Rongmin Zhu - Academia.edu (original) (raw)

Papers by Rongmin Zhu

arXiv (Cornell University), Nov 6, 2019

Let A, B be two rings and T = A M 0 B with M an A-B-bimodule. Given two complete hereditary cotor... more Let A, B be two rings and T = A M 0 B with M an A-B-bimodule. Given two complete hereditary cotorsion pairs (A A , B A) and (C B , D B) in A-Mod and B-Mod respectively. We define two cotorsion pairs (Φ(A A , C B), Rep(B A , D B)) and (Rep(A A , C B), Ψ(B A , D B)) in T-Mod and show that both of these cotorsion pairs are complete and hereditary. Given two cofibrantly generated model structures M A and M B on A-Mod and B-Mod respectively. Using the result above, we investigate when there exist a cofibrantly generated model structure M T on T-Mod and a recollement of Ho(M T) relative to Ho(M A) and Ho(M B). Finally, some applications are given in Gorenstein homological algebra.

Advances in Mathematics(China), Jul 3, 2019

arXiv (Cornell University), Nov 12, 2019

Let A and B be abelian categories and F : A → B an additive and right exact functor which is perf... more Let A and B be abelian categories and F : A → B an additive and right exact functor which is perfect, and let (F, B) be the left comma category. We give an equivalent characterization of Gorenstein projective objects in (F, B) in terms of Gorenstein projective objects in B and A. We prove that there exists a left recollement of the stable category of the subcategory of (F, B) consisting of Gorenstein projective objects modulo projectives relative to the same kind of stable categories in B and A. Moreover, this left recollement can be filled into a recollement when B is Gorenstein and F preserves projectives.

arXiv (Cornell University), Dec 29, 2014

Let A and B be rings, U a (B, A)-bimodule and T = (A 0 U B) be the triangular matrix ring. In thi... more Let A and B be rings, U a (B, A)-bimodule and T = (A 0 U B) be the triangular matrix ring. In this paper, we characterize the Gorenstein homological dimensions of modules over T , and discuss when a left T-module is strongly Gorenstein projective or strongly Gorenstein injective module.

arXiv (Cornell University), Sep 2, 2021

arXiv (Cornell University), Apr 2, 2023

Czechoslovak Mathematical Journal

DergiPark (Istanbul University), Dec 10, 2021

Let R be a ring, F a subbifunctor of the functor Ext 1 R (−, −), W F a self-orthogonal class of l... more Let R be a ring, F a subbifunctor of the functor Ext 1 R (−, −), W F a self-orthogonal class of left R-modules respect to F. We introduce W F-Gorenstein modules G(W F) as a generalization of W-Gorenstein modules (Geng and Ding, 2011, [14]), F-Gorenstein projective and F-Gorenstein injective modules (Tang, 2014 [27]). We introduce the notion of relative singularity category D W F (R) with respect to W F. Moreover, we give a necessary and sufficient condition such that the stable category G(W F) and the relative singularity category D W F (R) are triangle-equivalence.

Publicationes Mathematicae Debrecen

Let A, B be two rings and T = ( A M 0 B ) with M an A-B-bimodule. We first construct a semi-compl... more Let A, B be two rings and T = ( A M 0 B ) with M an A-B-bimodule. We first construct a semi-complete duality pair DT of T -modules using duality pairs in A-Mod and B-Mod respectively. Then we characterize when a left T -module is Gorenstein DT -projective, Gorenstein DT -injective or Gorenstein DT -flat. These three class of T -modules will induce model structures on T -Mod. Finally we show that the homotopy category of each of model structures above admits a recollement relative to corresponding stable categories. Our results give new characterizations to earlier results in this direction.

arXiv: Category Theory, 2019

Let UUU be a silting object in a derived category over a dg-algebra AAA, and let BBB be the endom... more Let UUU be a silting object in a derived category over a dg-algebra AAA, and let BBB be the endomorphism dg-algebra of UUU. Under some appropriate hypotheses, we show that if UUU is good, then there exist a dg-algebra CCC, a homological epimorphism BrightarrowCB\rightarrow CBrightarrowC and a recollement among the (unbounded) derived categories mathbfD(C,d)\mathbf{D}(C,d)mathbfD(C,d) of CCC, mathbfD(B,d)\mathbf{D}(B,d)mathbfD(B,d) of BBB and mathbfD(A,d)\mathbf{D}(A,d)mathbfD(A,d) of AAA. In particular, the kernel of the left derived functor −otimesmathbbLBU-\otimes^{\mathbb{L}}_{B}U−otimesmathbbLBU is triangle equivalent to the derived category mathbfD(C,d)\mathbf{D}(C,d)mathbfD(C,d). Conversely, if −otimesmathbbLBU-\otimes^{\mathbb{L}}_{B}U−otimesmathbbLBU admits a fully faithful left adjoint functor, then UUU is good. Moreover, we establish a criterion for the existence of a recollement of the derived category of a dg-algebra relative to two derived categories of weak non-positive dg-algebras. Finally, some applications are given related to good cosilting objects, good 2-term silting complexes, good tilting complexes and modules, which recovers a...

