Assigned Exercises (Winter Quarter)

Math 8204
Winter 1998

Assigned exercises

Last updated Thursday, March 5, 1998.

Exercise 1 (Due Monday, January 26)
Exercise 2 (Due Monday, January 26)
Exercise 3 (Due Monday, March 16)
Exercise 4 (Due Monday, March 16)
Exercise 5 (Due Monday, March 16)
Exercise 6 (Due Monday, March 16)
Exercise 7 (Due Monday, March 16)
Exercise 8 (Due Monday, March 16)

Exercise 1 (Due Monday, January 26) Let R be the field of real numbers, and let m be a maximal ideal in R[x₁,..., x_n].

Consider the quotient ring R[x₁,..., x_n]/m as a vector space over R, and show that the dimension of this vector space is equal to either 1 or 2.
Show that:
- If the vector space dimension is 1, then m= (x₁ - a₁,..., x_n - a_n) for some (a₁,..., a_n) in Rⁿ.
- If the vector space dimension is 2, then m has an n-element generating set consisting of one quadratic polynomial and n - 1 linear polynomials.
  HINT: By renumbering, we can assume that the field extension R[x₁,..., x_n]/m is generated by x₁ [or more precisely, by the element that is represented by x₁ modulo m]. In particular, the quadratic polynomial can be a polynomial in x₁ alone . . .
Let J be the extension of m to C[x₁,..., x_n]. Thus:
J = m· R[x₁,..., x_n]. Show that:
- If the vector space dimension is 1, then V(J) consists of one point with real coordinates.
- If the vector space dimension is 2, then V(J) consists of 2 points which are the complex conjugates of each other.

In subsequent exercises you may cite the Nullstellensatz whenever that seems appropriate.

To set the stage for the next exercise, we note that as corollary of the Nullstellensatz that if X is an affine variety, then the natural map

A(X) - > $\Oh$ (X), from the coordinate ring to the ring of regular functions, is an isomorphism. (More generally, if f $\in$ A(X), then the natural map A(U_f) - > $\Oh$ (U_f) is an isomorphism. Here, U_f is the open subset of X consisting of all points where f is nonzero.)

Last quarter, we saw that if X and Y are isomorphic affine varieties, then there is an isomorphism of rings of regular functions: $\Oh$ (X) $\cong$ $\Oh$ (Y). It follows that we have an isomorphism of coordinate rings:

A(X) $\cong$ A(Y).

We want to ask whether the converse of this is true, i.e. whether two affine varieties whose coordinate rings are isomorphic (as K-algebras, of course), are actually isomorphic. Recall that, more generally, that if $\varphi$ : X - > Y is a regular map, then we have an isomorphism of rings of regular functions: $\varphi$ *: $\Oh$ (Y) - > $\Oh$ (X), given by composition of functions. It follows that we have a homomorphism of coordinate rings:

$\varphi$ *: A(Y) - > A(X). (By "abuse of notation" we're using the same symbol to denote both homomorphisms. Note also, that the process of passing from a regular map to a ring homomorphism is contravariant, i.e., the direction of the arrow is reversed.)

Exercise 2 (Due Monday, January 26) Let K be an algebraically closed field, and let X and Y be affine varieties over K. We consider the coordinate rings A(X) = K[x₁,...,x_n]/I(X), and A(Y) = K[x₁, ...,x_n]/I(Y).

Let f: A(Y) - > A(X) be a homomorphism of K-algebras. Show that there is a regular map $\varphi$ : X -> Y such that $\varphi$ * = f.
HINT. For definiteness, suppose that X A^m and Y Aⁿ. Let $\xbar$ _i $\in$ A(Y) be the element represented by x_i, for i = 1,...,n. Use the elements f( $\xbar$ ₁) ,..., f( $\xbar$ _n) (considered as regular functions ... ) to define a map from X to Aⁿ. (REMARK. This process is not supposed to be difficult; however it may be confusing. An important part of the work consists of convincing oneself that the appropriate choice between X and Y has been made, and that the functions which are supposed to be giving the coordinates of the map are functions on the variety where one wants them to live.)
Let X, Y, and Z be affine varieties. Consider homomorphisms f: A(X) - > A(Y) and g: A(Y) - > A(Y). Show that if f corresponds to $\varphi$ : Y - > X, and g corresponds to $\psi$ : Z -> Y, then g^of corresponds to $\varphi$ ^o $\psi$ . EQUIVALENTLY, if $\varphi$ * = f and $\psi$ * = g, then ( $\varphi$ ^o $\psi$ )* = g^of.
Show that if X and Y are affine varieties and A(X) $\cong$ A(Y), then X $\cong$ Y.

Exercise 3 (Due Monday, March 16) = Exercise 6.2 in the text.

Exercise 4 (Due Monday, March 16) = Exercise 6.4 in the text.

Exercise 5 (Due Monday, March 16) = Exercise 6.5 in the text. (Note that the statement about $\sigma$ _H was proved in class.)

Exercise 6 (Due Monday, March 16) With the same notation as in the previous exercise, show that:

if p $\in$ H, then $\sigma$ _H $\cap$ $\sigma$ _p is a projective line contained in G(1,3);
if p $\notin$ H, then $\sigma$ _H $\cap$ $\sigma$ _p is empty.

Exercise 7 (Due Monday, March 16) = Exercise 6.8 in the text.

To establish the context for the next exercise, let G(k,n) be the Grassmannian of k-planes in Pⁿ. Then dim G(k,n) = dim G(k+1,n+1) = (k+1)(n-k), and G(k,n) has an open covering in which each open set is isomorphic to A^(k+1)(n-k).

Exercise 8 (Due Monday, March 16) Let $\sigma$ $\subset$ G(k,n) $\times$ Pⁿ be the variety of incident k-planes in Pⁿ. Show that $\sigma$ has an open covering such that each open set in the covering is isomorphic to A^N, where N = (k+1)(n-k) + k = k(n-k) + n.

A hint for Exercise 8 is linked here.

Comments and inquiries to: roberts@math.umn.edu

Back to my homepage: http://www.math.umn.edu/~roberts

Last updated Thursday, March 5, 1998.

Math 8204 Winter 1998

Assigned exercises

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Math 8204
Winter 1998