Restricted Flat Dimension

• Restricted flat dimension is a homological invariant that measures flatness by restricting tests to modules with finite flat dimension.
• It encompasses two formulations: a depth/grade approach for finitely generated modules (large Rfd and small rfd) and a Tor-theoretic perspective for arbitrary modules.
• The invariant connects to local cohomology, Cohen–Macaulay and Gorenstein dimensions, and supports structural properties such as the Govorov–Lazard direct-limit framework.

Searching arXiv for papers on restricted flat dimension and related usages. Restricted flat dimension is a homological invariant for modules over commutative noetherian rings that measures vanishing of $\operatorname{Tor}$ only against modules of finite flat dimension, rather than against all modules. In the commutative-algebra literature, the term appears in two closely related forms: a depth/grade-theoretic formulation for finitely generated modules, including a large invariant $\operatorname{Rfd}$ and a small invariant $\operatorname{rfd}$, and a $\operatorname{Tor}$-theoretic formulation that tests against the class $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$ (Ando, 2021, Dey et al., 27 Aug 2025). The invariant is closely tied to local cohomology, annihilator theorems, Cohen–Macaulay and Gorenstein homological dimensions, and approximation properties such as the Govorov–Lazard property. The same phrase is also used in several unrelated areas, including differential-flatness theory, extremal combinatorics over $F_2^r$, and quasi-isometry invariants of right-angled Artin groups; these usages are terminologically analogous but mathematically distinct (Ollivier et al., 2017, Blokhuis et al., 2013, Huang, 2014).

1. Algebraic definitions and basic formulations

For a commutative Noetherian ring $R$ and a finitely generated $R$-module $M$, the large restricted flat dimension is defined by

$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$

and equivalently by

$\operatorname{Rfd}$0

The small restricted flat dimension is

$\operatorname{Rfd}$1

and satisfies

$\operatorname{Rfd}$2

One always has

$\operatorname{Rfd}$3

and $\operatorname{Rfd}$4 if and only if $\operatorname{Rfd}$5, equivalently $\operatorname{Rfd}$6 (Ando, 2021).

A second formulation, emphasized in later work, is available for arbitrary $\operatorname{Rfd}$7-modules over a commutative noetherian ring: $\operatorname{Rfd}$8 where

$\operatorname{Rfd}$9

This is contrasted with the classical flat dimension

$\operatorname{rfd}$0

so the restricted invariant differs by testing $\operatorname{rfd}$1 only against modules of finite flat dimension (Dey et al., 27 Aug 2025).

This restricted-testing perspective is the core conceptual distinction. A plausible implication is that restricted flat dimension interpolates between ordinary flat dimension and more specialized relative homological dimensions, especially in settings where finite-flat-dimension modules form a robust test class.

2. Large versus small restricted flat dimension

The distinction between $\operatorname{rfd}$2 and $\operatorname{rfd}$3 is structural rather than cosmetic. The large invariant is expressed through local depth defects, while the small invariant is expressed through grade defects. Their comparison is immediate from the definitions, but equality requires additional hypotheses (Ando, 2021).

The depth/grade formalism leads to a local viewpoint. For a fixed prime $\operatorname{rfd}$4, one studies the localized conditions

$\operatorname{rfd}$5

These conditions are not automatically equivalent. However, if $\operatorname{rfd}$6, then $\operatorname{rfd}$7, and if $\operatorname{rfd}$8, then again $\operatorname{rfd}$9, where $\operatorname{Tor}$0 is the annihilator condition described below (Ando, 2021).

The later Govorov–Lazard work adopts the notation

$\operatorname{Tor}$1

and treats bounded restricted flat dimension as a hierarchy of module classes. Under Cohen–Macaulay hypotheses with a pointwise dualizing module, these classes coincide with bounded Cohen–Macaulay flat dimension: $\operatorname{Tor}$2 This identifies restricted flat dimension with a recognized relative dimension theory in a substantial class of rings (Dey et al., 27 Aug 2025).

3. Local cohomology characterization

A central result connects restricted flat dimension to vanishing patterns of local cohomology. For an ideal $\operatorname{Tor}$3, the standard criterion states

$\operatorname{Tor}$4

Using this, one obtains the equivalence

$\operatorname{Tor}$5

Equivalently,

$\operatorname{Tor}$6

This is one of the key operational characterizations of the small restricted flat dimension (Ando, 2021).

The significance of this equivalence is that a grade-theoretic inequality becomes a uniform family of local-cohomological vanishing conditions. This suggests that restricted flat dimension is naturally positioned between homological dimension theory and the annihilator/vanishing theory of local cohomology.

A further refinement gives a quantitative annihilator statement. If $\operatorname{Tor}$7 is prime and

$\operatorname{Tor}$8

then there exists $\operatorname{Tor}$9 such that

$$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$0

In particular, if $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$1, then one gets honest vanishing of $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$2 in those low degrees. The proof proceeds by induction on $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$3, using the syzygy sequence $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$4 and the fact that $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$5 (Ando, 2021).

4. Annihilators, resolving subcategories, and dominance

For a prime $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$6, one considers the subcategory

$$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$7

Corollary 3.3 of Ando’s paper asserts that $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$8 is a dominant resolving subcategory of $$F_{<\infty}=\{F\in \operatorname{Mod}R\mid \operatorname{fd}_R F<\infty\}$$9 (Ando, 2021). Here “dominant” means that for each $F_2^r$0, some syzygy of the residue field $F_2^r$1 lies in $F_2^r$2.

