On the Krull Dimensions of the Steenrod Algebra

Abstract

For any prime p, we show that the mod p Steenrod algebra (a local ring with nil maximal ideal) has infinite little Krull dimension. This contrasts sharply with the case of a commutative (or noetherian) local ring with nil maximal ideal which must have little Krull dimension equal to 0. Also, we show that the Steenrod algebra has no Krull dimension, classical Krull dimension, or Gabriel dimension.

Key Words:

• Krull dimension
• Steenrod algebra

2010 Mathematics Subject Classification:

• 55S10
• 16P60
• 16N60

ACKNOWLEDGMENTS

The author would like to thank his advisor Charles J. Odenthal for his guidance and encouragement. The author would also like to thank the anonymous reviewer for all his or her suggestions and comments.

Notes

For most of this article, p is odd, and we will deal with the p = 2 case in the appendix.

This is sometimes called the little classical Krull dimension.

³See [5, p. 116] and also [6, Theorem 3].

⁴See the appendix below.

Communicated by E. Kirkman.