insight - ScientificComputing - # Homological Algebra

Bounding the Finitistic Dimensions of DG-Modules over Commutative DG-Rings

Q: How do these findings concerning finitistic dimensions over DG-rings inform our understanding of derived algebraic geometry and its applications?

These findings provide a new perspective on the homological complexity of derived schemes and their singularities. Relationship between finitistic dimensions and singularities: In classical algebraic geometry, the finitistic dimensions of a ring (particularly the depth, which coincides with the small finitistic projective dimension for local rings) are closely related to the singularities of the corresponding affine scheme. A ring is regular (i.e., its corresponding affine scheme is non-singular) if and only if its global dimension is finite. Similarly, Cohen-Macaulay rings, which correspond to schemes with less severe singularities, are characterized by the equality of their depth and dimension. The results of the paper suggest an analogous relationship between the finitistic dimensions of a DG-ring and the singularities of the corresponding derived scheme. Understanding the derived category of a derived scheme: The derived category of quasi-coherent sheaves on a scheme is a fundamental object of study in derived algebraic geometry. The finitistic dimensions of a DG-ring provide information about the structure of its derived category. For instance, the existence of DG-modules with prescribed projective dimensions (Theorem B) indicates a certain richness and complexity in the derived category. Applications to specific geometric contexts: The vanishing results for derived Hochschild (co)homology (mentioned as an application in the abstract) are particularly relevant to studying deformations of algebraic structures and to understanding the Hochschild-Kostant-Rosenberg isomorphism in derived algebraic geometry.

Q: Could there be alternative methods, perhaps using different homological invariants, to characterize the finitistic dimensions of DG-modules, potentially leading to different bounds?

Yes, exploring alternative homological invariants is a promising direction for further research. Some possibilities include: Gorenstein homological dimensions: These dimensions, defined using Gorenstein projective, injective, and flat modules, have proven to be useful in non-commutative algebra. Investigating their behavior in the context of DG-rings could lead to new insights and potentially sharper bounds, especially for DG-rings with Gorenstein properties. Relative homological dimensions: One could consider relative homological dimensions, where resolutions are built using a specific class of DG-modules (e.g., those with a certain support condition). This approach could be particularly fruitful for studying DG-rings arising from specific geometric situations. Model-categorical approaches: Derived categories are naturally equipped with model structures, and exploring finitistic dimensions from this perspective might offer new tools and techniques.

Q: If we consider the analogy between commutative rings and affine schemes, what would be a geometric interpretation of the finitistic dimension of a DG-ring, and how does it relate to the geometry of the corresponding derived scheme?

While a definitive geometric interpretation of the finitistic dimension of a DG-ring is still an active area of research, we can draw some insights from the analogy with classical algebraic geometry: Measure of singularity severity: As mentioned earlier, the finitistic dimensions of a ring capture information about the singularities of the corresponding affine scheme. Similarly, the finitistic dimension of a DG-ring can be viewed as a measure of the "derived singularities" of the corresponding derived scheme. A higher finitistic dimension suggests a more complex and "more singular" derived scheme. Obstruction to "derived smoothness": Regular rings, which correspond to non-singular affine schemes, have finite global dimension. Analogously, one might expect that "smooth" derived schemes (e.g., those corresponding to smooth DG-algebras) would have some form of finiteness for their finitistic dimensions. The inequalities in Theorem C, showing that the finitistic dimension is bounded but not necessarily finite, suggest that the relationship between finitistic dimensions and "derived smoothness" is more subtle and requires further investigation. Relationship with derived depth: The paper establishes a connection between the small finitistic projective dimension and the "sequential depth" of a local DG-ring. In classical algebraic geometry, depth is a local invariant measuring the dimension of the "non-singular locus" around a point. The results of the paper suggest a similar interpretation for the derived depth of a DG-ring, potentially relating it to the "derived dimension" of the "non-singular locus" in the corresponding derived scheme.

Core Concepts

This paper establishes bounds on the projective dimension of DG-modules over commutative noetherian DG-rings with finite amplitude, effectively answering Bass's questions regarding finitistic dimensions in this context.

Abstract

Bibliographic Information: Bird, I., Shaul, L., Sridhar, P., & Williamson, J. (2024). Finitistic dimensions over commutative DG-rings. arXiv:2204.06865v4 [math.AC].
Research Objective: This paper investigates the homological dimensions of commutative noetherian non-positive DG-rings with finite amplitude, focusing on establishing bounds for projective dimensions and understanding the finitistic dimensions of such DG-rings.
Methodology: The authors employ techniques from homological algebra, particularly utilizing amplitude inequalities, dualizing DG-modules, and faithfully flat descent to derive bounds on projective dimensions. They also construct explicit examples of DG-modules with prescribed projective dimensions to demonstrate the sharpness of their bounds.
Key Findings: The paper proves that for a DG-module M of finite flat dimension over a commutative noetherian DG-ring A with bounded cohomology, the projective dimension of M is bounded above by the difference between the Krull dimension of H0(A) and the infimum of M. Furthermore, they establish that the big finitistic projective dimension of A is bounded below by the difference between the Krull dimension of H0(A) and the amplitude of A, and bounded above by the Krull dimension of H0(A). Notably, they demonstrate that both bounds are achievable.
Main Conclusions: The research provides a comprehensive understanding of finitistic dimensions in the context of commutative noetherian DG-rings with finite amplitude. The established bounds offer valuable insights into the structure of these rings and their module categories.
Significance: This work significantly contributes to the field of homological algebra, extending classical results on finitistic dimensions from commutative rings to the more general setting of DG-rings. The findings have implications for the study of derived algebraic geometry and other areas where DG-rings play a crucial role.
Limitations and Future Research: While the paper thoroughly addresses Bass's questions for commutative noetherian DG-rings with finite amplitude, it leaves open the question of whether the big and small finitistic projective dimensions coincide precisely when the DG-ring is locally Cohen-Macaulay. Further research could explore this question and investigate the behavior of finitistic dimensions over more general classes of DG-rings.

