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Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras

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Stephen Zito

Representation Theory · Submitted 4 Aug 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We give a complete classification of the projective dimension of a C-module inside the module category of B.

how it is stated · asserted 1.00 read the record →

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arXiv ID1608.01616 (version 1)
Pith IDCPMDTP6Z, Pith's permanent record of this paper and its review
CitationStephen Zito. "Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras." arXiv:1608.01616 (2016). https://pith.science/paper/CPMDTP6Z
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,138 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.15 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. contribution 0.74 0.86 0.04 asserted 0.82
s1 We investigate how various properties of a C-module are affected when considered in the module category of B. contribution 0.97 0.88 0.05 unclear 0.58
s2 We give a complete classification of the projective dimension of a C-module inside the module category of B. result 0.64 0.94 0.60 asserted 1.00
s3 If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. result 0.93 0.94 0.12 asserted 1.00
s4 In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module. result 1.00 0.95 0.30 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. We investigate how various properties of a C-module are affected when considered in the module category of B. We give a complete classification of the projective dimension of a C-module inside the module category of B. If a C-module M is rigid, we show two sufficient conditions for M to be a rigid B-module. In particular, if M is an indecomposable and rigid C-module, we prove M is always a rigid B-module.

Human review

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How the literature uses it

1 arXiv paper cite this in 2 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 3 sentences without a confident label

Cites as background 1 paper

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