The same work organizes the local theory into the implications

$F_2^r$3

where $F_2^r$4 denotes the membership condition $F_2^r$5. Under the extra hypotheses already noted—finite $F_2^r$6 at $F_2^r$7, Cohen–Macaulay defect at most $F_2^r$8, or equality of grade and depth on generalizations—these conditions become equivalent (Ando, 2021).

These results place restricted flat dimension inside the broader framework of resolving subcategories and approximation theory. The dominant character of $F_2^r$9 shows that the local-cohomological consequences of small restricted flat dimension are stable enough to control syzygies of residue fields, which is a strong structural property.

5. Govorov–Lazard property and approximation theory

Recent work strengthens the structural theory of bounded restricted flat dimension. Let

$R$0

Over a Cohen–Macaulay commutative noetherian ring admitting a pointwise dualizing module $R$1, for each integer $R$2 the class

$R$3

satisfies the Govorov–Lazard property: every module in the class is a direct limit of finitely generated modules in the same class (Dey et al., 27 Aug 2025).

The proof factors through the trivial extension ring $R$4. Under the stated hypotheses,

$R$5

and $R$6 is Gorenstein. The Govorov–Lazard property is then deduced from the bounded Gorenstein flat theory over the trivial extension and descended back to $R$7 (Dey et al., 27 Aug 2025).

The zero-dimensional case is especially explicit. If $R$8 is a commutative noetherian ring of finite Krull dimension, then

$R$9

satisfies the Govorov–Lazard property: every module in $R$0 is a direct limit of finitely generated modules in $R$1 (Dey et al., 27 Aug 2025). In this case $R$2 means precisely that

$R$3

Approximation-theoretic consequences also follow. If $R$4 is Cohen–Macaulay with a pointwise dualizing module, then

$R$5

is preenveloping in the category of finitely generated $R$6-modules; equivalently, every finitely generated module admits a morphism into a restricted flat module that is universal for maps into restricted flat modules. An analogous result holds in the finite-Krull-dimension “almost Cohen–Macaulay” case (Dey et al., 27 Aug 2025).

At the level of torsion theories, there is a hereditary Tor-pair

$R$7

where

$R$8

In finite dimension, Baer-criterion arguments identify

$R$9

forcing $M$0 to be generated by its finitely generated objects under direct limits (Dey et al., 27 Aug 2025).

6. Examples, special cases, and relations to other dimensions

Several explicit computations anchor the theory. If $M$1, then

$M$2

If $M$3 is local and $M$4 is a nonzerodivisor, then for $M$5 one has

$M$6

More generally, if $M$7 is generated by an $M$8-regular sequence of length $M$9, then

$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$0

(Ando, 2021).

The later structural theory places these examples into a broader network of relative dimensions. If $$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$1 is Gorenstein, then restricted flat dimension coincides with Gorenstein flat dimension, and the direct-limit theorem recovers the classical result that every Gorenstein flat module is a direct limit of finitely generated Gorenstein projectives (Dey et al., 27 Aug 2025). Over Cohen–Macaulay rings with a dualizing module, restricted flat dimension coincides with Cohen–Macaulay flat dimension (Dey et al., 27 Aug 2025).

The following table summarizes the main algebraic formulations appearing in the cited literature.

Invariant	Formula	Context
$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$2	$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$3	finitely generated modules (Ando, 2021)
$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$4	$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$5	finitely generated modules (Ando, 2021)
$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$6	$$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$7	arbitrary modules (Dey et al., 27 Aug 2025)

This suggests that “restricted flat dimension” is best understood not as a single formula but as a relative homological paradigm: a dimension theory obtained by restricting the test class used to detect flatness defects.

7. Other uses of the term and terminological caution

The phrase “restricted flat dimension” is not unique to commutative algebra. In differential-flatness theory, Ollivier and Sadik study what their exposition calls the “restricted flat dimension” phenomenon: for diffieties of differential dimension $$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$8, any parametrizable subsystem of a flat system is flat, equivalently exogenous and endogenous flatness coincide in differential dimension at most $$\operatorname{Rfd}_R M =\sup_{\mathfrak p\in \operatorname{Spec}R} \bigl\{\operatorname{depth}R_{\mathfrak p}-\operatorname{depth}_{R_{\mathfrak p}}M_{\mathfrak p}\bigr\},$$9 (Ollivier et al., 2017). In that setting, “dimension” refers to differential dimension, not a homological invariant.

In geometric group theory and CAT(0) geometry, Huang introduces a quasi-isometry invariant for right-angled Artin groups: $\operatorname{Rfd}$00 where $\operatorname{Rfd}$01 is built from coarse intersections of top-dimensional flats in the universal cover of the Salvetti complex (Huang, 2014). Here the terminology concerns coarse geometry of flats rather than module-theoretic flatness.

In extremal combinatorics over $\operatorname{Rfd}$02, Blokhuis and Lev analyze the minimum size $\operatorname{Rfd}$03 of a set containing a $\operatorname{Rfd}$04-flat through every point, and the dual quantity $\operatorname{Rfd}$05 for shift-blocking sets, with the identity

$\operatorname{Rfd}$06

The underlying object is affine and linear $\operatorname{Rfd}$07-flats in a finite vector space, again unrelated to homological flat dimension (Blokhuis et al., 2013).

These usages are mathematically disjoint. A common misconception is that all appearances of “restricted flat dimension” refer to a single invariant. The cited literature shows instead that the phrase is polysemous: in commutative algebra it denotes a relative homological dimension, whereas in control theory, combinatorics, and geometric group theory it refers to distinct notions of flatness and dimension (Ando, 2021, Ollivier et al., 2017, Huang, 2014, Blokhuis et al., 2013).