Notes on the $\mathbb A_{{\mathrm {inf}}}$-Cohomology of Integral p-Adic Hodge Theory

Abstract

We present a detailed overview of the construction of the $\mathbb A_{{\mathrm {inf}}}$-cohomology theory from the preprint Integral p-adic Hodge theory, joint with Bhatt and Scholze. We focus particularly on the p-adic analogue of the Cartier isomorphism via relative de Rham–Witt complexes.

Appendices

Appendix 1: $\mathbb A_{{\mathrm {inf}}}$ and Its Modules

The base ring for the cohomology theory constructed in [5] is Fontaine’s infinitesimal period ring $\mathbb A_{{\mathrm {inf}}}:=W(\mathcal {O}^\flat )$, where $\mathcal {O}$ is the ring of integers of a complete, non-archimedean, algebraically closed field $\mathbb C$ of mixed characteristic. Since $\mathcal {O}$ is a perfectoid ring (Example 2), the general theory developed in Sect. 3 (including Sect. 3.3) applies in particular to $\mathcal {O}$. Our goal here is firstly to present a few results which are particular to $\mathcal {O}$ in order to familiarise the reader, who may be encountering these objects for the first time, with $\mathcal {O}$ and $\mathbb A_{{\mathrm {inf}}}$; then we will explain some of the finer theory of modules over $\mathbb A_{{\mathrm {inf}}}$.

We begin by recalling from [24, Sect. 3] that $\mathcal {O}^\flat$ is the ring of integers of $\mathbb C^\flat :={{\,\mathrm{Frac}\,}}\mathcal {O}^\flat$ (footnote 10 shows that $\mathcal {O}^\flat$ is an integral domain), which is a non-archimedean, algebraically closed field of characteristic $p>0$, with the same residue field k as $\mathcal {O}$. The absolute value on $\mathbb C^\flat$ is given by multiplicatively extending the absolute value on $\mathcal {O}^\flat$ given by

$$\begin{aligned} \mathcal {O}^\flat =\varprojlim _{x\mapsto x^p}\mathcal {O}{\mathop {\longrightarrow }\limits ^{x\mapsto x^{(0)}}} \mathcal {O}{\mathop {\longrightarrow }\limits ^{|\cdot |}}\mathbb R_{\ge 0}, \end{aligned}$$

where the first arrow uses the convention introduced just before Lemma 5, and the second arrow is the absolute value on $\mathcal {O}$. The reader may wish to check that this is indeed an absolute value, i.e., satisfies the ultrametric inequality, that $\mathbb C^\flat$ is complete under it, and that the ring of integers is exactly $\mathcal {O}^\flat$. The existence of the canonical projection $\mathcal {O}^\flat \rightarrow \mathcal {O}/p\mathcal {O}$ implies that $\mathcal {O}^\flat$ and $\mathcal {O}$ have the same residue field. Hensel’s lemma shows that $\mathbb C^\flat$ is algebraically closed.⁵⁶

Now we turn to $\mathbb A_{{\mathrm {inf}}}$. Let $t\in \mathbb A_{{\mathrm {inf}}}$ be any element whose image in $\mathbb A_{{\mathrm {inf}}}/p\mathbb A_{{\mathrm {inf}}}=\mathcal {O}^\flat$ belongs to $\mathfrak m^\flat \setminus \{0\}$; examples include $t=[\pi ]$, where $\pi \in \mathfrak m^\flat \setminus \{0\}$, and $t=\xi$, where $\xi$ is any generator of ${\text {Ker}}\,\theta$. Then p, t is a regular sequence, and $\mathbb A_{{\mathrm {inf}}}$ is a $\langle p,t \rangle$-adically complete local ring whose maximal ideal equals the radical of $\langle p,t \rangle$; in short, $\mathbb A_{{\mathrm {inf}}}$ “appears two-dimensional and Cohen–Macaulay”.

In fact, as we will explain the result of this appendix, modules (more precisely, finitely presented modules which become free after inverting p) over $\mathbb A_{{\mathrm {inf}}}$ even behave as though $\mathbb A_{{\mathrm {inf}}}$ were a two-dimensional, regular local ring.⁵⁷ Further details may be found in [5, Sect. 4.2].

