Let $f:X\to Y$ be an affine morphism of locally Noetherian schemes. By this, we know that $Rf_*\mathcal{F}=f_*\mathcal{F}$ for any quasi-coherent sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules (the equality holds in the derived category of $\mathcal{O}_Y$-modules).

Let $Y'$ be a closed subset of $Y$. Choose a coherent ideal $I \subseteq \mathcal{O}_Y$ defining the closed subscheme with underlying topological space $Y'$. Let $X':=f^{-1}(Y')$ and consider the ideal $J=f^\ast(I)\mathcal{O}_X$ of $\mathcal{O}_X$. Let $\hat{X}$ (resp. $\hat{Y}$) be the formal completion of $X$ at $X'$ (resp. $Y$ at $Y'$). Then, $f$ induces an adic morphism of formal schemes $$ \hat{f}:\hat{X}\longrightarrow \hat{Y}.$$

I would like to know if the formal completion of the above statement is true, namely:

Question: Do we have $R\hat{f}_*\mathcal{F}=\hat{f}_*\mathcal{F}$ for a coherent sheaf $\mathcal{F}$ on $\hat{X}$?

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    $\begingroup$ I think there are some typos: $I$ should be an ideal in $\mathcal{O}_Y$, and $f_*(I)$ should be $f^*(I)$ or $f^{-1}(I)$. I also think this map is not adic in general: for that, you need to have that $f^{-1}(I)\mathcal{O}_X$ is an ideal of definition, i.e. locally it contains a power of $J$ and some power of it is contained in $J$. Equivalently (since everything is locally Noetherian), $f^{-1}(I)\mathcal{O}_X$ and $J$ have the same radical. $\endgroup$ Apr 18 at 15:19
  • $\begingroup$ @PiotrAchinger thank you very much, I have edited my question accordingly $\endgroup$
    – Stabilo
    Apr 18 at 15:28
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    $\begingroup$ Sorry, what is a quasi-coherent sheaf on a formal scheme? $\endgroup$ Apr 18 at 15:55
  • $\begingroup$ @PiotrAchinger I changed to coherent in the sens of ringed space $\endgroup$
    – Stabilo
    Apr 18 at 16:34
  • $\begingroup$ @PiotrAchinger By the way, I believe that the "quasicoherent sheaves" on the formal scheme $\operatorname{Spf}(A_I^\wedge)$ should be the category of derived $I$-complete $A$-modules. $\endgroup$
    – Z. M
    May 3 at 9:14

I never liked the definition of formal completion (as defined in EGA or Hartshorne), so I'll use the following definition from Brian's formal GAGA paper.

Definition: Let $X$ be a locally Noetherian scheme, and $I \subseteq \mathcal{O}_X$ a coherent ideal. The formal completion of $X$ along $V(I)$, is the ringed topos $(X_{\text{Zar}}, \varprojlim \mathcal{O}_X/I^{n+1})$.

Note by Theorem 1.5 of the same paper, the category of modules on this definition of formal completion, is equivalent to the category of modules on the usual definition of formal completion as given in Hartshorne or EGA: Namely, the ringed space whose underlying topological space is $X_0 := V(I)$, equipped with the sheaf of rings that sends a Zariski open $U_0 \hookrightarrow X_0$ to $\varprojlim \Gamma(U_n, \mathcal{O}_{U_n})$. Here $U_n$ is the unique Zariski open in $X_n$ lifting $U_0$.

Ok, so now back to your question. We have locally Noetherian schemes $X,Y$ and a morphism of schemes $f : X \to Y$. In addition, we have a coherent ideal $I \in \text{Coh}(Y)$, and we complete $Y$ along $Y' := V(I)$. In addition, we also complete $X$ along $|f|^{-1}(Y')$.

Now we have a map of ringed topoi $\hat{f} : \widehat{X} \to \widehat{\hspace{1mm}Y}$ with underlying functor $u_\hat{f}$ on objects simply given by pullback. Furthermore, by the usual theory, this gives rise to a right derived functor $$R\hat{f}_\ast : D(\widehat{X}) \to D(\widehat{\hspace{1mm}Y}),$$ on the level of unbounded derived categories of modules.

We want to show for any $\mathcal{F} \in \text{Coh}(\widehat{X})$ that $R^i\hat{f}_\ast \mathcal{F} = 0$ for all $i > 0$. Now since the affines $\operatorname{Spec} B \to Y$ form a base for the Zariski topology on $Y$, and $R^i \hat{f}_\ast \mathcal{F}$ is the sheafification of the presheaf that sends a Zariski open $(\operatorname{Spec} B \hookrightarrow Y)$ to $H^i(u_\hat{f}(\operatorname{Spec} B), \mathcal{F}) = H^i(X_B, \mathcal{F})$ (see here), it is enough to show this last cohomology group is zero.