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Quotes

"In this paper we study the finitistic dimensions of commutative noetherian non-positive DG-rings with finite amplitude."
"We prove that any DG-module M of finite flat dimension over such a DG-ring satisfies proj dimA(M) ≤dim(H0(A)) −inf(M)."
"dim(H0(A)) −amp(A) ≤FPD(A) ≤dim(H0(A))."

Key Insights Distilled From

Finitistic dimensions over commutative DG-rings

by Isaac Bird, ... at arxiv.org 10-08-2024

https://arxiv.org/pdf/2204.06865.pdf

Finitistic dimensions over commutative DG-rings

Deeper Inquiries

How do these findings concerning finitistic dimensions over DG-rings inform our understanding of derived algebraic geometry and its applications?

These findings provide a new perspective on the homological complexity of derived schemes and their singularities.

Relationship between finitistic dimensions and singularities: In classical algebraic geometry, the finitistic dimensions of a ring (particularly the depth, which coincides with the small finitistic projective dimension for local rings) are closely related to the singularities of the corresponding affine scheme.  A ring is regular (i.e., its corresponding affine scheme is non-singular) if and only if its global dimension is finite. Similarly, Cohen-Macaulay rings, which correspond to schemes with less severe singularities, are characterized by the equality of their depth and dimension. The results of the paper suggest an analogous relationship between the finitistic dimensions of a DG-ring and the singularities of the corresponding derived scheme.

Understanding the derived category of a derived scheme: The derived category of quasi-coherent sheaves on a scheme is a fundamental object of study in derived algebraic geometry. The finitistic dimensions of a DG-ring provide information about the structure of its derived category. For instance, the existence of DG-modules with prescribed projective dimensions (Theorem B) indicates a certain richness and complexity in the derived category.

Applications to specific geometric contexts: The vanishing results for derived Hochschild (co)homology (mentioned as an application in the abstract) are particularly relevant to studying deformations of algebraic structures and to understanding the Hochschild-Kostant-Rosenberg isomorphism in derived algebraic geometry.

Could there be alternative methods, perhaps using different homological invariants, to characterize the finitistic dimensions of DG-modules, potentially leading to different bounds?

Yes, exploring alternative homological invariants is a promising direction for further research. Some possibilities include:

Gorenstein homological dimensions: These dimensions, defined using Gorenstein projective, injective, and flat modules, have proven to be useful in non-commutative algebra. Investigating their behavior in the context of DG-rings could lead to new insights and potentially sharper bounds, especially for DG-rings with Gorenstein properties.

Relative homological dimensions: One could consider relative homological dimensions, where resolutions are built using a specific class of DG-modules (e.g., those with a certain support condition). This approach could be particularly fruitful for studying DG-rings arising from specific geometric situations.

Model-categorical approaches:  Derived categories are naturally equipped with model structures, and exploring finitistic dimensions from this perspective might offer new tools and techniques.

If we consider the analogy between commutative rings and affine schemes, what would be a geometric interpretation of the finitistic dimension of a DG-ring, and how does it relate to the geometry of the corresponding derived scheme?

While a definitive geometric interpretation of the finitistic dimension of a DG-ring is still an active area of research, we can draw some insights from the analogy with classical algebraic geometry:

Measure of singularity severity: As mentioned earlier, the finitistic dimensions of a ring capture information about the singularities of the corresponding affine scheme. Similarly, the finitistic dimension of a DG-ring can be viewed as a measure of the "derived singularities" of the corresponding derived scheme. A higher finitistic dimension suggests a more complex and "more singular" derived scheme.

Obstruction to "derived smoothness":  Regular rings, which correspond to non-singular affine schemes, have finite global dimension. Analogously, one might expect that "smooth" derived schemes (e.g., those corresponding to smooth DG-algebras) would have some form of finiteness for their finitistic dimensions. The inequalities in Theorem C, showing that the finitistic dimension is bounded but not necessarily finite, suggest that the relationship between finitistic dimensions and "derived smoothness" is more subtle and requires further investigation.

Relationship with derived depth: The paper establishes a connection between the small finitistic projective dimension and the "sequential depth" of a local DG-ring. In classical algebraic geometry, depth is a local invariant measuring the dimension of the "non-singular locus" around a point. The results of the paper suggest a similar interpretation for the derived depth of a DG-ring, potentially relating it to the "derived dimension" of the "non-singular locus" in the corresponding derived scheme.