Remark 11

In light of the goal, it is sensible to recall the structure of modules over any two-dimensional regular local ring $\varLambda$, such as $\varLambda =\mathcal {O}_K[[T]]$ where $\mathcal {O}_K$ is a discrete valuation ring. Let $\pi ,t\in \varLambda$ be a system of local parameters and $\mathfrak m=\langle \pi ,t \rangle$ its maximal ideal.

• (i) Most importantly, any vector bundle on the punctured spectrum ${{\,\mathrm{Spec}\,}}\varLambda \setminus \{\mathfrak m\}$ extends uniquely to a vector bundle on ${{\,\mathrm{Spec}\,}}\varLambda$.

• (ii) Finitely generated modules over $\varLambda$ are perfect, i.e., admit finite length resolutions by finite free $\varLambda$-modules. (Proof. Immediate from the regularity of $\varLambda$.)

• (iii) If M is any finitely generated $\varLambda$-module, then there is a functorial short exact sequence

  $$\begin{aligned} 0\longrightarrow M_{{\mathrm {tor}}}\longrightarrow M\longrightarrow M_{{\mathrm {free}}}\longrightarrow \overline{M}\longrightarrow 0 \end{aligned}$$

  of $\varLambda$-modules, where $M_{{\mathrm {tor}}}$ is torsion, $M_{{\mathrm {free}}}$ is finite free, and $\overline{M}$ is killed by a power of $\mathfrak m$.

  Proof. $M_{{\mathrm {tor}}}$ is by definition the torsion submodule of M, whence $M/M_{{\mathrm {tor}}}$ restricts to a torsion-free coherent sheaf on the punctured spectrum ${{\,\mathrm{Spec}\,}}\varLambda \setminus \{\mathfrak m\}$; but the punctured spectrum is a regular one-dimensional scheme, so this torsion-free coherent sheaf is necessary a vector bundle, and so extends to a vector bundle on ${{\,\mathrm{Spec}\,}}\varLambda$ by (i); this vector bundle corresponds to a finite free $\varLambda$-module $M_{{\mathrm {free}}}$ which contains $M/M_{{\mathrm {tor}}}$, with the ensuing quotient $\overline{M}$ being supported at the closed point of ${{\,\mathrm{Spec}\,}}\varLambda$.

• (iv) Finite projective modules over $\varLambda [\tfrac{1}{\pi }]$ are finite free.

  Proof. Let N be a finite projective $\varLambda [\tfrac{1}{\pi }]$-module, and pick a finitely generated $\varLambda$-module $N'\subseteq N$ satisfying $N'[\tfrac{1}{\pi }]=N$. Then $N'_\mathfrak p$ is a projective module over $\varLambda _\mathfrak p$ for every non-maximal prime ideal $\mathfrak p\subseteq \varLambda$: indeed either $\pi \notin \mathfrak p$, in which case $N'_\mathfrak p$ is a localisation of the projective module N, or $\mathfrak p=\langle \pi \rangle$, in which case $\varLambda _\mathfrak p$ is a discrete valuation ring and it is sufficient to note that $N_\mathfrak p'$ obviously has no $\pi$-torsion. This means that $N'$ restricts to a vector bundle on the punctured spectrum, whose unique extension to a finite free $\varLambda$-module $N''$ satisfies $N''[\tfrac{1}{\pi }]=N$.

Theorem 11

• (i) (Kedlaya) Any vector bundle on the punctured spectrum

  $$\begin{aligned} {{\,\mathrm{Spec}\,}}\mathbb A_{{\mathrm {inf}}}\setminus \{\text {the max. ideal of }\mathbb A_{{\mathrm {inf}}}\} \end{aligned}$$

  extends uniquely to a vector bundle on ${{\,\mathrm{Spec}\,}}\mathbb A_{{\mathrm {inf}}}$.

• (ii) If M is a finitely presented $\mathbb A_{{\mathrm {inf}}}$-module such that $M[\tfrac{1}{p}]$ is finite free over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$, then M is perfect (again, this means that M admits a finite length resolution by finite free $\mathbb A_{{\mathrm {inf}}}$-modules).

