Ono and Yoshino proved an Auslander–Reiten (AR) principle in the unbounded derived category D(Mod R)D(\mathrm{Mod}\,R) of a commutative Noetherian ring — a mechanism behind the classical AR duality and its Iyama–Wemyss generalization — for smashing localizing subcategories (those whose localization functor preserves direct sums), and asked (Question 2.5) whether the same conclusions hold for an arbitrary localizing subcategory. Nakamura and Yoshino subsequently constructed colocalization functors γW\gamma_W attached to arbitrary subsets WW of Spec⁡R\operatorname{Spec}R and answered the question positively under the hypothesis dim⁡W<∞\dim W < \infty, which covers every localizing subcategory over a ring of finite Krull dimension. The remaining open problem is whether the AR principle and its companion local duality principle hold for γW\gamma_W when dim⁡W=∞\dim W = \infty, a regime in which the known boundedness arguments provably fail.

Background

Let RR be a commutative Noetherian ring and let D=D(Mod R)D = D(\mathrm{Mod}\,R) be its unbounded derived category, the localization of the category of chain complexes of RR-modules at quasi-isomorphisms; RHom⁡R\operatorname{RHom}_R denotes the right derived Hom functor and cohomological grading is used throughout. A full triangulated subcategory of DD is called localizing if it is closed under arbitrary direct sums and direct summands. The small support of a complex XX is supp⁡(X)={p∈Spec⁡R:X⊗RLκ(p)≠0}\operatorname{supp}(X) = \{\mathfrak p \in \operatorname{Spec}R : X \otimes^{\mathbf L}_R \kappa(\mathfrak p) \neq 0\}, where κ(p)\kappa(\mathfrak p) is the residue field of the local ring RpR_{\mathfrak p}. A theorem of Neeman (1992) classifies the localizing subcategories of DD: they correspond bijectively to subsets W⊆Spec⁡RW \subseteq \operatorname{Spec}R via W↦LW={X:supp⁡(X)⊆W}W \mapsto L_W = \{X : \operatorname{supp}(X) \subseteq W\}, and the inclusion LW↪DL_W \hookrightarrow D admits a right adjoint γW\gamma_W, a colocalization functor. The subcategory LWL_W is smashing precisely when the corresponding localization functor preserves direct sums, and this happens exactly when WW is specialization-closed (closed under enlarging prime ideals); in that case γW\gamma_W is the right derived functor R⁡ΓW\operatorname{R}\Gamma_W of the section (local cohomology) functor with support in WW. For a subset WW, dim⁡W\dim W denotes the supremum of the lengths of chains of distinct prime ideals contained in WW. For a complex XX, σ>nX\sigma_{>n}X and σ≤nX\sigma_{\le n}X denote the naive truncations with respect to cohomological degrees, and Dfg−(R)D^-_{fg}(R) is the subcategory of right-bounded complexes with finitely generated cohomology modules.

The classical Auslander–Reiten duality states that for maximal Cohen–Macaulay modules M,NM, N (finitely generated modules whose depth equals dim⁡R\dim R) over a Cohen–Macaulay local ring RR possessing a canonical module (a finite module that serves as a dualizing object for RR), if the non-free loci of MM and NN meet only in the closed point, then Hom⁡R(M,N)∨≅Ext⁡R1(N,τM)\operatorname{Hom}_R(M,N)^\vee \cong \operatorname{Ext}^1_R(N, \tau M), where (−)∨(-)^\vee is the Matlis dual (duality with respect to the injective hull of the residue field) and τM\tau M is a syzygy-based dual of MM. Iyama and Wemyss generalized this to singular loci of dimension at most one. Ono and Yoshino identified a single mechanism behind these dualities, their AR Principle (Theorem 2.3): if L\mathbb{L} is a smashing localizing subcategory of DD with right adjoint γ\gamma, II is a bounded complex of injective RR-modules with Ii=0I^i = 0 for i>ni > n, and σ≤−1X∈L\sigma_{\le -1}X \in \mathbb{L}, then the counit γI→I\gamma I \to I induces an isomorphism σ>nRHom⁡R(X,γI)≅σ>nRHom⁡R(X,I)\sigma_{>n}\operatorname{RHom}_R(X, \gamma I) \cong \sigma_{>n}\operatorname{RHom}_R(X, I). They also recorded a companion local duality form (Remark 2.4): under the same smashing hypothesis, for X∈Dfg−(R)X \in D^-_{fg}(R) and II a bounded injective complex, RHom⁡R(X,γI)≅γRHom⁡R(X,I)\operatorname{RHom}_R(X, \gamma I) \cong \gamma\operatorname{RHom}_R(X, I), a common generalization of local duality theorems of Hartshorne and Foxby. Question 2.5 of that paper asks whether both conclusions survive when L\mathbb{L} is an arbitrary localizing subcategory, not necessarily smashing.

