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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