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Formal completions and idempotent completions of triangulated categories of singularities. (English) Zbl 1216.18012

In this paper, the main result says that if \(X\) and \(X'\) are two schemes satisfying the mild condition (ELF), defined below, and if the formal completions of \(X\) and \(X'\) along their singularities are isomorphic, then the idempotent completions of the triangulated categories of singularties are equivalent. Here the condition (ELF) means that the scheme is separated, Noetherian of finite Krull dimension, and for any coherent sheaf \(\mathcal{F}\) there exists a locally free sheaf \(\mathcal{E}\) and an epimorphism \(\mathcal{E} \rightarrow \mathcal{F}\). The triangulated category of singularities \(\mathbb{D}_{\text{Sg}}(X)\) of \(X\) is the quotient of the bounded derived category of the abelian category of coherent sheaves on the scheme, by the full triangulated subcategory of perfect complexes.
Then the author proves that for a morphism \(f : X \rightarrow X'\) which is an isomorphism infinitely near a closed subscheme \(Z \subset X\) containing the singular locus of both schemes satisfying (ELF) \(X\) and \(X'\), then the induced functor \(\bar{f}^*\) is fully faithful and, moreover, any object \(B \in \mathbb{D}_{\text{Sg}}(X')\) is a direct summand of some object coming from \(\mathbb{D}_{\text{Sg}}(X)\).
Let \(\mathfrak{Perf}(X)\) be the triangulated category of perfect complexes. \(\mathfrak{Perf}_{\text{sing} X}(X)\) be \(\mathfrak{Perf}(X) \cap \mathbb{D}^b_{\text{sing} X}(\text{coh} X)\) of perfect complexes supported in \(\text{sing} X\). In the paper the author proves and uses a proposition that the idempotent completions of \(\mathbb{D}_{\text{Sg}}(X)\) and \(\mathbb{D}^b_{\text{sing} X}(\text{coh} X) / \mathfrak{Perf}_{\text{sing} X}(X)\) are equivalent. In the last section, he proves that there is a difference between \(K_{-1}\) of \(\mathfrak{Perf}(X)\) and \(\mathfrak{Perf}_{\text{sing} X}(X)\). He also gives an example where the triangulated category of singularities of two schemes constructed differently, are equivalent.

MSC:

18E30 Derived categories, triangulated categories (MSC2010)
14B05 Singularities in algebraic geometry
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