Algebra

Abstract

A proper understanding of the meaning of the concept of involution in the form introduced in Chapter 3 requires certain ideas from (commutative) algebra collected in this chapter. Some classical references for all mentioned topics are [98, 99, 125]; older textbooks on commutative algebra are usually less constructive. The rather new books [185, 267, 268] put strong emphasis on computational issues (using the computer algebra systems Singular and CoCoa, respectively). Basic algebraic notions are introduced in [281, 415]; for the theory of non-commutative rings we mention [171, 276, 319].

The first section quickly reviews some basic algebraic structures like monoids, rings, algebras etc. Much of the material is elementary. Special emphasis is put on modules, as they are central for the algebraic analysis of the symbol of a differential equation. In particular, we introduce the Hilbert function and polynomial, as similar concepts appear for differential equations in Section 8.2. A simple method for the determination of the Hilbert function may be considered as one motivation for the introduction of involutive bases.

The second section reviews a few basic concepts from homological algebra. We introduce (co)homology modules for complexes of $\mathcal{R}$-modules and discuss resolutions. These notions are in particular used in Chapters 5 and 6. Section 10.5 requires in addition some knowledge about exact and derived functors. The third section concerns the more specialised topic of coalgebras and comodules; these concepts are only used in Chapter 6.