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The Stacks project

57.2 Conventions and notation

Let k be a field. A k-linear triangulated category \mathcal{T} is a triangulated category (Derived Categories, Section 13.3) which is endowed with a k-linear structure (Differential Graded Algebra, Section 22.24) such that the translation functors [n] : \mathcal{T} \to \mathcal{T} are k-linear for all n \in \mathbf{Z}.

Let k be a field. We denote \text{Vect}_ k the category of k-vector spaces. For a k-vector space V we denote V^\vee the k-linear dual of V, i.e., V^\vee = \mathop{\mathrm{Hom}}\nolimits _ k(V, k).

Let X be a scheme. We denote D_{perf}(\mathcal{O}_ X) the full subcategory of D(\mathcal{O}_ X) consisting of perfect complexes (Cohomology, Section 20.49). If X is Noetherian then D_{perf}(\mathcal{O}_ X) \subset D^ b_{\textit{Coh}}(\mathcal{O}_ X), see Derived Categories of Schemes, Lemma 36.11.6. If X is Noetherian and regular, then D_{perf}(\mathcal{O}_ X) = D^ b_{\textit{Coh}}(\mathcal{O}_ X), see Derived Categories of Schemes, Lemma 36.11.8.

Let k be a field. Let X and Y be schemes over k. In this situation we will write X \times Y instead of X \times _{\mathop{\mathrm{Spec}}(k)} Y.

Let S be a scheme. Let X, Y be schemes over S. Let \mathcal{F} be a \mathcal{O}_ X-module and let \mathcal{G} be a \mathcal{O}_ Y-module. We set

\mathcal{F} \boxtimes \mathcal{G} = \text{pr}_1^*\mathcal{F} \otimes _{\mathcal{O}_{X \times _ S Y}} \text{pr}_2^*\mathcal{G}

as \mathcal{O}_{X \times _ S Y}-modules. If K \in D(\mathcal{O}_ X) and M \in D(\mathcal{O}_ Y) then we set

K \boxtimes M = L\text{pr}_1^*K \otimes _{\mathcal{O}_{X \times _ S Y}}^\mathbf {L} L\text{pr}_2^*M

as an object of D(\mathcal{O}_{X \times _ S Y}). Thus our notation is potentially ambiguous, but context should make it clear which of the two is meant.


Comments (2)

Comment #5117 by J on

Typo: last paragraph not


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