Cotilting modules over commutative Noetherian rings

Šťovíček, Jan; Trlifaj, Jan; Herbera, Dolors

doi:10.1016/j.jpaa.2014.01.008

Cited by 11 publications

(9 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Even though there is an explicit duality between tilting modules and cotilting modules of cofinite type, the one way nature of the duality makes the tilting side harder to approach. For example, cotilting modules over commutative noetherian case are described in [ ŠTH14], but tilting modules were described only for special classes of noetherian rings. The crucial step in our approach is to show that a 1-cotilting class is of cofinite type if and only if it is closed under injective envelopes (Corollary 3.13).…”

Section: Introductionmentioning

confidence: 99%

One-tilting classes and modules over commutative rings

Hrbek

2016

Journal of Algebra

View full text Add to dashboard Cite

We classify 1-tilting classes over an arbitrary commutative ring. As a consequence, we classify all resolving subcategories of finitely presented modules of projective dimension at most 1. Both these collections are in 1-1 correspondence with faithful Gabriel topologies of finite type, or equivalently, with Thomason subsets of the spectrum avoiding a set of primes associated in a specific way to the ring. We also provide a generalization of the classical Fuchs and Salce tilting modules, and classify the equivalence classes of all 1-tilting modules. Finally we characterize the cases when tilting modules arise from perfect localizations.Comment: 19 pages, 1 figur

show abstract

Section: Introductionmentioning

confidence: 99%

One-tilting classes and modules over commutative rings

Hrbek

2016

Journal of Algebra

View full text Add to dashboard Cite

show abstract

“…In this section we construct to each cotilting class of cofinite type over a commutative ring a cotilting module inducing it. The construction generalizes ideas from [ ŠTH14].…”

Section: Construction Of the Corresponding Cotilting Modulesmentioning

confidence: 80%

“…The aforementioned connection with the derived functors of torsion and completion, as well as the Čech (co)homology, is explained in Section 7. In the following Section 8, we show how a cotilting module associated to any cotilting class of cofinite type over a commutative ring can be constructed, building on the idea from [ ŠTH14]. In the final Section 9, we characterize the cofinite-type cotilting classes amongst the general ones, and we construct new examples of ncotilting classes which are not of cofinite type, but which are in a sense difficult to tell apart from cofinite type cotilting classes.…”

Section: Introductionmentioning

confidence: 99%

Tilting classes over commutative rings

Hrbek

Šťovíček

2019

Forum Mathematicum

Self Cite

View full text Add to dashboard Cite

AbstractWe classify all tilting classes over an arbitrary commutative ring via certain sequences of Thomason subsets of the spectrum, generalizing the classification for noetherian commutative rings by Angeleri, Pospíšil, ŠÅ¥ovíček and Trlifaj (2014). We show that the n-tilting classes can equivalently be expressed as classes of all modules vanishing in the first n degrees of one of the following homology theories arising from a finitely generated ideal: {\operatorname{Tor}_{*}(R/I,-)}, Koszul homology, Čech homology, or local homology (even though in general none of those theories coincide). Cofinite-type n-cotilting classes are described by vanishing of the corresponding cohomology theories. For any cotilting class of cofinite type, we also construct a corresponding cotilting module, generalizing the construction of Šťovíček, Trlifaj and Herbera (2014). Finally, we characterize cotilting classes of cofinite type amongst the general ones, and construct new examples of n-cotilting classes not of cofinite type, which are in a sense hard to tell apart from those of cofinite type.

show abstract

“…In particular, we see that (4) As in the proof of 4.7(4), to construct the cotilting module we follow the process given in [15]. In particular a cotilting module that generates D d will be of the form In particular, a cotilting module inducing this class is Hom R (E(k), E(k)) R, the m-adic completion of R.…”

Section: The Definable Closure Of the Balanced Big Cohen-macaulay Mod...mentioning

confidence: 99%

Cotilting with balanced big Cohen-Macaulay modules

Bird¹

2019

Preprint

View full text Add to dashboard Cite

Over a d-dimensional Cohen-Macaulay local ring admitting a canonical module the definable closure of the class of balanced big Cohen-Macaulay modules is d-cotilting and is the smallest such class containing the maximal Cohen-Macaulay modules. We describe its cotilting structure and contrast it to the largest d-cotilting class containing the maximal Cohen-Macaulay modules. The enables an implicit classification of all d-cotilting classes containing the maximal Cohen-Macaulay modules.

show abstract

Cotilting modules over commutative Noetherian rings

Cited by 11 publications

References 13 publications

One-tilting classes and modules over commutative rings

One-tilting classes and modules over commutative rings

Tilting classes over commutative rings

Cotilting with balanced big Cohen-Macaulay modules

Contact Info

Product

Resources

About