Complete Intersection Flat Dimension

• Complete intersection flat dimension is a homological invariant defined via quasi-deformations that measures flat dimensional properties and generalizes complete intersection dimension.
• It extends classical invariants from finitely generated modules to arbitrary modules while providing concrete depth and dependency formulas.
• The invariant interacts with André–Quillen homology and supports ascent theories in local homomorphism contexts, solidifying its role in modern homological algebra.

Searching arXiv for papers on complete intersection flat dimension and closely related invariants. Complete intersection flat dimension is a homological invariant defined by quasi-deformations and flat dimension, designed as a flat analogue of complete intersection dimension. In the setting of a local Noetherian ring $(R,\mathfrak m,k)$, for a homologically bounded $R$-complex $X$, the invariant is

$$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$

where a quasi-deformation is a diagram $R\to R' \leftarrow Q$ with $R\to R'$ a flat extension and $Q\to R'$ a deformation (Sahandi et al., 2010). In the module-theoretic formulation used for arbitrary $R$-modules, one writes

$$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$

with $R\to R'$ flat and $R$0 surjective with kernel generated by a $R$1-regular sequence (Sharif, 2013). For finitely generated modules, complete intersection flat dimension agrees with complete intersection dimension, so $R$2 extends the Avramov–Gasharov–Peeva invariant beyond the finitely generated/projective setting (Sharif, 2013). Subsequent work introduced upper and weakly complete intersection flat dimensions, obtained by restricting the allowed flat maps in the quasi-deformation to weakly regular maps or to maps with complete intersection closed fiber (Sather-Wagstaff et al., 9 Aug 2025).

1. Definition through quasi-deformations

A deformation of a local ring is a presentation $R$3 where $R$4 is a local ring and $R$5 is a complete intersection ideal, equivalently $R$6 is generated by a $R$7-regular sequence (Sahandi et al., 2010). A quasi-deformation is then a diagram

$R$8

such that $R$9 is a flat extension and $X$0 is a deformation (Sahandi et al., 2010). If $X$1 is generated by a $X$2-regular sequence of length $X$3, the quasi-deformation has codimension $X$4 (Sahandi et al., 2010).

Within this framework, complete intersection flat dimension is obtained by measuring the flat dimension of the base change over the auxiliary ring $X$5, corrected by the projective or flat dimension of $X$6 over $X$7. For homologically bounded complexes the precise formula is

$X$8

and if the quasi-deformation has codimension $X$9, then $$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$0, so the quantity in the infimum is often written as $$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$1 (Sahandi et al., 2010). For arbitrary modules the same pattern appears with underived tensor product: $$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$2 (Sharif, 2013).

This definition is explicitly presented as a flat-version analogue of complete intersection dimension (Sahandi et al., 2010). For finitely generated modules one has

$$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$3

so complete intersection flat dimension extends complete intersection dimension from finitely generated modules to arbitrary modules (Sharif, 2013). A plausible implication is that $$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$4 should be regarded as the natural derived-flat enlargement of the classical module invariant.

2. Fundamental comparisons and localization behavior

The basic comparison established for homologically bounded complexes is

$$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$5

with equality whenever $$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$6 (Sahandi et al., 2010). Here the large restricted flat dimension is

$$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$7

and it admits the Chouinard-type formula

$$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$8

(Sahandi et al., 2010). Consequently, in the finite case one obtains the concrete expression

$$\operatorname{CIfd}_R X := \inf\Big\{ \operatorname{fd}_Q(R'\otimes_R^{\mathbf L} X)-\operatorname{fd}_Q R' \ \Big|\ R\to R' \leftarrow Q \text{ is a quasi-deformation} \Big\},$$9

(Sahandi et al., 2010).

The invariant localizes: $R\to R' \leftarrow Q$0 (Sahandi et al., 2010). It also sits in the chain

$R\to R' \leftarrow Q$1

with equality to the left of any finite number (Sahandi et al., 2010). Thus finite flat dimension implies finite complete intersection flat dimension, and finite complete intersection flat dimension collapses to restricted flat dimension (Sahandi et al., 2010).

