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

Synopsis

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

See also

Ways to use dualRingToric:

For the programmer

The object dualRingToric is a method function with options.