Faltings' finiteness theorem for local cohomology up to order n: the missing converse on $S_n$ formal fibers, and lifting along $I$-adic completion

Let $A$ be a commutative noetherian ring, $M$ a finite $A$-module, and $Z$ a closed or specialization-stable subset of $\operatorname{Spec} A$. The local cohomology modules $H^i_Z(M)$ are generally not finitely generated, and Faltings' 1981 Endlichkeitssatz identifies how far structural hypotheses on $A$ restore finite generation in degrees up to $n$: on a GB-ring (Ratliff's going-between condition) whose localizations have $S_n$ formal fibers, the inequality $\operatorname{depth} M_{\mathfrak p} + \operatorname{codim}(\mathfrak p, Z) > t$ for all $\mathfrak p \notin Z$ forces $H^i_Z(M)$ to be finite for $i \le t \le n$; conversely, the weak theorem up to order 1 already forces the GB property, and for $n \le 2$ the methods of Faltings' Satz 4 also force the $S_n$ condition on formal fibers. The paper's closing questions remain open in their central parts: (a) does the finiteness theorem up to order $n$, required for all essentially-of-finite-type algebras, force $S_n$ formal fibers for $n \ge 3$ — completing the characterization of the rings satisfying the weak or strong theorem up to order $n$; and (b) if $A$ is complete and separated in the $I$-adic topology and $A/I$ satisfies a finiteness or annihilator theorem, must $A$ satisfy it too? The companion annihilator-theorem side is, by contrast, complete in the all-orders form: Kawasaki proved that the conditions (C1)–(C3) (universal catenarity, Cohen–Macaulay formal fibers of all localizations, open Cohen–Macaulay loci) are sufficient as well as necessary, which answers the paper's question (c) in the all-orders form; the finite-order non-local analogue of Satz 4 nevertheless remains open (part (3) below).

Background

Local cohomology. Let $A$ be a commutative noetherian ring and $X = \operatorname{Spec} A$. A subset $Z \subseteq X$ is stable under specialization if it contains, with any prime $\mathfrak p$, every prime containing $\mathfrak p$; closed subsets $V(\mathfrak a) = \{\mathfrak p \supseteq \mathfrak a\}$ and their unions are of this kind. For such $Z$ the functor $\Gamma_Z(M) = \{m \in M : \operatorname{Supp}(Am) \subseteq Z\}$ is left exact, and its right derived functors $H^i_Z(-)$ are the local cohomology functors with support in $Z$; for $Z = V(\mathfrak a)$ one writes $H^i_{\mathfrak a}(-)$. These modules measure the failure of sections of $M$ to extend across the complement of $Z$, but they are almost never finitely generated: for a non-zerodivisor $x$ in a domain $A$ one has $H^1_{(x)}(A) \cong A_x/A$, which is not a finite $A$-module. Determining when finite generation can nevertheless be forced — for which rings, in which degrees, and under which hypotheses on the module — is a foundational problem going back to Grothendieck's SGA 2.

The two theorems. Grothendieck's finiteness theorem covers rings that admit a dualizing complex: a bounded complex of injective modules with finitely generated cohomology inducing a duality on the derived category of finite modules (a noetherian ring admits one exactly when it is a homomorphic image of a Gorenstein ring — a ring of finite injective dimension over itself — of finite Krull dimension). Faltings (1978) proved the corresponding annihilation and finiteness statements for rings with dualizing complexes and for homomorphic images of regular rings; his 1981 paper determined how far the statements survive for arbitrary noetherian rings. Say that $A$ satisfies the weak finiteness theorem up to order $n$ if for every closed $Z \subseteq X$, every finite $A$-module $M$, and every $t \le n$, the inequality $\operatorname{depth} M_{\mathfrak p} + \operatorname{codim}(\mathfrak p, Z) > t$ for all $\mathfrak p \in X \setminus Z$ implies that $H^i_Z(M)$ is finitely generated for $i = 0, \ldots, t$; here $\operatorname{codim}(\mathfrak p, Z) = \min\{\operatorname{ht}(\mathfrak q/\mathfrak p) : \mathfrak q \in V(\mathfrak p) \cap Z\}$, with value $\infty$ when the set is empty, where $\operatorname{ht}(\mathfrak q/\mathfrak p)$ is the height of the prime $\mathfrak q/\mathfrak p$ (the supremum of the lengths of the chains of prime ideals descending from it), and depth is the length of a maximal regular sequence in $A_{\mathfrak p}$, a regular sequence being a sequence of elements each of which is a non-zerodivisor modulo the ideal generated by its predecessors. The strong finiteness theorem up to order $n$ makes the same demand for all specialization-stable $Z$. The companion annihilator theorem for $A$ states that for specialization-stable $Y, Z \subseteq X$, a finite module $M$, and an integer $n$, the inequality $\operatorname{ht}(\mathfrak p/\mathfrak q) + \operatorname{depth} M_{\mathfrak q} \ge n$ for all $\mathfrak q \in X \setminus Y$ and $\mathfrak p \in V(\mathfrak q) \cap Z$ is equivalent to the existence of an ideal $\mathfrak b$ with $V(\mathfrak b) \subseteq Y$ annihilating $H^0_Z(M), \ldots, H^{n-1}_Z(M)$. The annihilator theorem is a strengthening of the finiteness theorem, so its validity is the stronger property.