• (iii) If M is a finitely presented $\mathbb A_{{\mathrm {inf}}}$-module such that $M[\tfrac{1}{p}]$ is finite free over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$, then there is functorial short exact sequence of $\mathbb A_{{\mathrm {inf}}}$-modules

  $$\begin{aligned} 0\longrightarrow M_{{\mathrm {tor}}}\longrightarrow M\longrightarrow M_{{\mathrm {free}}}\longrightarrow \overline{M}\longrightarrow 0 \end{aligned}$$

  such that: $M_{{\mathrm {tor}}}$ is a perfect $\mathbb A_{{\mathrm {inf}}}$-module killed by a power of p; $M_{{\mathrm {free}}}$ is a finite free $\mathbb A_{{\mathrm {inf}}}$-module; and $\overline{M}$ is a perfect $\mathbb A_{{\mathrm {inf}}}$-module killed by a power of the ideal $\langle p,t \rangle$.

• (iv) Finite projective modules over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$ are finite free.

Proof

We have nothing to say about (i) here, and refer instead to [5, Lemma 4.6]. We will also only briefly comment on the remaining parts of the theorem, since these self-contained results may be easily read in [5, Sect. 4.2].

(ii) By clearing denominators in a basis for $M[\tfrac{1}{p}]$ to construct a finite free $\mathbb A_{{\mathrm {inf}}}$-module $M'\subseteq M$ satisfying $M'[\tfrac{1}{p}]=M[\tfrac{1}{p}]$, we may reduce to the case that M is killed by a power of p, i.e., M is a $\mathbb A_{{\mathrm {inf}}}/p^r\mathbb A_{{\mathrm {inf}}}=W_r(\mathcal {O}^\flat )$-module for some $r\gg 0$. By an induction on r, using that $W_r(\mathcal {O}^\flat )$ can be shown to be coherent [5, Proposition 3.24] (this is not a trivial result), one can reduce to the case $r=1$, in which case it easily follows from the classification of finitely presented modules over the valuation ring $\mathcal {O}^\flat$: they have the shape $(\mathcal {O}^\flat )^n\oplus \mathcal {O}^\flat /a_1\mathcal {O}^\flat \oplus \cdots \oplus \mathcal {O}^\flat /a_m\mathcal {O}^\flat$, for some $n\ge 1$ and $a_i\in \mathcal {O}^\flat$, and so in particular are perfect.

(iii) This is proved similarly to the analogous assertion in the previous remark.

(iv) This is proved exactly as the analogous assertion in the previous remark, once it is checked that the localisation $\mathbb A_{{\mathrm {inf}},\langle p \rangle }$ is a discrete valuation ring.

Corollary 5

Let M be a finitely presented $\mathbb A_{{\mathrm {inf}}}$-module such that $M[\tfrac{1}{p}]$ is finite free over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$. If either $M\otimes _{\mathbb A_{{\mathrm {inf}}}}W(k)$ or $M\otimes _{\mathbb A_{{\mathrm {inf}}}}\mathcal {O}$ is p-torsion-free (equivalently, finite free over W(k) or $\mathcal {O}$ respectively), then M is a finite free $\mathbb A_{{\mathrm {inf}}}$-module.

Proof

It follows easily from the hypothesis that the map $M\rightarrow M_{{\mathrm {free}}}$ in Theorem 11(iii) becomes an isomorphism after tensoring by W(k) or $\mathcal {O}$; hence $M[\tfrac{1}{p}]$ and $M\otimes _{\mathbb A_{{\mathrm {inf}}}}k$ have the same rank over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$ and k respectively. But an easy Fitting ideal argument shows that if N is a finitely presented module over a local integral domain R satisfying $\dim _{{{\,\mathrm{Frac}\,}}R}(N\otimes _R{{\,\mathrm{Frac}\,}}R)=\dim _{k(R)}(N\otimes _Rk(R))$, then N is finite free over R.

To state and prove the next corollary we use the elements $\xi ,\xi _r,\mu \in \mathbb A_{{\mathrm {inf}}}$ constructed in Sect. 3.3:

Corollary 6

Let M be a finitely presented $\mathbb A_{{\mathrm {inf}}}$-module, and assume:

• $M[\tfrac{1}{p\mu }]$ is a finite free $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p\mu }]$-module of the same rank as the W(k)-module $M\otimes _{\mathbb A_{{\mathrm {inf}}}}W(k)$.

