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Pullback of a module along a morphism of ringed spaces
Definition
Let
be a morphism of ringed spaces, and let be an -module. The pullback of along is the -module
Here the ring map
is the morphism corresponding to under the inverse/direct-image adjunction Inverse image is left adjoint to direct image on sheaves. The inverse-image construction is taken with its algebraic structure: applying the neighbourhood-colimit construction to the restriction-compatible ring operations of , and then sheafifying, makes a sheaf of rings. The same construction makes an -module (Presheaves and sheaves of groups, rings, and modules).
Consequently is an -algebra, the displayed sheaf tensor product is well typed, and it carries the asserted -module structure.
Depends on
Used by
- Euler characteristic in a proper flat family is locally constant Corollary
- Twisting the exact sequence of an effective Cartier divisor Corollary
- Upper semicontinuity of fibre cohomology dimensions Corollary
- A proper nonprojective scheme from glued projective spaces Counterexample
- Cohomology and base-change map Definition
- Hilbert function and Euler characteristic on a projective scheme Definition
- An upper jump of h0 in a flat projective family Example
- Projective bundle of a trivial module Example
- Pullback carries a free module to the corresponding free module Example
- Differential of an S-morphism Lemma
- Dual and base change for finite locally free sheaves Lemma
- Euler characteristic of a closed point, and invariance under an invertible twist Lemma
- Finite projective complex for proper flat coherent cohomology Lemma
- Flat field extension commutes with coherent cohomology Lemma
- Line bundles on projective three-space and their restrictions Lemma
- Projection formula for a closed immersion and an invertible sheaf Lemma
- Pullback of a Cartier divisor computes the pullback of its line bundle Lemma
- Pullback of modules is right exact, and flat stalk maps make it exact Lemma
- Regular hyperplane step for coherent support induction Lemma
- Relative differentials commute with scheme base change Lemma
- Scheme pullback preserves quasi-coherence Lemma
- Support dimension under field extension Lemma
- Symmetric algebras are quasi-coherent and commute with pullback Lemma
- The invariant differentials of a group scheme Lemma
- The twisting sheaf of projective space is very ample and ample Lemma
- Universal finite projective cohomology complex over any base Lemma
- Inverse image of sheaves and pullback of modules are not the same construction Remark
- Cohomology and base change for proper flat coherent families Theorem
- Conormal sequence for a closed immersion Theorem
- Degree of the coherent Hilbert polynomial Theorem
- Descent of vanishing along a faithfully flat morphism Theorem
- Euler characteristic is a Hilbert polynomial Theorem
- Projective bundle represents line quotients Theorem
- Pullback of modules is left adjoint to pushforward Theorem
- Serre vanishing for coherent sheaves and ample twists Theorem
- Transitivity sequence for schemes Theorem
Dependency tree · two levels
15 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- The Stacks Project, Definition 6.26.1 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Section 2.7 and Chapter 6 (standard reference, not scraped)