Algebra Comutativa 2010

[dede]

Examen: Algebra Comutativa
Profesor: Dorin Popescu

1) \( I=(x_{1}x_{4},x_{1}x_{3}x_{5},x_{2}x_{3}x_{4},x_{2}x_{5}) \subset S=K[x_1,x_2,x_3,x_4,x_5] \) K corp. Calculati:
a) complexul simplicial asociat lui I
b) dim S/I, Ass S/I si dim \( S/p, p\in Ass S/I \)
c)\( depth_S S/I \)
d)\( pd_S I \)
e)\( pd_{S/I} K \)
f) S/I e Cohen Macaulay ?

2) \( J=(x^2y,y^2z,xz^2,xyz) \subset S=K[x,y,z] \) K corp. Calculati:
a) \( depth_S J \)
b) \( pd_S S/I \) si dim S/I
c) \( pd_{S/J} K \)
d) S/J e Cohen Macaulay ?

3) Enuntati si demonstrati Depth Lemma
4) Enuntati si demonstrati teorema Auslander Buchsbaum Serre.