Two-periodic semi-derived Hall algebras for categories without projective resolutions (e.g. coherent sheaves on smooth projective varieties) and their precise comparison with derived Hall algebras

Gorsky's semi-derived Hall algebra $\mathrm{SDH}(\mathcal{E})$ interpolates between the Ringel–Hall algebra of the category of bounded complexes $C^b(\mathcal{E})$ and the derived Hall algebras of Toën and Xiao–Xu, and its $\mathbb{Z}/2$-graded (two-periodic) version recovers Bridgeland's Hall algebra and its tilting invariance — but the two-periodic construction is only available when the exact category $\mathcal{E}$ has enough projectives and finite projective resolutions, and the precise relation to the Kapranov, Toën, and Xiao–Xu algebras was left as an expectation. Later work resolved the hereditary case (Lu–Peng) and the Frobenius/stable case (Gorsky), but the construction for categories of coherent sheaves on smooth projective varieties of dimension at least 2 — and a general comparison theorem with derived Hall algebras, together with the underlying $K_0$/Euler-form obstructions — remains open. The open problem asks for a two-periodic semi-derived Hall algebra in this remaining generality, with the full structural package: associativity, a basis indexed by the periodic derived category, derived/tilting invariance beyond equivalences induced by exact functors, and explicit comparison isomorphisms.

Background

The Ringel–Hall algebra of a finitary abelian (or exact) category $\mathcal{A}$ — one with finitely many isomorphism classes of objects of each finite total length, and finite Hom- and $\mathrm{Ext}^1$-spaces — is the vector space with basis the isomorphism classes of objects, whose structure constants count short exact sequences with fixed end terms; twisting by the Euler form $\langle A,B\rangle=|\mathrm{Hom}(A,B)|/|\mathrm{Ext}^1(A,B)|$ makes it associative, and Ringel and Green used it to realize the positive part of quantum groups. An exact category (in the sense of Quillen) is an additive category with a distinguished class of 'conflations' abstracting short exact sequences; from it one forms the category of bounded complexes $C^b(\mathcal{E})$ and the bounded derived category $D^b(\mathcal{E})$. A complex is acyclic if it is exact; these become invertible in Hall-type localizations, producing 'quantum tori' (group algebras of Grothendieck groups twisted by an Euler form). Kapranov earlier built a Hall-type algebra for the $\mathbb{Z}/2$-graded (two-periodic) part of a derived category, and Toën, then Xiao–Xu, defined derived Hall algebras directly on suitable Hom-finite triangulated categories; Bridgeland (Ann. of Math. 2013) then realized the whole quantum group of a quiver using the Hall algebra of two-periodic complexes with projective components, which Yanagida and Gorsky identified with the Drinfeld double (the canonical Hopf-algebra double construction) of the twisted extended Hall algebra of a hereditary category. Gorsky (arXiv:1303.5879) introduced the semi-derived Hall algebra $\mathrm{SDH}(\mathcal{E})$: an associative unital algebra between the Hall algebra of $C^b(\mathcal{E})$ and the derived Hall algebra of $D^b(\mathcal{E})$, free as a module over the quantum torus of acyclic complexes with a basis indexed by $D^b(\mathcal{E})$, and invariant under derived equivalences induced by exact functors. He also constructed a $\mathbb{Z}/2$-graded version $\mathrm{SDH}_{\mathbb{Z}/2}(\mathcal{E})$ isomorphic to Bridgeland's algebra, but only under hypothesis (C4'): $\mathcal{E}$ has enough projectives and every object has a finite projective resolution; he wrote that 'we do not know, at least for the moment, how to resolve all these difficulties for the categories of coherent sheaves', the difficulties being that the stupid truncation functors $\sigma_{\geq n}$ (which discard the components of a complex below degree $n$) used throughout the proofs are unavailable and it is unknown whether the $\mathbb{Z}/2$-graded analogues of key lemmas (Lemma 5.10, Corollaries 5.11–5.12) hold in general. He further left open (i) the precise relation between the semi-derived Hall algebras and the algebras of Kapranov and Toën–Xiao–Xu ('we expect that there are strong relations'), (ii) invariance under derived equivalences not induced by exact functors (Cramer's Drinfeld-double theorem for hereditary categories does not need that assumption), and (iii) structural issues in the two-periodic setting: writing $\mathcal{C}_{\mathbb{Z}/2}(\mathcal{E})$ for the category of two-periodic complexes, $\mathcal{C}_{\mathbb{Z}/2,ac}(\mathcal{E})$ for its full subcategory of acyclic complexes, and $\widetilde{\mathcal{E}}$ for the category obtained from the stalk complexes (complexes concentrated in a single degree) of $\mathcal{E}$ by closing under extensions inside $\mathcal{C}_{\mathbb{Z}/2}(\mathcal{E})$ and passing to quasi-isomorphism classes (Gorsky's Definition 9.1), Warning 9.9 shows that the canonical map $K_0(\mathcal{C}_{\mathbb{Z}/2,ac}(\mathcal{E}))\to K_0(\widetilde{\mathcal{E}})$ need not be injective and the Euler form cannot be defined on all of $K_0(\widetilde{\mathcal{E}})\times K_0(\widetilde{\mathcal{E}})$, so the grading/twist framework of the $\mathbb{Z}$-graded theory has no direct two-periodic counterpart. Subsequent work resolved important special cases: Lu–Peng built the two-periodic semi-derived Ringel–Hall algebra for arbitrary finitary hereditary abelian categories without enough projectives (including coherent sheaves on smooth projective curves and weighted projective lines — stacky projective curves with finitely many orbifold points of prescribed weights, the standard hereditary sheaf categories beyond projective curves), with a natural basis, a Drinfeld-double isomorphism, and tilting invariance; Gorsky himself handled Frobenius categories (exact categories where projective and injective objects coincide), proving a comparison with the Toën/Xiao–Xu derived Hall algebra of the stable category; Lin–Peng and Zhang established comparison theorems between modified Ringel–Hall, Bridgeland, and derived Hall algebras in hereditary and periodic settings; and Saito determined the Grothendieck groups of periodic derived categories, clarifying the torsion phenomena behind Warning 9.9. None of this covers coherent sheaves on smooth projective varieties of dimension at least 2, where the category is non-hereditary and has neither enough projectives nor finite projective resolutions.