For finitely generated modules over local Noetherian rings, finite projective dimension and finite flat dimension coincide, and these dimensions are equal when finite (Tavanfar, 2019). This is why many CI-dimension statements for finitely generated modules are translatable into flat-dimension language in that setting (Tavanfar, 2019). The same point is made in later work on $R\to R' \leftarrow Q$2-criteria: for finitely generated modules over Noetherian local rings, the paper’s $R\to R' \leftarrow Q$3-results can usually be read as $R\to R' \leftarrow Q$4-results as well (Martins et al., 12 Jan 2026).

3. Depth formula and dependency formula

A central application of finite complete intersection flat dimension is the depth formula for complexes. If $R\to R' \leftarrow Q$5 satisfy

$R\to R' \leftarrow Q$6

then

$R\to R' \leftarrow Q$7

(Sahandi et al., 2010). For modules this yields: if $R\to R' \leftarrow Q$8 are $R\to R' \leftarrow Q$9-modules with

$R\to R'$0

and $R\to R'$1, then

$R\to R'$2

(Sahandi et al., 2010).

The proof mechanism is explicit. Finite CIfd gives a quasi-deformation

$R\to R'$3

such that $R\to R'$4 has finite flat dimension over $R\to R'$5, up to the codimension correction (Sahandi et al., 2010). Over a deformation $R\to R'$6 of codimension $R\to R'$7, one has the comparison formulas

$R\to R'$8

$R\to R'$9

(Sahandi et al., 2010). These identities allow one to apply the classical finite-flat-dimension depth formula over $Q\to R'$0 and descend the result back to $Q\to R'$1 (Sahandi et al., 2010).

The same paper proves a dependency formula extending Jorgensen’s result. Under the same finiteness assumptions,

$Q\to R'$2

(Sahandi et al., 2010). A plausible implication is that finite complete intersection flat dimension is the exact hypothesis that permits controlled transfer of depth and amplitude calculations through complete-intersection deformations.

4. André–Quillen homology and homomorphism-theoretic criteria

Complete intersection flat dimension also appears in the study of local homomorphisms via André–Quillen homology. For a local homomorphism $Q\to R'$3, one considers $Q\to R'$4, meaning the CIfd of the target ring as an $Q\to R'$5-module (Sharif, 2013). In this setting, if $Q\to R'$6 is essentially of finite type and

$Q\to R'$7

then

$Q\to R'$8

(Sharif, 2013). Here $Q\to R'$9 is the first Koszul homology attached to a Cohen factorization of $R$0 (Sharif, 2013). This criterion shows that finite CIfd, together with a computable Koszul-homology condition, is equivalent to the sharp André–Quillen bound $R$1 in the essentially finite type case (Sharif, 2013).

A complementary result states that if $R$2 is surjective, $R$3 is complete intersection at $R$4, and $R$5, then

$R$6

(Sharif, 2013). Since $R$7 is finitely generated over $R$8 in that context, this conclusion is closely tied to finite complete intersection flat dimension (Sharif, 2013).

Later work sharpened the ascent picture by introducing upper and weakly complete intersection flat dimensions (Sather-Wagstaff et al., 9 Aug 2025). For an arbitrary $R$9-module $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$0,

$$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$1

$$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$2

and the inequalities

$$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$3

hold (Sather-Wagstaff et al., 9 Aug 2025). If $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$4 is essentially of finite type and complete intersection at the maximal ideal, then finiteness of $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$5, $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$6, or $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$7 ascends to the corresponding $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$8-module invariant under the hypotheses of Theorem 3.1 and Corollary 3.1 in that paper (Sather-Wagstaff et al., 9 Aug 2025). This directly identifies ascent along complete-intersection-type maps as a central structural property of the flat theory (Sather-Wagstaff et al., 9 Aug 2025).

5. Relation to complete intersection dimension, upper variants, and test modules

The literature distinguishes carefully between complete intersection flat dimension and several adjacent invariants. Complete intersection dimension is normally formulated using projective dimension over the auxiliary ring $$\operatorname{CI\!-\!fd}_R M := \inf \left\{ \operatorname{fd}_Q(R'\otimes_R M)-\operatorname{pd}_Q(R') \;\middle|\; R\to R' \leftarrow Q \text{ is a quasi-deformation} \right\},$$9, for modules or homologically finite complexes (Tavanfar, 2019). The papers on test modules do not define complete intersection flat dimension explicitly; instead they use complete intersection dimension or upper complete intersection dimension, together with projective-dimension detection by test modules (Majadas, 2012, Tavanfar, 2019).

In the finitely generated local setting, however, the distinction often collapses. The 2019 paper proves that if a local ring possesses a test module $R\to R'$0 such that

$R\to R'$1

then $R\to R'$2 is complete intersection (Tavanfar, 2019). The same source emphasizes that, because finite projective and flat dimensions agree for finitely generated modules over local Noetherian rings, many conclusions can be read as flat-dimension statements in that restricted setting (Tavanfar, 2019). This suggests that the test-module detection theorem is adjacent to, but not itself a theorem about, complete intersection flat dimension.

The 2012 paper of Majadas uses the stronger invariant upper complete intersection dimension,

$R\to R'$3

defined using weakly regular homomorphisms $R\to R'$4 and deformations $R\to R'$5 with $R\to R'$6 (Majadas, 2012). Its main theorem is that if $R\to R'$7 is a test $R\to R'$8-module and $R\to R'$9, then $R$00 is complete intersection (Majadas, 2012). That paper is explicit that it does not study complete intersection flat dimension as a separate invariant; rather, it connects a flat-dimension-detecting property of modules to upper complete intersection dimension (Majadas, 2012).

A later paper extending CI-dimension techniques to $R$01, $R$02, and CI-perfect modules similarly does not develop a separate theory of $R$03, but notes that, for finitely generated modules over Noetherian local rings, its $R$04-results can usually be read as $R$05-results (Martins et al., 12 Jan 2026). This recurring pattern indicates that complete intersection flat dimension is often the ambient generalization, while concrete theorems in the finitely generated case are still expressed in terms of complete intersection dimension.

6. Extensions, neighboring theories, and current perspective

Complete intersection flat dimension belongs to a broader family of refined flat-like invariants. One neighboring direction studies Gorenstein flat dimension over local homomorphisms. If $R$06 is a local homomorphism with $R$07 and $R$08 is a non-zero finitely generated $R$09-module with

$R$10

then $R$11 is totally reflexive over $R$12 and $R$13 is an exceptional complete intersection map (Faridian, 2024). That paper does not define CI-flat dimension, but it shows that a very small flat-like invariant over the source ring can force exceptional complete intersection structure of the map (Faridian, 2024). A plausible implication is that such detection theorems occupy the same conceptual landscape as CI-flat dimension theory.

Another adjacent direction concerns quasi-complete intersection ideals. The structure theorem that every q.c.i. ideal is obtained from nested complete intersection ideals by way of a flat base change is not a theorem about CI-flat dimension by name, but it provides a structural model that is highly compatible with CI-flat constructions (Kustin et al., 2018). The paper computes

$R$14

for the model quotient $R$15 over $R$16, while the target q.c.i. quotient is obtained by Tor-independent base change (Kustin et al., 2018). This suggests that complete intersection flat dimension should interact naturally with q.c.i. quotients, although that implication is not stated there.

Recent work on ascent has made the flat theory more explicit. The 2025 paper defines, compares, and studies

$R$17

for arbitrary modules, and proves ascent results along essentially of finite type local homomorphisms with complete intersection closed fiber (Sather-Wagstaff et al., 9 Aug 2025). This places complete intersection flat dimension in a more differentiated hierarchy, where the allowed left leg of the quasi-deformation determines the exact variant (Sather-Wagstaff et al., 9 Aug 2025). The same paper also uses this ascent theorem to extend André–Quillen dimension criteria from surjective local homomorphisms to essentially of finite type homomorphisms (Sather-Wagstaff et al., 9 Aug 2025).

Taken together, these results indicate a stable core picture. Complete intersection flat dimension is defined by quasi-deformations and auxiliary flat dimension; it coincides with complete intersection dimension on finitely generated modules; it controls depth formulas and dependency formulas for complexes; it interacts strongly with André–Quillen homology; and it admits upper and weak variants tailored to ascent problems (Sahandi et al., 2010, Sharif, 2013, Sather-Wagstaff et al., 9 Aug 2025). This suggests that the modern role of complete intersection flat dimension is to serve as the flexible flat-side invariant for complete-intersection homological algebra, especially beyond the finitely generated setting.