We also have a one to one correspondence between prime ideals in $\Gamma(V)$ and irreducible affine subvarieties of $V$. We know from the weak nullstellensatz that for an affine algebraic variety $V$, the points in $V$ have a one to one correspondence with the maximal ideals in $\Gamma(V)$. This also means that any affine algebraic variety naturally lives in some affine $n$-space, as the projection $k\rightarrow k/I$ induces an inclusion $V(I)\rightarrow k^n$. This is good, because these are exactly 1 the algebras we can use to study algebraic varieties. It turns out that every finitely generated $k$-algebra is of the form $k/I$, where $I$ is some ideal in $k$. We did not actually state any results about this correspondence, so before we tackle Noether normalization we need to do so.įirst off, Noether normalization holds for any finitely generated $k$-algebra, so it would be helpful for us to understand how all of these behave. Last time we covered some very introductory basics regarding algebraic varieties, and in particular we looked a bit at the correspondence between ideals in $k$ and affine algebraic sets in $k^n$. We will encounter some of these again in various forms below. Being a finitely generated $k$-algebra is often called being a finite type $k$-algebra, and being a finitely generated module over a ring $B$ is often called being finite over $B$, or being an integral extension of $B$. There are several ways to package this information. Then there exists a non-negative integer $d$ and algebraically independent elements $y_1, \ldots, y_d$, such that $A$ is a finitely generated module over $k$.
Lemma (Noether normalization for algebras): Let $A$ be a finitely generated commutative $k$-algebra. We restrict ourselves to infinite fields $k$, and perhaps also algebraically closed ones. As we saw in the last post, algebraic geometry is very much focused on finitely generated $k$-algebras, and polynomial rings, so the fact that geometric information arises from the result is hopefully not very surprising. Such an algebra is not necessarily a $k$-vector space, but, Noether normalization tells us that it is a finitely generated module over a polynomial ring over $k$.
Noether normalization tells us information about algebras, in particular finitely generated algebras over a field $k$.
Dim notion definition update#
So, I thought I would update last years post with my new knowledge, as well as generalize the intuition to schemes - which we introduced in the last post. One year later, I’m still not comfortable, but a bit more than last year. When I wrote it I didn’t yet understand all the pieces, as I was not very comfortable working with algebraic geometry. The first post on this blog is titled “geometric intuition”, and discusses the geometry behind Noether’s normalization lemma.
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