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Strong generators in \(\mathbf{D}^{\mathrm{perf}}(X)\) and \(\mathbf{D}^b_{\mathrm{coh}}(X)\). (English) Zbl 1478.18014

In 2003, A. Bondal and M. van den Bergh [Mosc. Math. J. 3, No. 1, 1–36 (2003; Zbl 1135.18302)] proved that the bounded derived categories of coherent sheaves on smooth proper commutative and noncommutative varieties have strong generators, hence saturated. Strong generators are particularly useful in triangulated categories proper over a noetherian, commutative ring \(R\). A category that admits a strong generator was then called regular. Bondal and Van den Bergh proposed a conjecture as follows:
“One could generalize the condition over the scheme to be quasicompact and separated.”
This paper is devoted to proving this conjecture which is presented in the following Theorem:
{Theorem.} Let \(X\) be a quasicompact, separated scheme. Then \(\mathbf{D}_{\mathrm{perf}}(X)\) is regular if, and only if, \(X\) can be covered by open affine subschemes \(\mathrm{Spec}(R_i)\), with each \(R_i\) of finite global dimension.
The author also proves that for a noetherian scheme \(X\) of finite type over an excellent scheme of dimension \(\leq 2\), the derived category \(\mathbf{D}_{\mathrm{coh}}^b(X)\) is strongly generated.

MSC:

18G80 Derived categories, triangulated categories
18G20 Homological dimension (category-theoretic aspects)

Citations:

Zbl 1135.18302
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References:

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