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The Stacks project

Lemma 15.90.10. Let R be a ring, let f \in R, and let R \to R' be a ring map which induces isomorphisms R/f^ nR \to R'/f^ nR' for n > 0. The sequence (15.90.9.1) is

  1. exact on the right,

  2. exact on the left if and only if M[f^\infty ] \to (M \otimes _ R R')[f^\infty ] is injective, and

  3. exact in the middle if and only if M[f^\infty ] \to (M \otimes _ R R')[f^\infty ] is surjective.

Thus M is glueable for (R \to R', f) if and only if M[f^\infty ] \to (M \otimes _ R R')[f^\infty ] is bijective. If (R \to R', f) is a glueing pair, then M is glueable for (R \to R', f) if and only if M[f^\infty ] \to (M \otimes _ R R')[f^\infty ] is injective. For example, if (R, f) is a glueing pair, then M is glueable if and only if M[f^\infty ] \to (M \otimes _ R R^\wedge )[f^\infty ] is injective.

Proof. We will use the results of Lemma 15.90.6 without further mention. The functor M \otimes _ R - is right exact (Algebra, Lemma 10.12.10) hence we get (1).

The kernel of M \to M \otimes _ R R_ f = M_ f is M[f^\infty ]. Thus (2) follows.

If the sequence is exact in the middle, then elements of the form (x, 0) with x \in (M \otimes _ R R')[f^\infty ] are in the image of the first arrow. This implies that M[f^\infty ] \to (M \otimes _ R R')[f^\infty ] is surjective. Conversely, assume that M[f^\infty ] \to (M \otimes _ R R')[f^\infty ] is surjective. Let (x, y) be an element in the middle which maps to zero on the right. Write y = y'/f^ n for some y' \in M. Then we see that f^ n x - y' is annihilated by some power of f in M \otimes _ R R'. By assumption we can write f^ nx - y' = z for some z \in M[f^\infty ]. Then y = y''/f^ n where y'' = y' + z is in the kernel of M \to M/f^ nM. Hence we see that y can be represented as y'''/1 for some y''' \in M. Then x - y''' is in (M \otimes _ R R')[f^\infty ]. Thus x - y''' = z' \in M[f^\infty ]. Then (x, y'''/1) = (y''' + z', (y''' + z')/1) as desired.

If (R \to R', f) is a glueing pair, then (15.90.9.1) is exact in the middle for any M by Algebra, Lemma 10.12.10. This gives the penultimate statement of the lemma. The final statement of the lemma follows from this and the fact that (R, f) is a glueing pair if and only if (R \to R^\wedge , f) is a glueing pair. \square


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