Nakamura and Yoshino then developed the theory of the colocalization functors γW\gamma_W for arbitrary (not necessarily specialization-closed) subsets W⊆Spec⁡RW \subseteq \operatorname{Spec}R, together with the dual localization functors λW\lambda_W and cosupport. Their Local Duality Principle (Theorem 4.1) establishes γWRHom⁡R(X,Y)≅RHom⁡R(X,γWY)\gamma_W\operatorname{RHom}_R(X,Y) \cong \operatorname{RHom}_R(X, \gamma_W Y) for arbitrary WW under either of two hypotheses imposing dim⁡W<∞\dim W < \infty, or for generalization-closed WW (closed under shrinking prime ideals) with no dimension restriction; and their Theorem 4.6 establishes the AR Principle for arbitrary WW with dim⁡W<∞\dim W < \infty, answering Question 2.5 in the positive over every ring of finite Krull dimension. The engine of both proofs is a boundedness result (Corollary 3.15): if dim⁡W\dim W is finite, γW\gamma_W shifts injective resolutions by at most dim⁡W\dim W degrees. The same authors show this fails badly in general: for dim⁡W=∞\dim W = \infty, γW\gamma_W need not even send left-bounded complexes to left-bounded complexes (their Example 5.5), because γW\gamma_W is then typically not the right derived functor of any additive functor on Mod R\mathrm{Mod}\,R. Whether the conclusions of the AR Principle and the Local Duality Principle nevertheless remain true in that regime is the unresolved core of the question.

Problem Statement

Let RR be a commutative Noetherian ring, D=D(Mod R)D = D(\mathrm{Mod}\,R) its unbounded derived category, and let W⊆Spec⁡RW \subseteq \operatorname{Spec}R be a subset that need not be specialization-closed, so that the localizing subcategory LW={X:supp⁡(X)⊆W}L_W = \{X : \operatorname{supp}(X) \subseteq W\} need not be smashing; let γW:D→LW\gamma_W : D \to L_W be the right adjoint of the inclusion (the colocalization functor of Nakamura–Yoshino). Assume dim⁡W=∞\dim W = \infty, which is possible only over rings of infinite Krull dimension.

(i) AR Principle clause: let I∈DI \in D be a bounded complex of injective RR-modules with Ii=0I^i = 0 for all i>ni > n, and let X∈DX \in D satisfy σ≤−1X∈LW\sigma_{\le -1}X \in L_W. Does the counit γWI→I\gamma_W I \to I of the adjunction induce a natural isomorphism

σ>nRHom⁡R(X,γWI)≅σ>nRHom⁡R(X,I) ?\sigma_{>n}\operatorname{RHom}_R(X, \gamma_W I) \cong \sigma_{>n}\operatorname{RHom}_R(X, I)\,?

(ii) Local duality clause: assume in addition that WW is not generalization-closed; let X∈Dfg−(R)X \in D^-_{fg}(R) and let II be a bounded complex of injective RR-modules. Is there a natural isomorphism RHom⁡R(X,γWI)≅γWRHom⁡R(X,I)\operatorname{RHom}_R(X, \gamma_W I) \cong \gamma_W \operatorname{RHom}_R(X, I)?

Both clauses are known to hold when dim⁡W<∞\dim W < \infty (and clause (ii) also for generalization-closed WW of any dimension); the problem is exactly Question 2.5 of Ono–Yoshino minus what Nakamura–Yoshino settled. The two clauses are independent: a proof of one of (i) or (ii) leaves the other open.

References

  1. Maiko Ono and Yuji Yoshino, An Auslander–Reiten principle in derived categories, Journal of Pure and Applied Algebra 221 (2017), no. 6, 1268–1278, DOI 10.1016/j.jpaa.2016.09.009, https://doi.org/10.1016/j.jpaa.2016.09.009 (preprint: arXiv:1805.05676, https://arxiv.org/abs/1805.05676)
  2. Tsutomu Nakamura and Yuji Yoshino, A local duality principle in derived categories of commutative Noetherian rings, Journal of Pure and Applied Algebra 222 (2018), no. 9, 2580–2595, DOI 10.1016/j.jpaa.2017.10.008, https://doi.org/10.1016/j.jpaa.2017.10.008 (preprint: arXiv:1610.07715, https://arxiv.org/abs/1610.07715)
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