Dimension

[Caccio Dj]

Hello everybody. My name is nicola, i'm new in the forum. I have recently started using macaulay2 and would like to know more about the concept of dimension (i.e. what kind of definition is used). In particular, how to calculate the dimension of a variety.

I thank anyone who answers me.

[Baptiste Calmès]

Generally speaking, the notion of dimension used in commutative algebra and algebraic geometry is the Krull dimension.
It is defined for a ring, and thus for an affine variety.

A variety being defined as being locally affine varieties, its dimension is then the maximum of Krull dimensions of these affine varieties. Of course, one can show it is independent of which decomposition you choose.

Practically speaking, computing it for a given variety thus depends on how your variety is described in the first place.
In the most obvious case, if your variety is Spec(A) where A is a ring, then use this:
https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2-1.19.1/share/doc/Macaulay2/Macaulay2Doc/html/_dim_lp__Ring_rp.html

Best,

Baptiste

> Le 6 avr. 2022 à 10:55, Caccio Dj <nicola...@live.it> a écrit :
>
> Hello everybody. My name is nicola, i'm new in the forum. I have recently started using macaulay2 and would like to know more about the concept of dimension (i.e. what kind of definition is used). In particular, how to calculate the dimension of a variety.
> I thank anyone who answers me.
>

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