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VersalDeformations :: checkComparisonTheorem

checkComparisonTheorem -- checks if the Piene-Schlessinger or Di Dennaro comparison theorem holds



Let $S$ be a polynomial ring in $n$ variables, and $F$ be a matrix representing a map of graded free modules over $S$. Let $E$ be the target of $F$, $M$ the cokernel, and $K$ the image. Denote by $\bar E,\bar{M},\bar{K}$ the corresponding sheaves. We are interested in comparing the degree zero local Quot functor parametrizing quotients of the module $E$ specializing to $M$, with the local Quot functor parametrizing quotients of the locally free sheaf $\bar E$ specializing to $\bar M$. In the special case that $E=S$, this means that we are comparing degree zero embedded deformations of the affine cone of $X=V(K)\subset \mathbb{P}^n$ with embedded deformations of $X$.

Let $d_1,\ldots,d_m$ be the degrees of the generators of the source of $F$. The comparison theorem of Piene and Schlessinger states that in the case $E=S$ and $K$ saturated, the above-mentioned functors are isomorphic if the natural maps $M_{d_i}\to H^0(\mathbb{P}^n,\bar{M}(d_i))$ are isomorphisms. This is equivalent to requiring that $H^1((\mathbb{P}^n,\bar{K}(d_i))=0$ for each $i$. More generally, the theorem of Di Gennaro may be used. Consider arbitrary $E$ as above, and suppose that $K$ is a truncation of a saturated submodule. Again, the above-mentioned functors are isomorphic if $H^1((\mathbb{P}^n,\bar{K}(d_i))=0$ for each $i$. See [PS85] and [DG89].

This method tests if the above hypotheses of Di Gennaro's comparison test are fulfilled. Inputing an ideal has the same effect as inputing gens F. In the following example, the comparison theorem does not hold for the ideal I, but does for the partial truncation J.

i1 : S = QQ[a..d];
i2 : I = ideal(a,b^3*c,b^4);

o2 : Ideal of S
i3 : J=ideal b^4+ideal (ambient basis(3,I))

             4   3   2    2    2      2                   2            2
o3 = ideal (b , a , a b, a c, a d, a*b , a*b*c, a*b*d, a*c , a*c*d, a*d )

o3 : Ideal of S
i4 : checkComparisonTheorem I

o4 = false
i5 : checkComparisonTheorem J

o5 = true

Ways to use checkComparisonTheorem :

For the programmer

The object checkComparisonTheorem is a method function.