Change-of-Rings Theorems for the Small Finitistic Dimension

Abstract: In this paper, we study the small finitistic dimension of a commutative ring from the viewpoint of finitistic flat homological algebra. Using the class $FPR(R)$ of modules admitting finite projective resolutions, we investigate the finitistic flat ($FT$-flat) dimension and establish several of its basic properties. We prove change-of-rings results for the $FT$-flat dimension, including quotient and polynomial extension results, as well as localization inequalities. As applications, we obtain characterizations of the small finitistic dimension in terms of $FT$-flat dimension, derive quotient and polynomial extension theorems for the small finitistic dimension, and establish local upper bounds in terms of the small finitistic dimensions of localizations.

• The paper introduces FT‐flat dimensions to generalize classical projective and flat dimensions, enabling rigorous change-of-rings results for fdim(R).
• It provides explicit upper and lower bounds for fdim(R) through analyses of localization, quotients, polynomial extensions, and triangular matrix rings.
• The work offers new characterizations for commutative rings and lays the groundwork for further exploration of homological invariants like Gorenstein dimensions.

Change-of-Rings Theorems for the Small Finitistic Dimension

Introduction and Context

The paper "Change-of-Rings Theorems for the Small Finitistic Dimension" (2604.18958) addresses the homological invariant known as the small finitistic dimension, denoted $f\!\dim(R)$, for a commutative ring $R$. Traditionally defined as the supremum of projective dimensions of finitely generated $R$-modules of finite projective dimension, its computation and behavior under ring-theoretic constructions are nontrivial, especially in the non-Noetherian setting. The authors adopt Glaz's refinement, characterizing $f\!\dim(R)$ as the supremum taken over the class $FPR(R)$ of $R$-modules that admit finite projective resolutions by finitely generated projectives.

A central innovation is the use of the "finitistic flat" ($FT$-flat) dimension, which generalizes the classical flat dimension in the context of modules with finite projective resolutions. This perspective unifies homological techniques and enables the derivation of change-of-rings theorems for $f\!\dim(R)$, analogous to classical results for global and weak global dimensions.

$FT$-Flat Dimension and its Structural Properties

The authors introduce and systematically study $FT$-flat modules: for an $R$0-module $R$1, $R$2 is $R$3-flat if $R$4 for every $R$5. The $R$6-flat dimension $R$7 is the minimal length of a finite $R$8-flat resolution of $R$9, or infinity if none exists. The equivalence $R$0 (originally shown in [WZKXS2020]) justifies the focus on $R$1-flat dimension.

The paper establishes key closure properties of $R$2 under short exact sequences, paralleling analogous results for projective modules but requiring more technical care due to the restriction to finitely generated projectives. This is formalized in Lemma 2.1.

A significant technical point is that $R$3 is not, in general, preserved under localization; an explicit example with $R$4 and $R$5 shows that $R$6 need not be contained in $R$7.

Change-of-Rings Theorems for $R$8-Flat Dimension

The main homological results generalize the classical change-of-rings inequalities to $R$9-flat dimension. If $f\!\dim(R)$0 is a ring homomorphism, the following hold for any $f\!\dim(R)$1-module $f\!\dim(R)$2:

• $f\!\dim(R)$3,
• If $f\!\dim(R)$4 is flat as an $f\!\dim(R)$5-module, $f\!\dim(R)$6.

These mirror the classical results for projective and flat dimensions, but the techniques require careful tracking of resolutions in $f\!\dim(R)$7 (Theorem 2.8). The arguments employ dimension shifting and induction on resolution length, using the fact that the class $f\!\dim(R)$8 is closed under appropriate operations.

A further result is an upper bound for the $f\!\dim(R)$9-flat dimension of a module in terms of the suprema of the $FPR(R)$0-flat dimensions of localizations at maximal (or prime) ideals (Prop. 2.11):

$FPR(R)$1

However, the reverse inequality does not generally hold due to the aforementioned non-local nature of $FPR(R)$2.

Small Finitistic Dimension: Characterizations and Change-of-Rings Results

A centerpiece of the paper is the extension of these methods to $FPR(R)$3, yielding several new and exact characterizations (Theorem 3.1). Among these:

• $FPR(R)$4 if and only if for every maximal ideal $FPR(R)$5, $FPR(R)$6,
• or, equivalently, if and only if for all $FPR(R)$7, $FPR(R)$8.

The authors prove explicit change-of-rings theorems for $FPR(R)$9 paralleling results for global and weak global dimension:

• Quotients: For any $R$0 neither a zero-divisor nor a unit, setting $R$1, if $R$2, then $R$3 (Theorem 3.3).
• Polynomial Extensions: For $R$4 with $R$5 and $R$6,

$R$7

(Theorem 3.4).

• Localizations: $R$8 (Prop. 3.7).

These results hold in full generality for commutative rings, including non-Noetherian ones, and recover the known classical bounds as special cases. The proofs utilize the formal apparatus of $R$9-flat dimensions, reduction to projective and flat resolutions, and an analysis of the syzygy modules involved.

Triangular Matrix Rings

The paper also advances the study of the small finitistic dimension under ring extensions by considering upper triangular matrix rings $FT$0 with $FT$1-bimodule $FT$2 (Section 4). Assuming $FT$3 is projective as left $FT$4-module and right $FT$5-module, the authors derive precise bounds:

$FT$6

Even when $FT$7, $FT$8 may be $FT$9, which is illustrated via a local finite-dimensional algebra example. For the specific case of the upper triangular matrix ring $f\!\dim(R)$0, an explicit recursive bound is established: $f\!\dim(R)$1.

Implications and Future Perspectives

The systematic homological analysis and change-of-rings theorems for the small finitistic dimension in this work significantly broaden the set of computable cases for this invariant, especially in non-Noetherian or non-coherent settings. By embedding $f\!\dim(R)$2 within the framework of $f\!\dim(R)$3-flat dimension, various structural and functorial properties become transparent, and the previously intractable behavior under localization, quotients, and extensions is clarified.

Practically, these theorems enable the reduction of $f\!\dim(R)$4 computation to simpler factors, supporting both explicit calculation in examples and structural results (e.g., rings with $f\!\dim(R)$5 are characterized by all modules in $f\!\dim(R)$6 being projective).

From a theoretical standpoint, the machinery built in this paper suggests further investigations:

• Sharpness and failure cases for the localization inequality—especially where $f\!\dim(R)$7 is not "well behaved."
• Extensions to other classes of rings (e.g., semihereditary, coherent) or to invariants beyond projective dimensions (e.g., Gorenstein versions).
• Applications to homological conjectures, such as the finitistic dimension conjecture, in broader settings.

Conclusion

This work establishes a rigorous change-of-rings theory for the small finitistic dimension of commutative rings, leveraging $f\!\dim(R)$8-flat resolutions to systematically extend classical homological algebra. The results synthesize quotient, localization, and extension theorems for $f\!\dim(R)$9, filling a notable gap in the literature. The study of triangular and upper triangular matrix rings further demonstrates the efficacy and limitations of these techniques. The paper sets a foundation for further explorations of invariants related to projective dimensions under general ring-theoretic constructions and provides essential tools for future research in homological and commutative algebra.