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The Stacks project

Situation 99.5.1. Let S be a scheme. Let f : X \to B be a morphism of algebraic spaces over S. Assume that f is of finite presentation. We denote \mathcal{C}\! \mathit{oh}_{X/B} the category whose objects are triples (T, g, \mathcal{F}) where

  1. T is a scheme over S,

  2. g : T \to B is a morphism over S, and setting X_ T = T \times _{g, B} X

  3. \mathcal{F} is a quasi-coherent \mathcal{O}_{X_ T}-module of finite presentation, flat over T, with support proper over T.

A morphism (T, g, \mathcal{F}) \to (T', g', \mathcal{F}') is given by a pair (h, \varphi ) where

  1. h : T \to T' is a morphism of schemes over B (i.e., g' \circ h = g), and

  2. \varphi : (h')^*\mathcal{F}' \to \mathcal{F} is an isomorphism of \mathcal{O}_{X_ T}-modules where h' : X_ T \to X_{T'} is the base change of h.


Comments (2)

Comment #890 by Matthew Emerton on

In condition (3), should proper over instead be proper over ?

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  • 1 comment(s) on Section 99.5: The stack of coherent sheaves

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