For any module M over a Noetherian ring R there is a map $u: M \to H$ that is versal for maps from M to free modules; that is, such that any map from M to a free module factors through u. Such a map may be constructed by choosing a set of s generators for Hom(M,R), and using them as the components of a map $u: M \to H := R^s$.
(NOTE: In the paper of Eisenbud, Huneke and Ulrich cited below, the versal map is described with the term ``universal'', which is misleading, since the induced map from H is generally not unique.)
Suppose that $M$ has a free presentation $F \to G$, and let $u1$ be the map $u1: G\to H$ induced by composing $u$ with the surjection $p: G \to M$. By definition, the Rees algebra of $M$ is the image of the induced map $Sym(u1): Sym(G)\to Sym(H)$, and thus can be computed with symmetricKernel(u1). The map u is computed from the dual of the first syzygy map of the dual of the presentation of $M$.
We first give a simple example looking at the syzygy matrix of the cube of the maximal ideal of a polynomial ring.
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A more complicated example.
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Here is an example from the paper "What is the Rees Algebra of a Module" by David Eisenbud, Craig Huneke and Bernd Ulrich, Proc. Am. Math. Soc. 131, 701-708, 2002. The example shows that one cannot, in general, define the Rees algebra of a module by using *any* embedding of that module, even when the module is isomorphic to an ideal; this is the reason for using the map provided by the routine versalEmbedding. Note that the same paper shows that such problems do not arise when the ring is torsion-free as a ZZ-module, or when one takes the natural embedding of the ideal into the ring.
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As a module (or ideal), $Hom(I,R^1)$ is minimally generated by 3 elements, and thus a versal embedding of $I$ into a free module is into $R^3$.
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it is injective:
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It is easy to make two other embeddings of $I$ into free modules. One is the natural inclusion of $I$ into $R$ as an ideal:
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Another is the map defined by multiplication by x and y.
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We can compose $ui, inci$ and $gi$ with a surjection $R\to i$ to get maps $u:R^1 \to R^3, inc: R^1 \to R^1$ and $g:R^1 \to R^2$ having image $i$.
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We now form the symmetric kernels of these maps and compare them. Note that since symmetricKernel defines a new ring, we must bring them to the same ring to make the comparison. First the map u, which would be used by reesIdeal:
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Next the inclusion:
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Finally, the map g1:
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The following test yields ``true'', as implied by the theorem of Eisenbud, Huneke and Ulrich.
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But the following yields ``false'', showing that one must take care in general, which inclusion one uses.
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The object versalEmbedding is a method function.