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Modules, comodules and cotensor products over Frobenius algebras

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Lowell Abrams

Rings and Algebras · Submitted 8 Jun 1998

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

The authors' main claim We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module.

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arXiv IDmath/9806044 (version 3)
Pith ID2SWNQK43, Pith's permanent record of this paper and its review
CitationLowell Abrams. "Modules, comodules and cotensor products over Frobenius algebras." arXiv:math/9806044 (1998). https://pith.science/paper/2SWNQK43
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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, 5,047 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.42 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M.

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 characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. result 0.52 0.92 0.24 asserted 1.00
s1 We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. result 0.97 0.95 0.18 asserted 1.00
s2 This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. contribution 0.34 0.82 0.16 asserted 1.00
s3 We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module. result 0.97 0.93 0.39 asserted 1.00
s4 Some properties of D are described. contribution 0.66 0.84 0.03 asserted 0.63
s5 Finally, we show that when A is a symmetric algebra, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M. result 0.99 0.93 0.25 asserted 1.00

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

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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 characterize noncommutative Frobenius algebras A in terms of the existence of a coproduct which is a map of left A^e-modules. We show that the category of right (left) comodules over A, relative to this coproduct, is isomorphic to the category of right (left) modules. This isomorphism enables a reformulation of the cotensor product of Eilenberg and Moore as a functor of modules rather than comodules. We prove that the cotensor product M \Box N of a right A-module M and a left A-module N is isomorphic to the vector space of homomorphisms from a particular left A^e-module D to N \otimes M, viewed as a left A^e-module. Some properties of D are described. Finally, we show that when A is a symmetric algebra, the cotensor product M \Box N and its derived functors are given by the Hochschild cohomology over A of N \otimes M.

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