The ring-theoretic conditions. A chain of prime ideals is saturated if each inclusion is immediate. $A$ is a GB-ring ('going between', a condition introduced and studied by Ratliff) if for every ring $B$ finite as an $A$-module, every saturated chain of primes of $B$ contracts to a saturated chain of $A$. $A$ is universally catenary if every finitely generated $A$-algebra is catenary (all saturated chains of primes between two fixed primes have the same length). For $\mathfrak p \in X$, the formal fibers of the localization $A_{\mathfrak p}$ are the fibers of the completion map $\operatorname{Spec} \widehat{A_{\mathfrak p}} \to \operatorname{Spec} A_{\mathfrak p}$, where $\widehat{A_{\mathfrak p}} = \varprojlim_r A_{\mathfrak p}/(\mathfrak p A_{\mathfrak p})^r$ is the completion (the inverse limit of the quotients by the powers of the maximal ideal); explicitly, they are the rings $\widehat{A_{\mathfrak p}} \otimes_{A_{\mathfrak p}} \kappa(\mathfrak q)$ for $\mathfrak q \in \operatorname{Spec} A_{\mathfrak p}$, where $\kappa(\mathfrak q)$ is the residue field at $\mathfrak q$. A ring satisfies Serre's condition $S_n$ when the depth of each of its localizations is at least the minimum of $n$ and its dimension. Excellent rings (Grothendieck's class of noetherian rings with geometrically regular formal fibers and open regularity loci, comprising fields, $\mathbb{Z}$, and all finitely generated algebras over them) and algebras of finite type over a field have Cohen–Macaulay formal fibers (a ring being Cohen–Macaulay when the depth of each of its localizations equals its dimension), but general noetherian rings need not, and completion without such hypotheses is badly behaved — this is the source of the difficulty of the subject. An $A$-algebra is essentially of finite type if it is a localization of a finitely generated $A$-algebra; requiring the theorems for all essentially-of-finite-type $A$-algebras is the stability that makes these classes natural.

Faltings' 1981 results. The paper proves: a local-global principle (Satz 1) reducing finiteness of the modules $H^i_Z(M)$ to the local case; sufficiency (Satz 2): a GB-ring whose localizations have $S_n$ formal fibers satisfies the weak finiteness theorem up to order $n$, and if in addition the $S_{n+1}$ locus $\{\mathfrak p : A_{\mathfrak p} \text{ is } S_{n+1}\}$ is Zariski-open, the strong one; and necessity (Satz 3): the weak theorem up to order 1 forces the GB property, and the strong theorem up to order $n$ for all essentially-of-finite-type $A$-algebras forces $A$ to be universally catenary with open $S_{n+1}$ loci. The companion Satz 4, for $A$ local, is a complete characterization: all essentially-of-finite-type $A$-algebras satisfy the annihilator theorem (weak or strong) up to order $n$ if and only if $A$ is universally catenary and its formal fibers satisfy $S_n$. The paper closes with the questions that this record is about.

