I: Varieties

I.1: Affine Varieties

1.1 #to_work

Let \(Y\) be the plane curve \(y=x^2\) (i.e., \(Y\) is the zero set of the polynomial \(f=\) \(y-x^2\) ). Show that \(A(Y)\) is isomorphic to a polynomial ring in one variable over \(k\).
Let \(Z\) be the plane curve \(x y=1\). Show that \(A(Z)\) is not isomorphic to a polynomial ring in one variable over \(k\).

* Let \(f\) be any irreducible quadratic polynomial in \(k[x, y]\), and let \(W\) be the conic defined by \(f\). Show that \(A(W)\) is isomorphic to \(A(Y)\) or \(A(Z)\). Which one is it when?

\(A(Y) = k[x,y]/\left\langle{y-x^2}\right\rangle \cong k[t, t^2] \cong k[t]\).

\(A(Z) = k[x, y]/\left\langle{xy-1}\right\rangle \cong k[x^{\pm 1}]\). This is not isomorphic to any polynomial ring, since it contains an invertible element \(x^{-1}\) which is not in \(k\).

1.2 The Twisted Cubic Curve #completed

Let \(Y \subseteq \mathbf{A}^3\) be the set \(Y = \left\{{(t, t^2,t^3) {~\mathrel{\Big\vert}~}t\in k}\right\}\).

Show that \(Y\) is an affine variety of dimension 1.
Find generators for the ideal \(I\left(Y\right)\).
Show that \(A(Y)\) is isomorphic to a polynomial ring in one variable over \(k\).

We say that \(Y\) is given by the parametric representation \(x=t . y=t^2, z=t^3\).

Useful facts: \(\sqrt{I} = \sqrt{\prod p_i^{a_i}} = \prod p_i\) in a UFD when \(I\) is a principal ideal factored into irreducibles. An ideal is also radical iff the quotient is reduced, and \(\left\langle{f}\right\rangle\) is radical when \(f\) is irreducible.

For dimension in part 1, note \begin{align*}A(Y) = k[x,y]/\left\langle{y-x^2, z-x^3}\right\rangle \cong k[t, t^2,t^3] \cong k[t].\end{align*} One can now use that \(\operatorname{krulldim}k[t] = 1\) and Prop 1.7 (\(\dim Y = \operatorname{krulldim}A(Y)\)) to conclude \(\dim Y = 1\). This also shows part (3)
To see that \(Y\) is an affine variety, it STS it is an irreducible closed subset of \({\mathbf{A}}^3\). It is closed since \(Y = V(y-x^2, z-x^3)\). It is irreducible iff \(I(Y) = \left\langle{y-x^2, z-x^3}\right\rangle\) is prime by Cor 1.4. It is prime, since \(k[x,y]/I(Y) \cong k[t]\) is a domain and \(I(Y)\) is radical since it is generated by irreducibles.

Let \(Y\) be the algebraic set in \(\mathbf{A}^3\) defined by the two polynomials \(x^2-yz\) and \(x z-x\). Show that \(Y\) is a union of three irreducible components. Describe them and find their prime ideals.

So \(Y =V(x, y) \cup V(x, z) \cup V(x^2-y)\) is a union of two lines and a parabola.

1.4 #completed

If we identify \(\mathbf{A}^2\) with \(\mathbf{A}^1 \times \mathbf{A}^1\) in the natural way, show that the Zariski topology on \(\mathbf{A}^2\) is not the product topology of the Zariski topologies on the two copies of \(\mathbf{A}^1\).

Alternative: consider \(V(xy-1) \subseteq {\mathbf{A}}^2\). If this had the product topology, the projection maps \(\pi_x, \pi_y: {\mathbf{A}}^2\to {\mathbf{A}}^1\) would be closed, but \(\pi_x(V(xy-1)) = {\mathbf{A}}^1\setminus\left\{{0}\right\}\) which is open and not closed since its complement \(0 = V(x)\) is closed and not open.

1.5 #to_work

Show that a \(k\)-algebra \(B\) is isomorphic to the affine coordinate ring of some algebraic set in \(\mathbf{A}^n\). for some \(n\), if and only if \(B\) is a finitely generated \(k\)-algebra with no nilpotent elements.

1.6 #to_work

Any nonempty open subset of an irreducible topological space is dense and irreducible. If \(Y\) is a subset of a topological space \(X\), which is irreducible in its induced topology, then the closure \(\mkern 1.5mu\overline{\mkern-1.5muY\mkern-1.5mu}\mkern 1.5mu\) is also irreducible.

1.7 #to_work

Show that the following conditions are equivalent for a topological space \(X\) :
- \(X\) is noetherian:
- Every nonempty family of closed subsets has a minimal element:
- \(X\) satisfies the ascending chain condition for open subsets:
- Every nonempty family of open subsets has a maximal element.
A noetherian topological space is quasi-compact, i.e., every open cover has a finite subcover.

Any subset of a noetherian topological space is noetherian in its induced topology.
A noetherian space which is also Hausdorff must be a finite set with the discrete topology.

1.8 #to_work

Let \(Y\) be an affine variety of dimension \(r\) in \(\mathbf{A}^n\). Let \(H\) be a hypersurface in \(\mathbf{A}^n\), and assume that \(Y \nsubseteq H\). Then every irreducible component of \(Y \cap H\) has dimension \(r-1\). ¹

1.9 #to_work

Let \(a \subseteq A=k\left[x_1, \ldots, x_n\right]\) be an ideal which can be generated by \(r\) elements. Then every irreducible component of \(Z(a)\) has dimension \(\geqslant n-r\).

1.10 #to_work

If \(Y\) is any subset of a topological space \(X\), then \(\operatorname{dim} Y \leqslant \operatorname{dim} X\).
If \(X\) is a topological space which is covered by a family of open subsets \(\left\{L_1 ;\right.\), then \(\operatorname{dim} X=\sup \operatorname{dim} U_i\).

Give an example of a topological space \(X\) and a dense open subset \(U\) with \(\operatorname{dim} L^{\prime}<\operatorname{dim} X\).
If \(Y\) is a closed subset of an irreducible finite-dimensional topological space \(X\), and if \(\operatorname{dim} Y=\operatorname{dim} X\), then \(Y=X\).

Give an example of a noetherian topological space of infinite dimension.

1.11 * #to_work

Let \(Y \subseteq \mathbf{A}^3\) be the curve given parametrically by \(x=t^3, y=t^4, z=t^5\). Show that \(I(Y)\) is a prime ideal of height 2 in \(k[x, y ;-]\) which cannot be generated by 2 elements. We say \(Y\) is not a local complete intersection-cf. (Ex. 2.17).

1.12 #to_work

Give an example of an irreducible polynomial \(f \in \mathbf{R}[x, y]\). whose zero set \(Z(f)\) in \(\mathbf{A}_{\mathbf{R}}^2\) is not irreducible (cf. 1.4.2).

Footnotes

(See (7.1) for a generalization.)