HW #2, due Wednesday 2/4 (sixth problem added)

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.

Wednesday 1/28

The Hilbert series for a general monomial ideal, by inclusion-exclusion.
The ideal membership problem, for monomial ideals.
The notation a == b mod I, and the generators of I as substitution rules in such congruences.
Term orders on monomials, which we will require to be well-orderings.
Examples: the lexicographic order, and the graded lexicographic order.
Given a term order, any polynomial p has a most expensive term, init p.

The reduction algorithm for testing ideal membership:
if p has a monomial m that's a multiple of some g_i,
replace p by p - (m/init g_i) g_i.
(It's enough to look only at p's most expensive monomial, init p.)
By the well-ordering condition, this terminates.
If the result is zero, then yes, the original p was in the ideal.

Scary example: I = < x-y, x-z >, with x the most expensive term. Then y-z is in the ideal, but this algorithm can't show it.

Monday 1/26

The Hilbert series of R/I for I a polynomial ideal, as the generating function of the Hilbert function.

The elements of R/I_X, while no longer well-defined as polynomials, are still well-defined as functions on X.

Given S a subspace of R, when is there a well-defined multiplication on R/S? Answer: S an ideal.

(Actually, we only showed that if S is an ideal, then the multiplication on R/S is well-defined. But necessity is easy: the zero element of R/S is the coset 0 + S = S itself, and we want C*0 = 0 for any element C in R/S. Upstairs in R, that says r*s in S for any r in any C, any s in S.)

HW #1, due Wednesday 1/28

1. A monomial ideal is one generated by monomials.
Let R = C[x,y].
Find all monomial ideals I such that there are exactly 4 monomials not in I.

(Note that you never need to have two generators where one divides the other.)

2. Let I,J be ideals in a ring R. Show that I intersect J is an ideal
(by checking each of conditions 1-3).

3. Let R = C[x,y], I = < x^2 y^20, x^15 y^4 >, J = < x^8 y^9 >.
Find generators for I intersect J.

4. Let Y = {0,1}, two elements of C. Find the ideal I_Y in C[x].

5. Compute the Hilbert function of C[x,y] / < x^a y^b >, for arbitrary a,b.

6. Let I be a monomial ideal in C[x,y]. Say that h_I(N)=0, for some large N. What can you say about the generators of I? (Assuming no redundant generators are used, meaning multiples of other generators.)

Wednesday 1/21

Partitions of sets, as indicating partial knowledge.
Cosets of a vector subspace.
Quotient vector spaces. Computing their dimension, in the finite-dim case.
The nth graded piece of a polynomial ring.
The Hilbert function of a monomial ideal (or really, of R/I).
Computing it by counting the leftover monomials.

Monday 1/19

We defined polynomial rings (with complex coefficients), and ideals therein.
Monomial ideals.

Two natural ways to produce ideals:

1) Any subset X of C^n has an I_X of those polynomials that vanish at every point of X.
Example: if X is the y-axis in C^2, then I_X in C[x,y] is all multiples of x.

2) Given any set in a polynomial ring R, we can generate an ideal from it by taking all R-linear combinations.

Big theorem to come: any ideal has a finite generating set.

(Warning: in the links above, the ring R may not be a polynomial ring, and may not even be commutative. That leads to a bunch of extra worries, like "left ideals" vs. "right ideals", that are not our problem.)

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