecomposable sheaves on \(\mathbb P^1.\) For \(n>1\) the tangent sheaf on \(\mathbb P^n\) is locally free, of rank \(n\) and it is indecomposable, so that one cannot expect an analogue of Grothendieck's result for \(\mathbb P^n\) in these cases. Also, the classification of indecomposable locally free sheaves is more complicated than in lower rank situations. \textit{A. Ishii} and \textit{H. Uehara} [J. Differ. Geom. 71, No. 3, 385--435 (2005; Zbl 1097.14013)] proved a result analogue to Grothendieck's result for the fundamental cycle \(Z_{A_n}\) of the Kleinian singularity \(A_n\): A non-zero sheaf \(\mathcal F\) on a scheme \(Y\) is said to be pure if the support of any non-trivial subsheaf of \(\mathcal F\) has the same dimension as \(Y.\) This is equivalent to a locally free sheaf on \(Y\) when it is smooth and one-dimensional. Then Ishii and Uehara's result says that a pure sheaf \(\mathcal E\) on \(Z_{A_n}\) decomposes uniquely upto isomorphism into a direct sum of invertible sheaves on connected subtrees of \(Z_{A_n}.\) The first main result of the present article says that if \(Z\) is the fundamental cycle of a Kleinian singularity except for \(A_n,\) then \[ \operatorname{max}\{\operatorname{rank}_Z\mathcal E|\mathcal E\text{ is an indecomposable pure sheaf on }Z\}=\infty. \] Notice that the author defines the rank on the reducible scheme the following way: Let \(Z^\prime\) be a \(1\)-dimensional closed subscheme of the fundamental cycle \(Z\) of a Kleinian singularity and \(\iota^\prime:Z^\prime\rightarrow X\) be the embedding to the minimal resolution \(X\) of the singularity. Then the rank of a sheaf \(\mathcal E\) on \(Z^\prime\) is defined by \(\operatorname{rank}_{Z^\prime}:=\operatorname{min}\{a\in\mathbb Z_{\geq 0}|c_1(\iota^\prime_\ast\mathcal E)\leq a\cdot Z^\prime\}.\) Applying this first result to the result of Ishii and Uehara, the first part of that result can be restated to say that the maximum of the rank of indecomposable pure sheaves is \(1.\) Thus the main result above is a counterexample to the expectation that the maximum rank of indecomposable pure sheaves are bounded for the other Kleinian singularities. The second main aim of the paper is to study the classification of \(\mathcal O_X\)-rigid pure sheaves on \(Z.\) This is related to the classification of spherical objects in a certain triangulated category: Let \(X\) be the minimal resolution of a Kleinian singularity. The fundamental cycle is the schematic fibre of the singularity. Then there is a natural embedding \(\iota:Z\rightarrow X.\) Let \(D_Z(X)\) denote the bounded derived category of coherent sheaves on \(X\) supported on \(X.\) A coherent sheaf \(\mathcal E\) on \(Z\) is said to be \(\mathcal O_X\)-rigid if the push forward \(\iota_\ast\mathcal E\) is rigid, i.e. \(\operatorname{Ext}^1_X(\iota_\ast\mathcal E,\iota_\ast\mathcal E)=0.\) Ishii and Uehara proved that the cohomology of spherical objects in \(D_Z(X)\) is the push-forward \(\iota_\ast\mathcal E\) of pure \(\mathcal O_X\)-rigid sheaves \(\mathcal E.\) For \(A_n\) the classification of spherical objects is a consequence of their result. The first main result of this paper shows that to classify spherical objects for the other Kleinian singularities following the approach of Ishii and Uehara might not work. The second main result of the present paper shows progress following a different approach; verbatim: Theorem 1.3. Let \(\mathcal E\) be an indecomposable pure sheaf on the reduced scheme \(Z_r\) of the fundamental cycle of a Kleinian singularity except for \(A_n.\) If \(\mathcal E\) is \(\mathcal O_X\)-rigid, then \(\operatorname{rank}_{Z_r}\mathcal E<3\) and the inequality is best possible. The article is very well-written, and the results proved are classical and easy to formulate. However, the proofs involve original advanced methods of spherical objects in derived (triangulated) categories, and this gives a new progress to their classification and the applications.