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dualRingToric -- computes the Koszul dual of a multigraded polynomial ring or exterior algebra

Synopsis

• Usage:
dualRingToric R
• Inputs:
• R, , either a standard polynomial ring, or an exterior algebra
• Optional inputs:
• Variable => , default value x
• SkewVariable => , default value e
• Outputs:
• , which is the Koszul dual of R

Description

This method computes the Koszul dual of a polynomial ring or exterior algebra. In particular, if $S = k[x_0, \ldots, x_n]$ is a $\mathbb{Z}^m$-graded ring for some $m$, and $\operatorname{deg}(x_i) = d_i$, then the output of dualRingToric is the $\mathbb{Z}^{m+1}$-graded exterior algebra on variables $e_0, \ldots, e_n$ with degrees $(-d_i, -1)$.

 i1 : R = ring(hirzebruchSurface(2, Variable => y)) o1 = R o1 : PolynomialRing i2 : E = dualRingToric(R, SkewVariable => f) o2 = E o2 : PolynomialRing, 4 skew commutative variable(s)

On the other hand, if $E$ is a $\mathbb{Z}^{m+1}$-graded exterior algebra on $n+1$ variables $e_0, \ldots, e_n$ with $\operatorname{deg}(e_i) = (-d_i, -1)$, then dualRingToric E is the $\mathbb{Z}^m$-graded polynomial ring $k[x_0, \ldots, x_n]$ with $\operatorname{deg}(x_i) = d_i$.

 i3 : RY = dualRingToric E o3 = RY o3 : PolynomialRing i4 : degrees RY == degrees R o4 = true

This method preserves the coefficient ring of the input ring.

 i5 : S = ZZ/101[x,y,z, Degrees => {1,1,2}] o5 = S o5 : PolynomialRing i6 : E' = dualRingToric S o6 = E' o6 : PolynomialRing, 3 skew commutative variable(s) i7 : coefficientRing E' === coefficientRing S o7 = true