I have been thinking about the concept of specialization of algebras defined over the field of rational functions $k = \mathbb{C}(t)$ (I'm using $\mathbb{C}$ for my work, but the question can be asked over any field). In my examples the algebras are quadratic, but I don't think that this is essential. The main question I want to ask is this: suppose I have a PBW-algebra over $k = \mathbb{C}(t)$ that is specializable in an appropriate sense (described below). When can I conclude that the specialized algebra has the same size as the original?

## Setup

The setup uses the language of reduction systems as in George Bergman's (excellent) paper The Diamond Lemma for Ring Theory. Start with a finite set $X = \{ x_1, \dots, x_n \}$ of generators, let $\langle X \rangle$ denote the free semigroup with unit on $X$ (i.e. the words on the alphabet $X$), and let $k \langle X \rangle$ be the free algebra on $X$.

A reduction system is a set $S$ of pairs $\sigma = (W_\sigma, f_\sigma)$, where $W_\sigma \in \langle X \rangle$ and $f_\sigma \in k \langle X \rangle$. Then the associated $k$-algebra is $$ A = k \langle X \rangle / (W_\sigma - f_\sigma \mid \sigma \in S). $$ Now we assume that we have a semigroup partial order $\leq$ on $\langle X \rangle$ (with descending chain condition), compatible with multiplication in the sense that $ A \leq A'$ implies that $BAC \leq BA'C$ for all $B,C \in \langle X \rangle$. Furthermore we assume that for each $\sigma = (W_\sigma, f_\sigma)\in S$, the element $f_\sigma \in k \langle X \rangle$ is a linear combination of monomials that are less than $W_\sigma$.

The idea is that you replace (reduce) certain monomials (the $W_\sigma$) with lower-order terms (the $f_\sigma$). A monomial $w \in \langle X \rangle$ is defined to be irreducible with respect to $S$ if $w$ does not contain any $W_\sigma$ as a subword. The classes of the irreducible monomials always span the algebra $A$; the Diamond Lemma gives a criterion (all ambiguities must be resolvable) in order for the classes of irreducible monomials to be independent. In that case we say that $A$ is a PBW-algebra and that $X$ is a set of PBW-generators.

## Specialization

Say that the algebra $A$ is specializable (at $0$) if each $f_\sigma$ is of the form $$ f_\sigma = \sum_{w \in \langle X \rangle} c_\sigma^w w, $$ where each coefficient $c_\sigma^w$ lies in the local subring $k_0 \subseteq k$ consisting of rational functions that do not have a pole at $0$.

Then we define a reduction system $S_0$ for the specialized algebra $A_0$ by simply evaluating all of the relations for $A$ at $t = 0$. Formally, for each $\sigma = (W_\sigma, f_\sigma) \in S$ we define $$ \sigma_0 = (W_\sigma, f_\sigma(0)) \in S_0,$$ where $$f_\sigma(0) = \sum_{w \in \langle X \rangle} c_\sigma^w(0) w.$$ This is well-defined because we assumed that none of the rational functions $c_\sigma^w$ had a pole at zero. Finally, define the specialization of $A$ at $0$ to be the $\mathbb{C}$-algebra $$ A_0 = \mathbb{C} \langle X \rangle / (W_\sigma - f_\sigma(0)). $$ Note that the specialized reduction system $S_0$ has the same set of irreducible words as $S$ did.

## Question

If the irreducible words are independent in $A$, are they independent in $A_0$? Another way of putting it is this: if $X$ was a set of PBW-generators for the original $\mathbb{C}(t)$-algebra $A$, is it also a set of PBW-generators for the $\mathbb{C}$-algebra $A_0$?