# geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring. What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric interpretation?

Gorenstein rings have more complicated geometric meaning: They are local rings of affine varieties which may be singular at the point, but the maximal $$R$$-regular sequence in the maximal ideal i.e. the associated complete intersection subscheme is an irreducible space.
Regular rings $$\subset$$ CI rings $$\subset$$ Gorenstein rings
The simplest example of a complete intersection ring which is not regular is the ring $$k[x]/(x^{2})$$. The maximal ideal is the ideal $$(\overline{x})$$ and of course $$x$$ is a regular element in $$k[x]$$. For a ring which is Gorenstein but not CI, consider the $$0$$-dimensional local ring $$k[x,y,z]/(x^{2},y^{2},xz,yz,z^{2}-xy)$$. The unique maximal ideal here is the ideal $$(x,y,z)$$ but you can see that the ideal $$(x^{2},y^{2}, xz,yz, z^{2}-xy)_{(\overline{x}, \overline{y}, \overline{z})}$$ is not generated by a regular sequence.
• Of course in general it is hard to give a general recipe of how to construct such examples. In special cases there are some results which can be helpful. For example a ring of dimension $0$ is complete intersection iff the Fitting ideal of its maximal ideal is non-zero. Obviously this gives a possibility to construct singular rings with CI. Mar 6, 2014 at 21:21