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The André–Quillen cohomology of commutative monoids

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Abstract

We observe that Beck modules for a commutative monoid are exactly modules over a graded commutative ring associated to the monoid. Under this identification, the Quillen cohomology of commutative monoids is a special case of the André–Quillen cohomology for graded commutative rings, generalizing a result of Kurdiani and Pirashvili. To verify this we develop the necessary grading formalism. The partial cochain complex developed by Pierre Grillet for computing Quillen cohomology appears as the start of a modification of the Harrison cochain complex suggested by Michael Barr.

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Notes

  1. But Quillen inadvertently omits the exactness condition.

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Acknowledgements

We are grateful to Pierre Grillet for forwarding us, in response to a letter from us outlining the results presented here, an early copy of a paper in which a similar story is worked out. He uses somewhat different language—his “multi” objects are our graded objects—but he did not make the connection with Harrison homology that we establish here. This work was carried out under the auspices of a program, supported by MIT’s Jameel World Education Laboratory, designed to foster collaborative research projects involving students from MIT and Palestinian universities. We acknowledge with thanks the contributions made by early participants in this program—Mohammad Damaj and Ali Tahboub of Birzeit University and Hadeel AbuTabeekh of An-Najah National University—as well as the support of Palestinian faculty—Reema Sbeih and Mohammad Saleh at Birzeit and Khalid Adarbeh and Muath Karaki at NNU. We thank Professor Victor Kac for pointing out to us the relevance of [14]. The first author acknowledges support by the MIT UROP office. Finally, we thank the referee for such a careful reading of the document.

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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Bhavya Agrawalla & Haynes Miller

  2. Department of Mathematics, Birzeit University, Ramallah, Palestine

    Nasief Khlaif

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  1. Bhavya Agrawalla
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  2. Nasief Khlaif
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Correspondence to Haynes Miller.

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Communicated by Mark V. Lawson.

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Agrawalla, B., Khlaif, N. & Miller, H. The André–Quillen cohomology of commutative monoids. Semigroup Forum (2024). https://doi.org/10.1007/s00233-024-10423-z

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