But this is none other than the i-th cohomology of the sheaf $\mathcal{F}$ on the ringed site $$\widehat{X_B} = (X_B, \varprojlim \mathcal{O}_{X_B}/((f^\ast I)|_{X_B})^{n+1}).$$ As discussed in the beginning of my answer, this is none other than the cohomology of $\mathcal{F}$ on the affine formal scheme $\widehat{X_B}$ in the usual sense. The desired result now follows from the following proposition.

Proposition: Let $A$ be a Noetherian ring an $I \subseteq A$ a finitely generated ideal. Define $X := \operatorname{Spec} A$, $X_n := \operatorname{Spec} A/I^{n+1}$, and $\widehat{X}$ the formal completion of $X$ along $X_0$ (in the sense of Hartshorne). Then for any $\mathcal{F} \in \text{Coh}(\widehat{X})$, we have $$ H^i(\widehat{X}, \mathcal{F}) = 0$$ for all $i > 0$.

To prove this, let us recall that there is a natural equivalence of categories (Theorem 2.3 of Brian's paper) $$\text{Coh}(\widehat{X}) \stackrel{\sim}{\to} \varprojlim \text{Coh}(X_n).$$ The functor from left to right is given by reduction mod $I^{n+1}$ (for all $n \geq 0$), and in the other direction we send an adic system to its inverse limit. So now using this equivalence of categories, we must show for $\{\mathcal{F}_n\} \in \varprojlim \text{Coh}(X_n)$ that $$H^i(\widehat{X}, \varprojlim \mathcal{F}_n) = 0.$$ To this end, we invoke the milnor exact sequence: $$0 \to R^1 \varprojlim H^{i-1}(X_n, \mathcal{F}_n) \to H^i(\widehat{X}, \varprojlim \mathcal{F}_n) \to \varprojlim H^i(X_n, \mathcal{F}_n) \to 0.$$ If $i > 1$, the left and right terms of this sequence vanish by the usual stuff (higher cohomology of any quasi-coherent on an affine is zero). So now we only need to worry about the $i=1$ case. The right dude is still zero so we only need to show that $$R^1 \varprojlim H^0(X_n, \mathcal{F}_n) = 0.$$

Now I claim that the adic system $\{H^0(X_n, \mathcal{F}_n)\}$ has all transition maps surjective, a fortiori Mittag-Leffler and therefore the $R^1$ thingy must be zero. But now the obstruction to surjectivity of $$H^0(X_{n+1}, \mathcal{F}_{n+1}) \to H^0(X_n, \mathcal{F}_n)$$ lies in some $H^1$ thingy which is zero for the same reason as before. Boom! The left and right terms in the Milnor exact sequence vanish for all $i > 0$ and we win.

  • 3
    $\begingroup$ A small nitpick is that you need to do the argument on the formal scheme $\hat Y$ whose underlying topological space is $Y'$. I think what you are saying is: $\hat Y$ has plenty of formal affine opens and their inverse images by $\hat f$ are formal affine open in $\hat X$ and we have theorems A and B for coherent modules on Noetherian affine formal schemes. $\endgroup$
    – Johan
    Apr 19 at 4:16
  • $\begingroup$ @Johan I am using the fact that for a locally Noetherian scheme $Y$ with an ideal $I \subseteq \mathcal{O}_Y$, the formal scheme $(\widehat{Y})$ (obtained by completing along $V(I)$) has the same category of coherent sheaves as that of the ringed topos $(X_{\text{Zar}}, \varprojlim \mathcal{O}_X/I^n)$. This is proven in Brian's formal GAGA paper here. $\endgroup$ Apr 19 at 4:32
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    $\begingroup$ EGA does not explicitly state theorems A and B in this setting; they were after something much more interesting:) Theorem B is an immediate consequence of EGA, III, Cor 3.4.4 applied to the identity morphism on a Noetherian affine formal scheme (by which I mean the formal spectrum of an adic Noetherian topological ring). Enjoy! $\endgroup$
    – Johan
    Apr 19 at 15:28
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    $\begingroup$ OK, now I understand why you said what you said (I was thrown off by the definition of completion in those notes by Brian; it is nonstandard I would say). Yes, it is fine as far as I can tell. Thanks! $\endgroup$
    – Johan
    Apr 19 at 17:16
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    $\begingroup$ @Stabilo No problem. I just added a proof of the cohomological vanishing result for formal completions of affine schemes. $\endgroup$ Apr 20 at 12:41

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