• There exists a Frobenius-semi-linear endomorphism of M which becomes an isomorphism after inverting $\xi$.

Then $M[\tfrac{1}{p}]$ is finite free over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$.

Proof

We must show that each Fitting ideal of the $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$-module $M[\tfrac{1}{p}]$ is either 0 or $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$; indeed, this means exactly that $M[\tfrac{1}{p}]$ is finite projective over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$, which is sufficient by Theorem 11(iv). Since Fitting ideals behave well under base change, it is equivalent to prove that the first non-zero Fitting ideal $J\subseteq \mathbb A_{{\mathrm {inf}}}$ of M contains a power of p. Again using that Fitting ideals base change well, our hypotheses imply that $J\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p\mu }]=\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p\mu }]$ and $JW(k)\ne 0$; that is, J contains a power of $p\mu$ and $J+W(\mathfrak m^\flat )$ contains a power of p, where $W(\mathfrak m^\flat ):={\text {Ker}}(\mathbb A_{{\mathrm {inf}}}\rightarrow W(k))$. Because of the existence of the Frobenius on M, we also know that J and $\varphi (J)$ are equal up to a power of $\varphi (\xi )$. In conclusion, we may pick $N\gg$ such that

• (i) $(p\mu )^N\in J$;

• (ii) $p^N\in J+W(\mathfrak m)$;

• (iii) $\varphi (\xi )^N\varphi (J)\subseteq J$ and $\varphi (\xi )^NJ\subseteq \varphi (J)$.

Since $W(\mathfrak m)$ is the p-adic completion of the ideal generated by $\varphi ^{-r}(\mu ^N)$, for all $r\ge 0$,⁵⁸ observation (ii) lets us write $p^N=\alpha +\beta \varphi ^{-r}(\mu ^N)+\beta 'p^{N+1}$ for some $\alpha \in J$ and $\beta ,\beta '\in \mathbb A_{{\mathrm {inf}}}$, and $r\ge 0$. Since $1-\beta 'p$ is invertible, we may easily suppose that $\beta '=0$, i.e., $p^N=\alpha +\beta \varphi ^{-r}(\mu ^N)$. Multiplying through by $p^N\xi _r^N$ gives $\xi _r^Np^{2N}=p^N\xi _r^N\alpha +\beta p^N\mu ^N$, which belongs to J by (i) and (ii).

We claim, for any $a,i\ge 1$, that

$$\begin{aligned} \xi _r^ap^{\mathrm {some~power}}\in J\implies \xi _r^{a-1}p^{\mathrm {some~other~power}}\in J. \end{aligned}$$

A trivial induction then shows that J contains a power of p, thereby completing the proof, and so it remains only to prove this claim. Suppose $\xi _r^ap^b\in J$ for some $a,b,i\ge 1$. Then $\varphi ^r(\xi _r)^ap^b\in \varphi ^r(J)$, and so $\varphi ^r(\xi _r)^{a+N}p^b\in J$ (since an easy generalisation of (iii) implies that $\varphi ^r(\xi _r)^N\varphi ^r(J)\subseteq J$). But $\varphi ^r(\xi _r)\equiv p^r$ mod $\xi _r$, so we may write $\varphi ^r(\xi _r)^{a+N}=p^{r(a+N)}+\alpha \xi _r$ for some $\alpha \in \mathbb A_{{\mathrm {inf}}}$ and thus deduce that $J\ni (p^{r(a+N)}+\alpha \xi _r)p^b=p^{r(a+N)+b}+\alpha \xi _rp^b$. Now multiply through by $\xi _r^{a-1}$ and use the supposition to obtain $\xi _r^{a-1}p^{r(a+N)+b}\in J$, as required.

We will also need the following to eliminate the appearance of higher Tors in the crystalline specialisation of the $\mathbb A_{{\mathrm {inf}}}$-cohomology theory:

Lemma 22

Let M be an $\mathbb A_{{\mathrm {inf}}}$-module such that $M[\tfrac{1}{p}]$ is flat over $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$. Then ${{\,\mathrm{Tor}\,}}_*^{\mathbb A_{{\mathrm {inf}}}}(M,W(k))=0$ for $*>1$.

