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.


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.