toricRR M
toricRR(M,L)
Let $A$ be a finitely generated free abelian group. Given an $A$-graded polynomial ring $S$ with $A \oplus \mathbb{Z}$-graded Koszul dual exterior algebra E, the BGG functor $\mathbf{R}$ sends an $S$-module $M$ to a free differential $E$-module with linear differential: see Section 3 of the paper accompanying this package for details. The free $E$-module underlying $\mathbf{R}(M)$ is $\bigoplus_{d \in A} M_d \otimes_k E^*(-d, 0)$, where $E^*$ denotes the dual of $E$ over the ground field $k$. This module has rank given by the dimension of $M$ as a $k$-vector space, which is typically infinite. Thus, this method usually only computes a finite rank quotient of $\mathbf{R}(M)$. Specifically: toricRR(M) is the quotient of $\mathbf{R}(M)$ given by those summands $M_d \otimes_k E^*(-d, 0)$ such that $d = e + a \operatorname{deg}(x_i)$, where $e$ is a generating degree of $M$, $a \in \{0,1\}$, and $0 \le i \le n$.
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There is also an optional input for a list L of degrees in $A$: toricRR(M, L) is the quotient $\bigoplus_{a \in L} M_d \otimes_k E^*(-d, 0)$ of $\mathbf{R}(M)$.
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For some lists of degrees L, the denominator of the quotient is not a differential module. Hence, the method sometimes enlarges the list of degrees in order to have a well-defined quotient. We demonstrate that phenomenon with the following example.
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A heft vector is necessary for the computation to produce a well-defined differential module.
The object toricRR is a method function.