1. Let I = < x^a y^b, x^c y^d >. What is the Hilbert series H_{R/I}? Don't just repeat the formula from class -- simplify your answer as much as possible, depending on circumstances.
2. Let I = < x^a y^b z^c, x^d y^e z^f, x^g y^h z^i >.
Assume H_{R/I} = 1/(1-t)^3 * (1 - t^{a+d+g}) (1 - t^{b+e+h}) (1 - t^{c+f+i}).
What does this tell you about a..i?
3. Let M be a 2x2 matrix with entries a,b,c,d, and let I be the ideal generated by the entries of M^2.
a) Show (Trace M)^3 is in I.
b) Show Trace M is not in I.
Hint: when taking an R-linear combination of the generators of I, break the {r_i} coefficients up into their homogeneous components, and work one degree at a time.
4. Let X be a line union a point, specifically {x=0} union {x=1,y=0}.
a) Show the ideal I_X contains < xy, x^2 - x >.
b) Show the opposite containment.
5. Let I,J be monomial ideals. Explain how to program up an algorithm that tests whether I contains J.
6. Given a subspace A of a vector space B, define the map i_A : A -> B that just takes any vector to itself, and the map p_A : B -> B/A that takes any b to its coset b+A.
The map i_A is one-to-one, meaning that no two vectors map the same place.
The map p_A is onto, meaning that every element of the target is p_A(something).
A map that is one-to-one and onto is called a correspondence (or depending on context, a bijection, or an isomorphism, or...).
The point of this problem is to see that "subspace is to one-to-one as quotient space is to onto".
Let T : V -> W be a linear map from one vector space to another.
Find a subspace W' of W, and a map C : V -> W', such that C is onto and T = i_W' o C.
Find a subspace V' of V, and a map D : V/V' -> W, such that D is one-to-one and T = D o p_V'.
You should be able to write down W',C,V',D explicitly in terms of T,V,W.
