A note on Olssonʼs Conjecture

Héthelyi, László; Külshammer, Burkhard; Sambale, Benjamin

doi:10.1016/j.jalgebra.2012.08.010

Cited by 15 publications

(45 citation statements)

References 34 publications

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“…In [Héthelyi et al 2013], we also verified Olsson's conjecture for defect groups of p-rank 2 provided p > 3. We use the opportunity to explore the case p = 3 in more detail.…”

Section: Olsson's Conjecturesupporting

confidence: 69%

“…In particular, C D (u) is a defect group of the block b u . Now the claim follows from [Héthelyi et al 2013, Proposition 2.5(ii)].…”

Section: Olsson's Conjecturementioning

confidence: 83%

“…We have seen in [Héthelyi et al 2013] that Olsson's conjecture holds for all controlled 2-blocks of defect at most 5. Using the table above, we remove the controlled condition.…”

Section: Olsson's Conjecturementioning

confidence: 99%

“…Keywords: 2-blocks, bicyclic defect groups, Brauer's k(B)-conjecture, Alperin's weight conjecture. [Robinson 1991, Theorem 3.4] (a stronger bound can be found in [Héthelyi et al 2013]). …”

Section: Introductionmentioning

confidence: 99%

“…In particular, we handle the 2-blocks of defect at most 5 and some 3-blocks with defect group of 3-rank 2 that were left over in [Héthelyi et al 2013]. …”

Section: Introductionmentioning

confidence: 99%

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2013

Algebra Number Theory

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We prove that limits of multiplicities associated to graded families of ideals exist under very general conditions. Most of our results hold for analytically unramified equicharacteristic local rings with perfect residue fields. We give a number of applications, including a "volume = multiplicity" formula, generalizing the formula of Lazarsfeld and Mustat ,ȃ , and a proof that the epsilon multiplicity of Ulrich and Validashti exists as a limit for ideals in rather general rings, including analytic local domains. We prove a generalization of this to generalized symbolic powers of ideals proposed by Herzog, Puthenpurakal and Verma. We also prove an asymptotic "additivity formula" for limits of multiplicities and a formula on limiting growth of valuations, which answers a question posed by the author, Kia Dalili and Olga Kashcheyeva. Our proofs are inspired by a philosophy of Okounkov for computing limits of multiplicities as the volume of a slice of an appropriate cone generated by a semigroup determined by an appropriate filtration on a family of algebraic objects.

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“…In [Héthelyi et al 2013], we also verified Olsson's conjecture for defect groups of p-rank 2 provided p > 3. We use the opportunity to explore the case p = 3 in more detail.…”

Section: Olsson's Conjecturesupporting

confidence: 69%

“…In particular, C D (u) is a defect group of the block b u . Now the claim follows from [Héthelyi et al 2013, Proposition 2.5(ii)].…”

Section: Olsson's Conjecturementioning

confidence: 83%

“…We have seen in [Héthelyi et al 2013] that Olsson's conjecture holds for all controlled 2-blocks of defect at most 5. Using the table above, we remove the controlled condition.…”

Section: Olsson's Conjecturementioning

confidence: 99%

“…Keywords: 2-blocks, bicyclic defect groups, Brauer's k(B)-conjecture, Alperin's weight conjecture. [Robinson 1991, Theorem 3.4] (a stronger bound can be found in [Héthelyi et al 2013]). …”

Section: Introductionmentioning

confidence: 99%

“…In particular, we handle the 2-blocks of defect at most 5 and some 3-blocks with defect group of 3-rank 2 that were left over in [Héthelyi et al 2013]. …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations