Hochschild dimensions of tilting objects (Q2911501)

If \(\mathcal{T}\) is a triangulated category, a generator is, roughly speaking, an object \(E\) such that any other object in \(\mathcal{T}\) can be built from \(E\) by taking shifts, direct sums, direct summands and a finite number of cones. The latter number is the generation time of \(E\) and the dimension of \(\mathcal{T}\) is the infimum over all generation times. This notion was introduced by \textit{R. Rouquier} [J.\ K-Theory 1, No.\ 2, 193--256 (2008); Erratum ibid. 257--258 (2008; Zbl 1165.18008)], where it was, in particular, proved that if \(\mathcal{T}\) is the bounded derived category \(D^b(X)\) of coherent sheaves on a smooth projective variety \(X\), then the dimension of \(D^b(X)\) is bounded below by the dimension of \(X\) and bounded above by twice the dimension of \(X\). \textit{D. Orlov} conjectured that the dimension of \(D^b(X)\) is, in fact, equal to \(\text{dim}(X)\) and proved this for curves [Mosc.\ Math.\ J.\ 9, No.\ 1, 143--149 (2009; Zbl 1197.18004)]. It is also known that the conjecture holds for affine varieties.NEWLINENEWLINEIn the paper under review the authors consider the case where \(X\) admits a tilting object. Recall that an object \(T\) in a \(k\)-linear (\(k\) field, for simplicity algebraically closed of characteristic zero) triangulated category \(\mathcal{T}\) is called tilting if \(\text{Hom}(T,T[i])=0\) for all \(i\neq 0\) and \(T\) generates \(\mathcal{T}\). Furthermore, recall that the Hochschild dimension of a \(k\)-algebra \(A\) is the projective dimension of \(A\) as an \(A\otimes_k A^{\mathrm {op}}\)-module. The main result is then as follows. If \(i_0\) is the largest \(i\) for which \(\text{Hom}(T,T\otimes \omega_X^\vee[i])\neq 0\), then the Hochschild dimension of \(\text{End}(T)\) is equal to \(\text{dim}(X)+i_0\). If \(i_0=0\), then the Hochschild dimension of \(\text{End}(T)\), the dimension of \(X\) and the dimension of \(D^b(X)\) are all equal. In particular, this applies to del Pezzo surfaces, toric surfaces with nef anti-canonical divisor and Fano threefolds of types \(V_5\) and \(V_{22}\).NEWLINENEWLINEThe paper is organised as follows. In Section 2 the authors recall some of the necessary background as well as extend some results on the Rouquier dimension to smooth and tame Deligne-Mumford stacks with quasi--projective coarse moduli spaces. In Section 3 first the main result is proved, before the examples where it is applicable are thoroughly investigated.