Later development. Takesi Kawasaki completed the annihilator-theorem side for arbitrary noetherian rings: the annihilator theorem holds for $A$ whenever (C1) $A$ is universally catenary, (C2) all formal fibers of all localizations of $A$ are Cohen–Macaulay, and (C3) every finitely generated $A$-algebra has open Cohen–Macaulay locus — and these conditions are necessary, the necessity direction being already in Faltings' 1981 paper. These conditions, called CM-excellence by Kęstutis Česnavičius, have become a standard framework: Kawasaki used them for the finiteness of Cousin cohomologies (the cohomology of Cousin complexes, chain complexes built from the residue fields at the points of the prime spectrum and graded by codimension) and, in earlier related work, for arithmetic Macaulayfication (modifying a ring or scheme outside a prescribed locus so as to make it Cohen–Macaulay there), Česnavičius for the Macaulayfication of noetherian schemes, and Ryo Takahashi proved Faltings' annihilator theorem for bounded complexes over CM-excellent rings, obtaining a complete classification of the t-structures (the datum on a triangulated category that determines which complexes behave as if concentrated in non-negative, respectively non-positive, degrees) of the bounded derived category. The finiteness-theorem converse and the lifting problem posed at the end of Faltings' paper, together with the finite-order non-local analogue of Satz 4, remain, by contrast, open.

Problem Statement

Let $A$ be a commutative noetherian ring, with the notions of weak/strong finiteness theorem up to order $n$, annihilator theorem, GB-ring, universal catenarity, formal fibers, Serre condition $S_n$, open $S_{n+1}$ loci, essentially-of-finite-type algebras, and Kawasaki's conditions (C1)–(C3) as defined in the background. The problem consists of the two questions with which Faltings' 1981 paper closes that remain open, together with the finite-order residue of the third.

(1) Characterization (converse). Complete the characterization of the noetherian rings for which the weak or the strong finiteness theorem up to order $n$ holds. Faltings proved sufficiency (GB-ring with $S_n$ formal fibers on all localizations, plus open $S_{n+1}$ loci for the strong version) and necessity in low order (order 1 forces the GB property; for $n \le 2$, requiring the finiteness theorem up to order $n$ for all essentially-of-finite-type $A$-algebras forces $S_n$ formal fibers, by the methods of Satz 4). Determine whether, for $n \ge 3$, the finiteness theorem up to order $n$ — required for all essentially-of-finite-type $A$-algebras — implies that the formal fibers of the localizations of $A$ satisfy $S_n$ or a similar condition, and more generally find exact necessary and sufficient conditions for the weak and the strong finiteness theorem up to order $n$. A counterexample — a noetherian ring satisfying the finiteness theorem up to order $n$ whose formal fibers fail $S_n$ — would refute the expected converse and show that the true conditions are strictly weaker. Part (1) asks for the general characterization — exact necessary and sufficient conditions for the weak and the strong finiteness theorem up to order $n$; the $n \ge 3$ implication above is the concrete open instance this subsumes.

(2) Lifting. Let $I$ be an ideal of a noetherian ring $A$ that is complete and separated in the $I$-adic topology, that is, such that the natural map $A \to \varprojlim_r A/I^r$ to the inverse limit of the quotients by the powers of $I$ is an isomorphism (surjectivity is completeness, injectivity is separatedness). If $A/I$ satisfies one of the finiteness theorems or the annihilator theorem, must $A$ satisfy it? The only known lifting result concerns dualizing complexes, whose existence ascends from $A/I$ to $A$ (Faltings 1978); since a dualizing complex implies the annihilator theorem, the question is whether the weaker ring-theoretic hypotheses (GB, $S_n$ formal fibers, universal catenarity, open loci — or Kawasaki's conditions (C1)–(C3)) ascend, or whether finiteness can be established directly over $A$. A negative answer would be an explicit complete separated $A$ with $A/I$ in a known-good class and one specialization-closed $Z$, one finite module $M$ and one degree $i$ in range for which $H^i_Z(M)$ fails finite generation (or the required annihilation).

(3) Residual of the non-local extension. Kawasaki's theorem characterizes, for arbitrary noetherian $A$, the validity of the annihilator theorem in all orders by (C1)–(C3), which answers the source question 'can Satz 4 be extended to non-local rings?' in the all-orders form. The finite-order analogue — a non-local version of Satz 4 with $S_n$ formal fibers for fixed $n$ — remains open. Determine whether this finite-order non-local analogue holds, by proving it or by exhibiting a counterexample.

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