2
$\begingroup$

Consider the complete intersection ideal ${\displaystyle (f,g_{1},g_{2},g_{3})\subset \mathbb {C} [x_{0},\ldots ,x_{n}]}$ and let $X$ be a projective variety defined by the (product) ideal sheaf ${\displaystyle {\mathcal {I}}=(f)(g_{1},g_{2},g_{3})}$.

How to calculate that

$${\displaystyle \bigoplus _{n\geq 0}{\frac {{\mathcal {I}}^{n}}{{\mathcal {I}}^{n+1}}}\cong {\frac {{\mathcal {O}}_{X}[a,b,c]}{(g_{2}a-g_{1}b,g_{3}a-g_{1}c,g_{3}b-g_{2}c)}}}$$

is an isomorphism of ${\mathcal {O}}_{{{\mathbb {P}}^{n}}}$-modules?

Source: example for graded algebra of affine cone.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.