Proxy smallness meets $t$-structures

Technical Summary: Proxy Smallness Meets t-Structures

1. Problem and Motivation
The paper addresses the limitations of "proxy smallness," a concept introduced by Dwyer, Greenlees, and Iyengar [DGI06a, DGI06b] as a weakening of compactness in triangulated categories. While proxy smallness successfully characterizes local complete intersection (LCI) singularities in the local setting (e.g., [Pol19]) and has been refined via tensor actions to characterize LCI schemes globally ([BILP22]), it fails to behave well under globalization when viewed through the lens of $t$-structures. Specifically, naive global analogues of proxy smallness do not capture the correct geometric properties when transitioning from local to global settings.

The authors aim to introduce a natural refinement of proxy smallness that integrates $t$-structures and tensor actions. The central goal is to develop a theory of "proxy smallness for $t$-structures" (specifically $t$-proxy smallness and $t$-$\otimes$-proxy smallness) to provide new characterizations of LCI schemes and to classify specific subcategories (preaisles) within the bounded derived category of coherent sheaves, $D^b_{\text{coh}}(X)$.

2. Methodology and Framework
The theory is developed within the context of rigidly compactly generated tensor triangulated categories, specifically focusing on $D_{\text{qc}}(X)$ for a Noetherian scheme $X$. The methodology relies on three primary technical pillars:

• Tensor Actions and Preaisles: The authors utilize the action of the category of perfect complexes, $\text{Perf}(X)$, on $D_{\text{qc}}(X)$ and related categories (such as the homotopy category of injectives $K(\text{Inj}(X))$). They define $\otimes$-preaisles (subcategories closed under positive shifts, extensions, direct summands, and tensoring with $\text{Perf}_{\leq 0}(X)$).
• Recollements: A crucial tool is Krause's recollement relating $D_{\text{qc}}(X)$, $K(\text{Inj}(X))$, and the singularity category $D_{\text{sg}}(X)$. The authors use this framework to "glue" $t$-structures and analyze how properties like proxy smallness behave across these categories.
• Pseudocoherent Generation: The paper extends results from the affine case to arbitrary Noetherian schemes by studying aisles generated by pseudocoherent complexes. A key technical result (Theorem 3.5) establishes that $\otimes$-aisles generated by pseudocoherent complexes are compactly generated, generalizing [ATJLS10].

3. Key Definitions and Contributions

• $t$-Proxy Smallness: An object $P$ is defined as $t$-proxy small if the cocomplete preaisle generated by $P$, denoted $\langle P \rangle^{(-\infty, 0]}$, is compactly generated and its compact objects lie within the preaisle generated by the compact objects of $P$. This restricts the "building" process to positive suspensions and extensions, unlike classical proxy smallness which allows arbitrary triangulated operations.
• $t$-$\otimes$-Proxy Smallness: This extends the concept to the presence of a tensor action. An object is $t$-$\otimes$-proxy small if the $\otimes$-preaisle generated by it satisfies the $t$-proxy small condition.
• Structural Characterization (Theorem 4.7): The authors characterize $t$-proxy small objects via recollements. An object $P$ is $t$-proxy small if and only if:
  1. The preaisle $\langle P \rangle^{(-\infty, 0]}$ is compactly generated.
  2. A specific endofunctor involving the derived Hom and tensor product preserves small coproducts.
  3. The standard $t$-structure of the dg-endomorphism algebra of $P$ glues along the relevant recollement.
• Distinction from Classical Proxy Smallness: The paper demonstrates that $t$-proxy smallness is a strictly stronger condition. Example 4.11 constructs an object in $D^b_{\text{coh}}(R)$ (where $R$ is a specific local Artinian ring) that is proxy small but not $t$-proxy small. This relies on module-theoretic counterexamples to conjectures by Eisenbud and Auslander found in [JŞ04] and [GP90].

4. Main Results

• Characterization of LCI Schemes (Proposition 1.2): The paper provides a new characterization of locally complete intersection schemes. For a Noetherian scheme $X$, the following are equivalent:

  1. $X$ is locally a complete intersection.
  2. Every object of $D^b_{\text{coh}}(X)$ is $\otimes$-proxy small.
  3. Every object of $D^b_{\text{coh}}(X)$ is $t$-$\otimes$-proxy small.
     The novelty lies in condition (3), which offers a $t$-structural refinement of existing characterizations. The proof utilizes a local-to-global property established via Lemma 6.8, showing that membership in a compactly generated aisle on $K(\text{Inj}(X))$ is a stalk-local condition.

• Classification of Preaisles (Theorem 1.1 and Theorem 5.4): The authors establish an injective map $\Theta$ from the collection of $\otimes$-suspended subcategories of $D^b_{\text{coh}}(X)$ (generated by $t$-$\otimes$-proxy small objects) to the Cartesian product of:

  • $\star$-suspended subcategories of the singularity category $D_{\text{sg}}(X)$.
  • Thomason filtrations on $X$.
    The map is given by $A \mapsto (\langle \pi(A) \rangle^{(-\infty, 0]}, \phi_A)$, where $\pi$ is the Verdier localization and $\phi_A$ is the Thomason filtration associated with the generated aisle in $D_{\text{qc}}(X)$.

• Explicit Classifications in Specific Cases:

  • Hypersurface Singularities (Theorem 1.3): If $X$ has only hypersurface singularities, $\Theta$ becomes a bijection between $\otimes$-suspended subcategories of $D^b_{\text{coh}}(X)$ and pairs $(W, \phi)$, where $W$ is a specialization closed subset of the singular locus and $\phi$ is a Thomason filtration satisfying a specific containment condition.
  • Zero Loci of Sections (Theorem 1.5): For schemes $X$ arising as zero loci of sections of vector bundles on regular schemes (satisfying resolution properties), the classification is refined to involve the singular locus of a related hypersurface scheme $Y$. This result provides a non-affine generalization of Takahashi's classification [Tak25] for suspended subcategories containing the structure sheaf.

5. Significance and Claims
The paper claims to provide a "natural refinement" of proxy smallness that is sensitive to the $t$-structure. Its significance is twofold:

1. Geometric Characterization: It offers a new, $t$-structural criterion for identifying locally complete intersection schemes, upgrading previous results to handle non-separated cases and providing a more robust local-to-global principle.
2. Classification Theory: It yields a topological classification of $\otimes$-preaisles on $D^b_{\text{coh}}(X)$ for specific classes of schemes (hypersurfaces and zero loci). This classification connects the geometry of the scheme (via Thomason filtrations) and its singularities (via the singularity category) to the structure of its derived category.

The authors emphasize that their approach is independent of the methods used in Takahashi's recent work [Tak25], offering an alternative proof and generalization to non-affine settings. Furthermore, the construction of counterexamples (Example 4.11) highlights the subtle differences between classical and $t$-structural notions of smallness, leveraging deep results in commutative algebra regarding the behavior of modules over non-LCI rings.