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I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus

1. The regular (non-singular) locus can fail to be open.
2. Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
3. Normalization need not be a module-finite extension.

Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies (and avoid these after some standard operations).

The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as are complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.

I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus

1. The regular (non-singular) locus can fail to be open.
2. Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
3. Normalization need not be a module-finite extension.

Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies and some others (and avoid these after some standard operations).

The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as are complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.

I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus

1. The regular (non-singular) locus can fail to be open.
2. Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
3. Normalization need not be a module-finite extension.

Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies (and avoid these after some standard operations).

The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as do complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.

I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus

1. The regular (non-singular) locus can fail to be open.
2. Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
3. Normalization need not be a module-finite extension.

Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies (and avoid these after some standard operations).

The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as are complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.

I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus

1. The regular (non-singular) locus can fail to be open.
2. Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the same length).
3. Normalization need not be a module-finite extension.

Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies (and avoid these after some standard operations).

The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as do complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.

I'd just like to briefly add that Noetherian rings can be surprisingly non-geometric. In particular, they can fail to be excellent. Thus

1. The regular (non-singular) locus can fail to be open.
2. Notions of dimension need not be reasonable (two maximal chains of primes with the same top and bottom members can be the different lengths).
3. Normalization need not be a module-finite extension.

Of course, excellent is just a hodge-podge of conditions that avoid these particular pathologies (and avoid these after some standard operations).

The usual rings (finite type over a field or $\mathbb{Z}$) are excellent, as do complete local rings. However, it can be hard to prove that an arbitrary ring is excellent.