E0040 definition notation E0001

Taylor algebras

The grid algebra \(B_k\), the Taylor algebra \(A_k\), and the deformation algebra \(C_k\) are three faces of the same polynomial ring. The discrete calculus lives in \(B_k\) (coverings), the differential calculus in \(A_k\) (partitions), and \(C_k\) interpolates over \(\ik[t]\).

(1) Definition (Grid and Taylor algebras). Let \(\ik\) be a commutative ring. For \(\nu \in \IN_0^k\), define:

  • The grid algebra is

    \[ B_k^\nu := \ik[\delta_1, \dots, \delta_k] \,/\, ((\delta_i)_{\nu_i + 1}), \]

    where \((\delta_i)_{\nu_i+1} = \delta_i(\delta_i - 1) \cdots (\delta_i - \nu_i)\) is the falling factorial. For \(\IQ \subseteq \ik\), this is the function algebra on the grid \(\prod_i \set{0, \dots, \nu_i}\).

  • The Taylor algebra is

    \[ A_k^\nu := \ik[\eps_1, \dots, \eps_k] \,/\, (\eps_i^{\nu_i + 1}). \]

  • For \(\nu = \mathbf{1}\), write \(B_k = B_k^{\mathbf{1}}\) and \(A_k = A_k^{\mathbf{1}}\). Both are free of rank \(2^k\) with basis \(\delta^S\) resp. \(\eps^S\) for \(S \subseteq [k]\).

(2) Definition (Multiplication rules at \(\nu = \mathbf{1}\)).

  • In \(B_k\): \(\delta^S \delta^T = \delta^{S \cup T}\) (idempotent — overlaps absorbed). A product \(\delta^{T_1} \cdots \delta^{T_p}\) contributes to the top face \(\delta^{[k]}\) iff \(T_1 \cup \cdots \cup T_p = [k]\), i.e. a covering.

  • In \(A_k\): \(\eps^S \eps^T = \eps^{S \cup T}\) if \(S \cap T = \emptyset\), and \(\eps^S \eps^T = 0\) otherwise (nilpotent — overlapping products vanish). A product \(\eps^{T_1} \cdots \eps^{T_p}\) is nonzero iff the \(T_i\) are pairwise disjoint, i.e. a partition.

(3) Definition (Deformation algebra). For \(\nu \in \IN_0^k\):

  • The deformation algebra is

    \[ C_k^\nu := \ik[t][x_1, \dots, x_k] \,\big/\, \big({\textstyle\prod_{j=0}^{\nu_i}} (x_i - jt)\big). \]

  • For \(\nu = \mathbf{1}\), \(C_k = C_k^{\mathbf{1}}\) has relation \(x_i^2 = t x_i\), giving \(x^S x^T = t^{|S \cap T|} x^{S \cup T}\). At \(t = 1\) all coverings contribute; at \(t = 0\) only partitions remain.

(4) Lemma (Flatness and fibers). \(C_k^\nu\) is free over \(\ik[t]\) of rank \(\prod_i (\nu_i + 1)\), with fibers

\[ C_k^\nu / (t) \cong A_k^\nu, \qquad C_k^\nu / (t - 1) \cong B_k^\nu. \]

Proof. The relations are monic of degree \(\nu_i + 1\) in \(x_i\), so \(C_k^\nu\) is free over \(\ik[t]\) with monomial basis \(\set{x^\alpha : 0 \leq \alpha \leq \nu}\). The fiber identifications follow by substituting \(t = 0\) and \(t = 1\).

Used by: