The Stacks project

Definition 86.9.1. Let $S$ be a scheme. Let $f : X \to Y$ be a proper, flat morphism of algebraic spaces over $S$ which is of finite presentation. A relative dualizing complex for $X/Y$ is a pair $(\omega _{X/Y}^\bullet , \tau )$ consisting of a $Y$-perfect object $\omega _{X/Y}^\bullet $ of $D(\mathcal{O}_ X)$ and a map

\[ \tau : Rf_*\omega _{X/Y}^\bullet \longrightarrow \mathcal{O}_ Y \]

such that for any cartesian square

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

where $Y'$ is an affine scheme the pair $(L(g')^*\omega _{X/Y}^\bullet , Lg^*\tau )$ is isomorphic to the pair $(a'(\mathcal{O}_{Y'}), \text{Tr}_{f', \mathcal{O}_{Y'}})$ studied in Sections 86.3, 86.4, 86.5, 86.6, 86.7, and 86.8.

Comments (2)

Comment #7480 by S on

does there exist a relative dualizing sheaf for a cohen Macaulay morphism of DM stacks where S is a discrete valuation ring and X is smooth?

Comment #7628 by on

Yes, for any CM morphism (suitably defined -- not necessarily representable by algebraic spaces) between any algebraic stacks, you can define and construct a relative dualizing sheaf.

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View Definition 86.9.1 as pdf