2
$\begingroup$

For a matroid $M$ let $C$ be the circuit ideal of $M$, that is, the Stanley-Reisner ideal of independence complex of $M$. Then there are simple ideal-theoretic operations that take $C$ to the facet ideal $F$ of $M$ (and vice versa). Moreover, Alexander duality takes $C$ to the cofacet ideal $F^*$ of $M$. So it is possible to "idealize" the matroid cryptomorphisms between (co)circuits to (co)facets.

Now drop the assumption on $C$ that it is square-free and its generators satisfy the circuit exchange axiom. All of the operations needed above are defined for the pair $(\tau, C)$ where $\tau$ is a monomial that is divisible by the least common multiple of the minimal generators of $C$. So for such a pair it is not hard to construct a tuple $(C, F, F^*, C^*)$ of monomial ideals that satisfy the same "idealized matroid cryptomorphisms" as matroids do. In particular, for $F$ appropriated defined we have $F^* = \{\tau/f | f \in G(F)\} = C^{[\mathbf{t}]}$ where $\tau = \mathbf{x}^{\mathbf{t}}$.

Question: Have such tuples of monomial ideals been explicitly considered in the literature?

Here are some examples where some of these ideals appear.

  • For a simplicial complex with Stanley-Reisner ideal $C$ and facet ideal $F$ the triples $(C, F, F^* := C^{[\boldsymbol{1}]})$ are considered in this paper (see Lemma 1.2).
  • If $F$ is a polymatroidal ideal minimally generated by $G$ and $\tau$ is any monomial divisible by $\mathrm{lcm} (G)$, then it's easy to show that $F^* = \langle \tau/f | f \in G \rangle$ is a polymatroidal ideal. Such ideals are natural generalizations of matroid facet ideals. While they have been widely studied, I have found no reference in the literature to a similar generalization of matroid circuit ideals.
  • Riemann-Roch ideals are those artinian monomial ideals $C$ for which there exists a $\mathbf{x}^{\mathbf{t}}$ such that $F^* = C^{[\mathbf{t}]} = \{\tau/f^* | f^* \in G(F^*)\}$. So they are similar in flavor to (identically) self-dual matroids.
$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.