Singularity categories of representations of algebras over local rings

Abstract

Let Λ be a finite-dimensional algebra with finite global dimension, R k = K[X]/(X k ) be the Z-graded local ring with k ≥ 1, and Λ k = Λ ⊗K R k . We consider the singularity category Dsg(mod Z (Λ k )) of the graded modules over Λ k . It is showed that there is a tilting object in Dsg(mod Z (Λ k )) such that its endomorphism algebra is isomorphic to the triangular matrix algebra T k−1 (Λ) with coefficients in Λ and there is a triangulated equivalence between Dsg(mod Z/kZ (Λ)) and the root category of T k−1 (Λ). Finally, a classification of Λ k up to the Cohen-Macaulay representation type is given.Let Λ be a finite-dimensional algebra with finite global dimension, R k = K[X]/(X k ) be the Z-graded local ring with X degree 1 where k is a positive integer, and Λ k = Λ ⊗ K R k . Then Λ k is a positively graded Gorenstein algebra. C. M. Ringel and M. Schmidmeier [49] investigate the Gorenstein projective modules over Λ k with Λ hereditary of type A 2 by using submodule categories, and describe its Auslander-Reiten quiver for k < 6. The structure of this kind of submodule categories has been studied by many people, see [48,49,52,53,54] and the references therein, and have been discovered to be related to weighted projective lines, see [28,29,30] and the references therein.Later, C. M. Ringel and P. Zhang [50] prove that the singularity category of Λ 2 is triangulated equivalent to the triangulated orbit category D b (mod(Λ))/Σ for Λ a path algebra KQ, where Q is an acyclic quiver. X.-H. Luo and P. Zhang generalize these works and introduce monic representations to describe the Gorenstein projective modules over A⊗ K KQ (also A⊗ K (KQ/I) with I generated by monomial relations) with A a finite-dimensional algebra [37,38]. Recently, D. Shen describes Gorenstein projective modules over the tensor product of two algebras in terms of their underlying one-sided modules [51], see also [21].In this paper, we mainly consider the singularity categories D sg (mod(Λ k )) and D sg (mod Z (Λ k )) over Λ k . K. Yamaura [58] proved that for a positively graded self-injective algebra, its stable category of the Z-graded modules admits a tilting object. Following him, we prove that the singularity category D sg (mod Z (Λ k )) of the Z-graded modules over Λ k has a tilting object with the same construction, in particular, its endomorphism algebra is isomorphic to the triangular matrix algebra T k−1 (Λ) with coefficients in Λ. In this way, we get that D sg (mod Z (Λ k )) is triangulated equivalent to the bounded derived category D b (mod(T k−1 (Λ))). Viewing Λ k as a Z/kZ-graded algebra naturally and considering the singularity category D sg (mod Z/kZ (Λ k )), we prove that the above triangulated equivalence induces a triangulated equivalence D sg (mod Z/kZ (Λ k )) ≃ R T k−1 (Λ) . Furthermore, when k = 2, this result recovers the result of [50]: D sg (mod(Λ 2 )) ≃ D b (mod(Λ))/Σ if Λ is hereditary. Finally, we classify Λ k up to the Cohen-Macaulay representation type, and give some examples to describe the Auslander-Reiten quivers for some ...

Hall algebras and quantum symmetric pairs I: Foundations

“…This together with Lemma 8.1 recovers the results on root categories in [44] (see also [34]) that Gproj(Λ) ≃ 𝐷 𝑠g (mod(Λ)) ≃ 𝐷 𝑏 (𝑘𝑄)∕Σ 2 .…”

“…Remark Theorem 3.18 for $$\imath$$quivers of diagonal type reads that $$\begin{equation*} \underline{\operatorname{Gproj}\nolimits }(\Lambda )\simeq D_{sg}(\operatorname{mod}\nolimits (\Lambda ))\simeq D^b({k}{Q^{\rm dbl}})/\Sigma \,\circ \,\Psi _{\operatorname{swap}\nolimits }. \end{equation*}$$This together with Lemma 8.1 recovers the results on root categories in [44] (see also [34]) that $$\underline{\operatorname{Gproj}\nolimits }(\Lambda )\simeq D_{sg}(\operatorname{mod}\nolimits (\Lambda ))\simeq D^b({k}Q)/\Sigma ^2$$.…”

“…From the above observation, similar to [34, Proposition 3.9, Theorem 3.11], we obtain the following result. Recall the shift functor $$\Sigma$$ of $$D^b({k}Q)$$ and the functor $$G$$ from ().…”

“…Proof The proof that $$(2)\,\,\circ \,\,G\simeq G\,\,\circ \,\,\Sigma ^2$$ and $$D_{sg}(\operatorname{mod}\nolimits ^{{\mathbb {Z}}/2}(\Lambda ^{\imath }))\simeq D^b({k}Q)/\Sigma ^2$$ is completely similar to that of [34, Proposition 3.9, Theorem 3.11], and hence is omitted. The equivalence $$D_{sg}(\operatorname{mod}\nolimits (\Lambda ))\simeq D_{sg}(\operatorname{mod}\nolimits ^{{\mathbb {Z}}/2}(\Lambda ^{\imath }))$$ follows by noting that $$\operatorname{mod}\nolimits (\Lambda )\cong \operatorname{mod}\nolimits ^{{\mathbb {Z}}/2}(\Lambda ^{\imath })$$.$$\Box$$…”

Singularity categories of derived categories of hereditary algebras are derived categories

“…We refer to [BGG,IO,K1,K2,Lu,MY,MU,Y] for recent results which realize stable categories as derived categories in different settings.…”