By Louis Comtet
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Additional info for Analyse Combinatoire
However in some cases it is possible to solve the classiﬁcation problem. 1. 2-dimensional systems Let R be a PID and Σ = (A, B) a reachable m-input 2-dimensional linear system over R. Since U1 (B) = R there exist invertible matrices P and Q such that P BQ = 1 0 0 d ... 0 ... 0 where d is a generator of U2 (B). Considering the relevant actions of the feedback group we obtain the following result. P ROPOSITION 97. Let Σ = (A, B) be a 2-dimensional reachable linear system over a principal ideal domain R.
An−1 b)P −1 = c(A). Taking an adequate feedback matrix one has the following result. P ROPOSITION 82. Let Σ = (A, b) be a reachable single-input n-dimensional system. Then Σ is feedback equivalent to the system ⎛ ⎛ 0 1 ⎜ ⎜0 0 ⎜ ⎜. ⎜. Σ =⎜ ⎜A = ⎜ . ⎝ ⎝0 0 0 0 0 1 .. ··· ··· .. 0 0 ··· ··· ⎛ ⎞⎞ ⎞ 0 0 ⎜ 0 ⎟⎟ 0⎟ ⎜ ⎟⎟ .. ⎟ ˆ ⎜ .. ⎟ ⎟ , b = ⎜ . ⎟⎟ . ⎝ 0 ⎠⎠ 1⎠ 0 1 In particular, there exists a unique class of reachable single-input n-dimensional systems. The next objective is to study the multi-input case (m > 1).
Let R be a commutative ring and M an R-module. (i) M is projective if M is a direct summand of a free R-module or equivalently the functor Hom(M, − ) is exact. (ii) M is ﬂat if the functor M ⊗R − is exact. See [5, Chapters I and II] and  for the main properties of these modules. When M is ﬁnitely generated one has the following characterization. T HEOREM 19. Let M be a ﬁnitely generated R-module. Then: (i) M is ﬂat if and only if Mp is a free Rp -module for all prime ideals p of R. Linear algebra over commutative rings 13 (ii) M is projective if and only if M is ﬂat and the rank function rk : Spec(R) → Z p → rankRp Mp is continuous.