Problem Statement

Construct a $\mathbb{Z}/2$-graded (two-periodic) semi-derived Hall algebra for finitary exact categories (in the sense of Gorsky's conditions (C1)–(C3): essentially small, idempotent complete, and linear over a finite field, with finite Hom- and $\mathrm{Ext}^p$-spaces and $\mathrm{Ext}^p(A,B)=0$ for all $p\gg 0$) that neither have enough projectives nor admit finite projective resolutions for all objects — the case Gorsky explicitly left open being $\mathcal{E}=\mathrm{coh}(X)$, the category of coherent sheaves on a smooth projective variety $X$ of dimension at least $2$ over a finite field — retaining the full structural package of the known cases: an associative unital algebra, free as a module over the quantum torus of acyclic two-periodic complexes with a basis indexed by the (two-periodic) derived category, invariance under derived equivalences including those not induced by exact functors between the underlying exact categories (in the spirit of Cramer's theorem), and the resolution or clean circumvention of the Warning 9.9 obstructions (non-injectivity of $K_0(\mathcal{C}_{\mathbb{Z}/2,ac}(\mathcal{E}))\to K_0(\widetilde{\mathcal{E}})$ and the impossibility of an Euler form on all of $K_0(\widetilde{\mathcal{E}})\times K_0(\widetilde{\mathcal{E}})$). Establish, in this generality, the precise comparison theorem expected by Gorsky: an explicit relation between the semi-derived Hall algebra and the derived Hall algebras of Toën and Xiao-Xu (and Kapranov's $\mathbb{Z}/2$-graded construction), specifying when they are isomorphic, when one is a twisted tensor factor of the other, and when the semi-derived algebra exists with no derived counterpart. The hereditary case (quiver representations, coherent sheaves on smooth projective curves and weighted projective lines, by Lu-Peng) and the Frobenius/stable case (by Gorsky) are already settled and serve as consistency requirements: the proposed construction must restrict to the known algebras there. These three components — the two-periodic construction with its full structural package, the comparison theorem covering all three regimes (isomorphism, twisted tensor factor, and existence with no derived counterpart), and the restriction to the hereditary and Frobenius/stable cases — are jointly posed as the target of the problem.

References

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