I came across the following ring $A$, which appears as a Chow ring. I am wondering if it has been studied before; in particular, I am looking for a reference where this object might have been described.

The graded ring $A^n \subset \mathbb{Z}[x_1,\dots,x_n]$ is the subring consisting of polynomials $p$ such that $$p(x_1,\dots,x_{n-1},0)=p(x_1,\dots,x_{i-1},0,x_i,\dots,x_{n-1})$$ for all $i\geq 1$. For $m\geq n$, there is a surjective homomorphism $A^m \to A^n$ given by setting the last $m-n$ variables to $0$. This is an isomorphism in degrees smaller than $n$. We define $$A = \varprojlim A^n$$ in the category of graded rings. In particular, the degree $k$ part of $A$ is $$A_{(k)}=A^k_{(k)}.$$

  • $\begingroup$ Could you tell us as the Chow ring of what it appears ? $\endgroup$
    – F. C.
    Apr 24, 2015 at 19:53
  • $\begingroup$ This is the Chow ring of the stack of expansions, or equivalently the moduli stack of aligned log structures, as considered e.g. in arXiv:1110.2976. $\endgroup$ Apr 25, 2015 at 20:16
  • $\begingroup$ So apparently this has led to arXiv:1806.10700. Looks like it might finally get algebraic geometers to care about quasisymmetric functions :) $\endgroup$ Jul 4, 2018 at 14:24
  • $\begingroup$ @darijgrinberg So far, it has made one algebraic geometer care about them. We will see :). $\endgroup$ Jul 5, 2018 at 8:50

1 Answer 1


The ring $A^n$ is the set of polynomials $f\in\mathbb{Z}[x_1,\ldots,x_n]$ such that the coefficient of $x_{i_1}^{a_1}\ldots x_{i_k}^{a_k}$ is equal to the coefficient of $x_{j_1}^{a_1}\ldots x_{j_k}^{a_k}$ whenever $i_1<\ldots<i_k$ and $j_1<\ldots<j_k$. Polynomials satisfying this condition are called quasi-symmetric, and $A$ is the ring of quasi-symmetric functions.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.