Proof

Let $[\mathfrak m^\flat ]\subseteq W(\mathfrak m^\flat )$ be the ideal of $\mathbb A_{{\mathrm {inf}}}$ which is generated by Teichmüller lifts of elements of $\mathfrak m^\flat$. We first observe that $\mathbb A_{{\mathrm {inf}}}/[\mathfrak m^\flat ]$ is p-torsion-free and has Tor-dimension $=1$ over $\mathbb A_{{\mathrm {inf}}}$: indeed, $[\mathfrak m^\flat ]$ is the increasing union of the ideals $[\pi ]\mathbb A_{{\mathrm {inf}}}$, for $\pi \in \mathfrak m^\flat \setminus \{0\}$, and the claims are true for $\mathbb A_{{\mathrm {inf}}}/[\pi ]\mathbb A_{{\mathrm {inf}}}$ since $p,[\pi ]$ is a regular sequence of $\mathbb A_{{\mathrm {inf}}}$.

Next, since $W_r(\mathfrak m^\flat )$ is generated by the analogous Teichmüller lifts in $W_r(\mathcal {O}^\flat )=\mathbb A_{{\mathrm {inf}}}/p^r\mathbb A_{{\mathrm {inf}}}$, for any $r\ge 1$ (c.f., footnote 20), the quotient $W(\mathfrak m^\flat )/[\mathfrak m^\flat ]$ is p-divisible. Combined with the previous observation, it follows that $W(\mathfrak m^\flat )/[\mathfrak m^\flat ]$ is uniquely p-divisible, i.e., an $\mathbb A_{{\mathrm {inf}}}[\tfrac{1}{p}]$-module, whence

$$\begin{aligned} {{\,\mathrm{Tor}\,}}_*^{\mathbb A_{{\mathrm {inf}}}}(W(\mathfrak m^\flat )/[\mathfrak m^\flat ],M)={{\,\mathrm{Tor}\,}}_*^{\mathbb A_{{\mathrm {inf}}}\left[ \tfrac{1}{p}\right] }\left( W(\mathfrak m^\flat )/[\mathfrak m^\flat ],M\left[ \tfrac{1}{p}\right] \right) , \end{aligned}$$

which vanishes for $*>0$ by the hypothesis on M. Combining this with the short exact sequence

$$\begin{aligned} 0\rightarrow W(\mathfrak m^\flat )/[\mathfrak m^\flat ]\rightarrow \mathbb A_{{\mathrm {inf}}}/[\mathfrak m^\flat ]\rightarrow \mathbb A_{{\mathrm {inf}}}/W(\mathfrak m^\flat )=W(k)\rightarrow 0 \end{aligned}$$

and the initial observation about the Tor-dimension of the middle term completes the proof.

Appendix 2: Two Lemmas on Koszul Complexes

Let R be a ring, and $g_1,\ldots ,g_d\in R$. The associated Koszul complex will be denoted by $K_R(g_1,\ldots ,g_d)=\bigotimes _{i=1}^d K_R(g_i)$, where $K_R(g_i):=[R\xrightarrow {g_i}R]$. Here we state two useful lemmas concerning such complexes, the second of which describes the behaviour of the décalage functor.

Lemma 23

Let $g\in R$ be an element which divides $g_1,\ldots ,g_d$, and such that $g_i$ divides g for some i. Then there are isomorphisms of R-modules

$$\begin{aligned} H^n(K_R(g_1,\ldots ,g_d))\cong R[g]^{\left( {\begin{array}{c}d-1\\ n\end{array}}\right) }\oplus R/gR^{\left( {\begin{array}{c}d-1\\ n-1\end{array}}\right) } \end{aligned}$$

for all $n\ge 0$.

Proof

[5, Lemma 7.10].

Lemma 24

Let $f\in R$ be a non-zero-divisor such that, for each i, either f divides $g_i$ or $g_i$ divides f. Then:

• If f divides $g_i$ for all i, then $\eta _f K_R(g_1,\ldots ,g_d)\cong K_R(g_1/f,\ldots ,g_d/f)$.

• If $g_i$ divides f for some i, then $\eta _f K_R(g_1,\ldots ,g_d)$ is acyclic.

Proof

[5, Lemma 7.9].