We show that compactly generated t-structures in the derived category of a commutative ring R are... more We show that compactly generated t-structures in the derived category of a commutative ring R are in a bijection with certain families of compactly generated t-structures over the local rings Rm where m runs through the maximal ideals in the Zariski spectrum Spec(R). The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of Spec(R). As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. In combination with a result due to Balmer and Favi, we conclude that the ⊗-Telescope Conjecture for a quasi-coherent and quasi-separated scheme is a stalk-local property. Furthermore, we generalize the results of Trlifaj and Şahinkaya and establish an explicit bijection between cosilting objects of cofinite type over R and compatible families of cosilting objects of cofinite type over all localizations Rm at maximal primes.

arXiv: Rings and Algebras, 2020

In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs... more In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement (mathcalA,mathcalC,mathcalB)(\mathcal{A},\mathcal{C},\mathcal{B})(mathcalA,mathcalC,mathcalB) of abelian categories with enough projective and injective objects. As a consequence, we investigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.

Let T be a right exact functor from an abelian category B into another abelian category A . Then ... more Let T be a right exact functor from an abelian category B into another abelian category A . Then there exists an abelian category, named comma category and denoted by (T ↓ A ). In this paper, we construct left Frobenius pairs (resp. strong left Frobenius pairs) over (T ↓ A ) using left Frobenius pairs (resp. strong left Frobenius pairs) over A and B. As a consequence, we obtain a recollement of (right) triangulated categories, generalizing the result of Xiong-Zhang-Zhang (J. Algebra 503 (2018) 21-55) about the recollement of additive (resp. triangulated) categories induced by monomorphism categories. This result is applied to the classes of flat modules and Gorenstein flat modules, the classes of Gorenstein projective modules and Gorenstein projective complexes, the class of Ding projective modules and the class of Gorenstein flat-cotorsion modules.

Journal of Algebra and Its Applications, 2021

Let [Formula: see text] be a ring, [Formula: see text] a small [Formula: see text]-preadditive ca... more Let [Formula: see text] be a ring, [Formula: see text] a small [Formula: see text]-preadditive category, and let [Formula: see text] be the category of [Formula: see text]-linear functors from [Formula: see text] to the left [Formula: see text]-module category [Formula: see text]. Given a cotorsion pair [Formula: see text] in [Formula: see text], we construct four cotorsion pairs [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] in [Formula: see text] and investigate when these cotorsion pairs are hereditary and complete. Moreover, under few assumptions, we show that there exist a recollement of homotopy categories induced by these cotorsion pairs. Finally, some applications are given in the category of [Formula: see text]-periodic chain complexes of [Formula: see text]-modules.

Electronic Research Archive, 2020

Let G(X) and G(Y) be Gorenstein subcategories induced by an admissible balanced pair (X , Y) in a... more Let G(X) and G(Y) be Gorenstein subcategories induced by an admissible balanced pair (X , Y) in an abelian category A. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of A induced by the balanced pair (X , Y). As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring R.

Communications in Algebra, 2019

Let X be a class of left R-modules, Y be a class of right R-modules. In this paper, we introduce ... more Let X be a class of left R-modules, Y be a class of right R-modules. In this paper, we introduce and study Gorenstein (X , Y)-flat modules as a common generalization of some known modules such as Gorenstein flat modules [9], Gorenstein n-flat modules [22], Gorenstein B-flat modules [7], Gorenstein AC-flat modules [2], Ω-Gorenstein flat modules [10] and so on. We show that the class of all Gorenstein (X , Y)-flat modules have a strong stability. In particular, when (X , Y) is a perfect (symmetric) duality pair, we give some functorial descriptions of Gorenstein (X , Y)-flat dimension, and construct a hereditary abelian model structure on R-Mod whose cofibrant objects are exactly the Gorenstein (X , Y)-flat modules. These results unify the corresponding results of the aforementioned modules.

TURKISH JOURNAL OF MATHEMATICS, 2016

Periodica Mathematica Hungarica, 2021