## Cotensor products of modules

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- by L. Abrams and C. Weibel PDF
- Trans. Amer. Math. Soc.
**354**(2002), 2173-2185 Request permission

## Abstract:

Let $C$ be a coalgebra over a field $k$ and $A$ its dual algebra. The category of $C$-comodules is equivalent to a category of $A$-modules. We use this to interpret the cotensor product $M \square N$ of two comodules in terms of the appropriate Hochschild cohomology of the $A$-bimodule $M \otimes N$, when $A$ is finite-dimensional, profinite, graded or differential-graded. The main applications are to Galois cohomology, comodules over the Steenrod algebra, and the homology of induced fibrations.## References

- Lowell Abrams,
*Modules, comodules, and cotensor products over Frobenius algebras*, J. Algebra**219**(1999), no. 1, 201–213. MR**1707668**, DOI 10.1006/jabr.1999.7901 - Frank W. Anderson and Kent R. Fuller,
*Rings and categories of modules*, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York-Heidelberg, 1974. MR**0417223** - Armand Brumer,
*Pseudocompact algebras, profinite groups and class formations*, J. Algebra**4**(1966), 442–470. MR**202790**, DOI 10.1016/0021-8693(66)90034-2 - P. Cartier, “Cohomology des coalgèbres (Exposés 4,5),” In
*Séminaire Sophus Lie, 1955–56*, Faculté des Sciences de Paris, 1957. - Saunders MacLane,
*Steinitz field towers for modular fields*, Trans. Amer. Math. Soc.**46**(1939), 23–45. MR**17**, DOI 10.1090/S0002-9947-1939-0000017-3 - Yukio Doi,
*Homological coalgebra*, J. Math. Soc. Japan**33**(1981), no. 1, 31–50. MR**597479**, DOI 10.2969/jmsj/03310031 - Samuel Eilenberg and John C. Moore,
*Homology and fibrations. I. Coalgebras, cotensor product and its derived functors*, Comment. Math. Helv.**40**(1966), 199–236. MR**203730**, DOI 10.1007/BF02564371 - John Fogarty,
*Invariant theory*, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0240104** - Saunders Mac Lane,
*Homology*, Die Grundlehren der mathematischen Wissenschaften, Band 114, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. MR**0156879** - John W. Milnor and John C. Moore,
*On the structure of Hopf algebras*, Ann. of Math. (2)**81**(1965), 211–264. MR**174052**, DOI 10.2307/1970615 - Rolf Schneider,
*Krümmungsschwerpunkte konvexer Körper. I*, Abh. Math. Sem. Univ. Hamburg**37**(1972), 112–132 (German). MR**307039**, DOI 10.1007/BF02993906 - Douglas C. Ravenel,
*Complex cobordism and stable homotopy groups of spheres*, Pure and Applied Mathematics, vol. 121, Academic Press, Inc., Orlando, FL, 1986. MR**860042** - Moss E. Sweedler,
*Hopf algebras*, Mathematics Lecture Note Series, W. A. Benjamin, Inc., New York, 1969. MR**0252485** - Earl J. Taft,
*Reflexivity of algebras and coalgebras*, Amer. J. Math.**94**(1972), 1111–1130. MR**309992**, DOI 10.2307/2373566 - Charles A. Weibel,
*An introduction to homological algebra*, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR**1269324**, DOI 10.1017/CBO9781139644136 - Lech Witkowski,
*On coalgebras and linearly topological rings*, Colloq. Math.**40**(1978/79), no. 2, 207–218. MR**547861**, DOI 10.4064/cm-40-2-207-218

## Additional Information

**L. Abrams**- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- Address at time of publication: Department of Mathematics, George Washington University, Washington, D.C. 20052
- Email: labrams@gwu.edu
**C. Weibel**- Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
- MR Author ID: 181325
- Email: weibel@math.rutgers.edu
- Received by editor(s): April 18, 2000
- Received by editor(s) in revised form: June 14, 2001
- Published electronically: February 1, 2002
- Additional Notes: The second author was partially supported by NSF grants.
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**354**(2002), 2173-2185 - MSC (2000): Primary 16E30; Secondary 16W30, 16E40
- DOI: https://doi.org/10.1090/S0002-9947-02-02883-0
- MathSciNet review: 1885648