Flat dimension, constructivity, and the Hilbert syzygy theorem

This paper is written in the spirit of constructive mathematics in the sense of Errett Bishop [3]. For a general introduction to constructive algebra see [11]. An algebraist with no background in constructive mathematics should have no trouble understanding the arguments, but might not see what obstacles are being overcome or sidestepped. From a procedural point of view, the di erence is that we do not use the law of excluded middle. From an intuitive point of view, the di erence is that our theorems are true under a computational interpretation. For example, a set is discrete if, for any two elements x and y, either x = y or x 6= y. Assuming the law of excluded middle, all sets are discrete; but in the absence of that law, there is no reason to believe that all sets are discrete. Moreover, the computational interpretation of a discrete set is one for which you have an algorithm that decides whether or not x = y. So, for example, the set of binary sequences is not discrete because (1) we cannot prove it is without using some form of the law of excluded middle, or (2) there is no algorithm that will decide, given an algorithm for a binary sequence, whether or not the sequence consists entirely of zeros. This is not to say that it is a theorem in constructive mathematics that the set of binary sequences is not discrete|on the contrary, any theorem in constructive mathematics is also a theorem in classical mathematics. As we can reduce questions about the equality of binary sequences to questions about the equality of real numbers, the real numbers constitute another example of a set that is not discrete. Arguments that require the axiom of choice are not constructively valid; in fact, the axiom of choice implies the law of excluded middle [6]. The failure of the axiom of choice comes up in the present context in that free modules need not be projective.