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2016 | OriginalPaper | Chapter

4 Recollement: The Theory of an Idempotent

Author : Benjamin Steinberg

Published in: Representation Theory of Finite Monoids

Publisher: Springer International Publishing

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Abstract

In this chapter we provide an account of the theory connecting the category of modules of a finite dimensional algebra A with the module categories of the algebras eAe and AAeA, for an idempotent e ∈ A, known as recollement  [BBD82, CPS88, CPS96]. We first learned of this subject from the monograph of Green [Gre80, Chapter 6]. A presentation much closer in spirit to ours is that of Kuhn [Kuh94b]. In the next chapter, we shall apply this theory to construct the irreducible representations of a finite monoid and in a later chapter we shall extend the results to finite categories.

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Literature
[BBD82]
go back to reference A.A. Beĭlinson, J. Bernstein, P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I (Luminy, 1981). Astérisque, vol. 100 (Soc. Math. France, Paris, 1982), pp. 5–171 A.A. Beĭlinson, J. Bernstein, P. Deligne, Faisceaux pervers, in Analysis and Topology on Singular Spaces, I (Luminy, 1981). Astérisque, vol. 100 (Soc. Math. France, Paris, 1982), pp. 5–171
[CPS88]
go back to reference E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988) E. Cline, B. Parshall, L. Scott, Finite-dimensional algebras and highest weight categories. J. Reine Angew. Math. 391, 85–99 (1988)
[CPS96]
go back to reference E. Cline, B. Parshall, L. Scott, Stratifying endomorphism algebras. Mem. Am. Math. Soc. 124 (591), viii+119 (1996) E. Cline, B. Parshall, L. Scott, Stratifying endomorphism algebras. Mem. Am. Math. Soc. 124 (591), viii+119 (1996)
[Gre80]
go back to reference J.A. Green, Polynomial Representations ofGLn. Lecture Notes in Mathematics, vol. 830 (Springer, Berlin, 1980) J.A. Green, Polynomial Representations ofGLn. Lecture Notes in Mathematics, vol. 830 (Springer, Berlin, 1980)
[Kuh94b]
go back to reference N.J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. II. K-Theory 8 (4), 395–428 (1994) N.J. Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra. II. K-Theory 8 (4), 395–428 (1994)
Metadata
Title
4 Recollement: The Theory of an Idempotent
Author
Benjamin Steinberg
Copyright Year
2016
DOI
https://doi.org/10.1007/978-3-319-43932-7_4