Nullstellen and Subdirect Representation

David Hilbert’s solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hilbert-NoetherBirkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere. The aim of this article is to analyze this linkage in the most elementary terms and then, based on our work of the 1980s, to present a general categorical framework for Birkhoff’s theorem. Introduction The first purpose of this article is to exhibit in elementary algebraic terms the linkage between Hilbert’s celebrated Nullstellensatz [6] for systems of polynomial equations and Birkhoff’s Subdirect Representation Theorem [2] for general algebras, as facilitated by Noether’s work [15] on the decomposition of ideals in rings. Hence, in Section 1 we give a brief tour of the development of the Nullstellensatz through Hilbert, Noether and Birkhoff. We do so not from a strictly historical perspective; rather, in today’s language we present six versions of the theorem as marked by these three great mathematicians and show how their proofs are interrelated, referring to them as the HNB Theorems. A seventh and an eighth version are given in Section 3, after a discussion of the categorical notion of subdirect irreducibility in Section 2. We illuminate the notion by examples, both traditional and unconventional, in particular in comma categories. We touch upon the dual notion only briefly, but refer the reader to substantial recent work by Matias Menni [14] on Lawvere’s concept of cohesion in this context. With a suitable notion of finitariness we formulate the all-encompassing seventh version of the HNB Theorem without recourse to any limits or colimits in the ambient category. The morphism version of it leads to atypical factorizations of morphisms, in the sense that even in standard categories like that of sets one obtains factorizations of maps in a constructive manner (without recourse to choice) which, however, may not be obtained in a functorial way. Acknowledgement I am indebted to Bill Lawvere for his suggestion to study the topic of this paper and his subsequent interest in and comment on a preliminary (but extended) version of this work, presented in part at The European Category Theory Meeting in Haute Bodeux (Belgium) in 2003. I am also grateful to László Márki for his careful reading of an earlier version of this paper and for detecting an inaccuracy that may be traced back to Birkhoff’s original paper (see 2.3(4) below).