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Results tagged with projective-resolution
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user 40297
4
votes
Accepted
Is the projective dimension of finite torsion-free modules over regular ring of dimension $n...
Yes. If $\operatorname{pd}_A(M)=n $, there exists a maximal ideal $\mathfrak{m}$ of $A$ such that $\operatorname{pd}_{A_{\mathfrak{m}}}(M_{\mathfrak{m}})=n $. By the Auslander-Buchsbaum theorem, this …
- 38k
8
votes
Accepted
Resolution of a torsion sheaf
Write the equation of $C$ as $\ XY-Z^2=0$, and consider the homomorphism $u: \mathcal{O}_{\mathbb{P}^2}(-1)^2\rightarrow \mathcal{O}_{\mathbb{P}^2}^2$ given by the matrix $\begin{pmatrix}
X & Z\\Z& Y
…
- 38k
2
votes
Accepted
Homological dimension of pure coherent sheaves and specialization
Take for $X$ the plane curve $X^3=Y^2T$, and for $F$ the ideal sheaf $(X-\pi ^2T,Y-\pi ^3T)$. Its restriction to the generic fiber is the ideal of a smooth point, hence is an invertible sheaf, while …
- 38k