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  • scheme
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  • descent
  • deformation
  • Serre-Swan
  • faithfully flat descent

faithfully flat

Classical flatness

Idea: modules are generalized bundles, and flatness is local triviality. Generally flats are colimits of free modules.

Flat modules

Flat Morphisms

Faithful flatness

Faithfully flat modules

Idea: faithfully flat iff tensoring reflects exactness, i.e. \(\xi\) is an exact SES iff \(\xi\otimes M\) is a SES. Ideas: - If \(F\) is an \(R{\hbox{-}}\)algebra with structure morphism \(f:R\to F\), then \(F\) is flat over \(R\) when \(\operatorname{Spec}F\to \operatorname{Spec}R\) is a submersion, i.e. the induced topology on \(\operatorname{Spec}R\) is a quotient topology. - Open covers are often faithfully flat (surjective) morphisms. - Can prove some statement about algebras/schemes are a faithfully flat base change.

Faithfully flat descent

• A special case of faithfully flat descent: Zariski descent.

• For $S\in \mathsf{Alg} {/ {R}} $ and $M\in {}{R}{\mathsf{Mod}} $, there is a base-change functor $\mathsf{Alg} _{/ {R}} \to \mathsf{Alg} _{/ {S}} $ where \(X\mapsto X\otimes_R S\) that preserves many properties: e.g. if \(M\in {}_{R}{\mathsf{Mod}} ^{\mathrm{fg}}\) then \(M\otimes_R S \in {}_{S}{\mathsf{Mod}} ^{\mathrm{fg}}\).

• The reverse implication will hold if $S_{/ {R}} $ is faithfully flat.

See Unsorted/descent.

One can reformulate faithfully flat descent as saying the pseudofunctor QCoh mapping \(U\) to quasicoherent sheaves on \(U\) is a fpqc stack over Aff \(S\). Consequence: the map taking \(U\) to the category of affine morphisms \(V \rightarrow U\) is a fpqc stack.

Flat Schemes

Derived Flatness

# Exercises

#examples/exercises

• Show that if \(R\) is Noetherian and \(M\in {}_{R}{\mathsf{Mod}} ^{\mathrm{fg}}\) then \(R\) is flat iff \(R\) is locally free.
• Show that \({}_{{\mathbf{Z}}}{\mathsf{Mod}} ^\flat\) consists of torsionfree abelian groups.
• Show that the completion of a local ring \(R\) at its maximal ideal is a faithfully flat \(R{\hbox{-}}\)algebra.
• Show that localizations can generally fail to be faithfully flat.
• Show that every field extension is faithfully flat.
• Show that a nonzero finitely generated flat module over a local ring is faithfully flat.
• Show that \(R\to R[x]\) is always a faithfully flat morphism.
• Show that the category of flat \(R{\hbox{-}}\)modules is closed under direct sums and summands.
• Show that free implies flat.
• Show that nonzero vector spaces are faithfully flat.
• Show that projections \(A\times B\to A\) are flat.
  • Show that if \(A\) is Noetherian, then every flat quotient map is a projection of this form.
• Show that if \(A\to B\) is a faithfully flat morphism of rings, then for $M\in {}_{A}{\mathsf{Mod}} $, \(M\) is (faithfully) flat iff \(B\otimes_A M\) is (faithfully) flat.
• Find a flat but not faithfully flat \({\mathbf{Z}}{\hbox{-}}\)algebra.
• Show that a flat morphism of local rings is faithfully flat.
• If \(f:R\to S\) is flat, show that \(R \left[ { \scriptstyle { { ({ (f^{-1}({\mathfrak{p}})) }^c) }^{-1}} } \right] \to S \left[ { \scriptstyle { { ({{\mathfrak{p}}}^c) }^{-1}} } \right]\) is faithfully flat.
• For \(R\to S\) faithfully flat, show that \(S\) Noetherian implies \(R\) is Noetherian, but the converse is not true.
• Show that if \(f:R\to S\) is faithfully flat then \(f^*: \operatorname{Spec}S\to \operatorname{Spec}R\) is surjective.
  • Conversely, show that if \(f\) is flat and \(f^*\) is surjective, then \(f\) is faithfully flat.
  • Show that if \(f\) is faithfully flat, then \(f^*\) is a quotient map of topological spaces.
• Show that being fiathfully flat is preserved under base change.
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