pragmatic 3

D: Artinian Rings

As always, we will be looking at homogeneous ideals of R = k[x_{0}, …, x_{n}]. As Juan and I have mentioned, one can often reduce questions about homogeneous ideals in polynomial rings to questions about homogeneous ideals I ⊂ R for which ht. I = n + 1 (the maximum possible). Definition-Proposition: For R, I as above, TFAE

1) I is an artinian ideal;

2) R/I is a ring of K-dim = 0; (i.e. an artinian graded k-algebra.)

3) \sqrt{I} = m = (x_{0}, …, x_{n});

4) m^{t} ⊂ I;

5) I_{s} = m_{s} for all s >> 0;

6) H_{R/I}(t) = 0 for all t >> 0.

7) dim_{k}(R/I) < ∞.

So, if I is an artinian ideal of R and A = R/I then

A = k ⊕ A_{1} ⊕ ⋯ ⊕ A_{s} where A_{s} ≠ 0. In this case, we can encapsulate the data of the Hilbert function of A in an s + 1-tuple (1, H_{A}(1), …, H_{A}(s))

which is called the h-vector of A. Examples:

1) Let A = k[x_{0}, …, x_{n}]/(x_{0}, …, x_{n})^{t}. In this case the h-vector of A is

(1, n + 1, \binom{n + 2}{2}, …, \binom{n + t − 1}{t − 1})

2) Let A = k[x, y]/(x^{2}, y^{2}). In this case the h-vector is (1, 2, 1).

-1- 6/30/00 3) Let A = k[x, y]/(x^{2}, xy, y^{3}). In this case the h-vector is also (1, 2, 1).

Let m̄ = A_{1} ⊕ ⋯ ⊕ A_{s} denote the maximal homogeneous ideal of A.

Definition: The socle of A, denoted soc(A), is defined by:

soc(A) := ann_{A}(m̄).

With the notation above, s is called the socle degree of A.

Remark:

1) Sticking with the notation above, we always have that A_{s} ⊆ soc(A) but soc(A) could be strictly larger. Consider the example 3 above. In that case

A ≃ k⊕ < x̄, ȳ > ⊕ < ȳ^{2} > .

Notice that x̄ ∈ A_{1} is also in soc(A). So, soc(A) = (x̄, ȳ^{2}).

On the other hand, in example 2 one easily sees that soc(A) is precisely A_{2}.

As always, when “containments” become “equalities”, things are more interesting!

Definitions: Let A as above, be an artinian ring.

1) If soc(A) = A_{s} then A is called an (artinian) level algebra.

2) If dim_{k} soc(A) = 1 then A is called an (artinian) Gorenstein k-algebra.

Notice that this definition forces the socle of A to be concentrated in the last graded piece of A (so A is level) and also forces that graded piece to have dimension 1.

3) The minimal number of generators of soc(A), which is the same as dim_{k} soc(A), is called the Cohen-Macaulay type of A. So, Gorenstein is the same as CM type = 1.

(Note: This is not the same definition of the CM type given by Juan, so I would like to sketch a justification of this alternative definition. )

In our situation R = k[x_{0}, …, x_{n}] and I is an ideal of height n + 1 and so I is a CM ideal. It follows that the homological dimension of R/I is n + 1. Consider a minimal free resolution for R/I:

0 → F_{n+1} → ⋯ → F_{1} → R → R/I → 0

and write F_{n+1} = ⊕R(−j)^{β_{n+1,j}}.

Now T or_{n+1}^{R}(R/I, k) = F_{n+1}⊗_{R}k = ⊕k(−j)^{β_{n+1,j}}, a graded vector space of dimension equal to the rank of F_{n+1}.

But, we can also calculate this T or using the Koszul complex, which is a resolution of k = R/(x_{0}, …, x_{n}). If we write that complex as:

0 → G_{n+1} ⟶^{d_{n+1}} ⋯ → R(−1)^{n+1} → R → k → 0

where

G_{n+1} = ⋀^{n+1}[R(−1)^{n+1}] ≃ R(−(n + 1))

and basis given by e_{0} ∧ … e_{n}

If we now tensor this Koszul complex with R/I and calculate the cohomology module in the last place, we are reduced to calculating the kernel of the map

F̄e_{0} ∧ … ∧ e_{n} ⟶ ∑_{i=0}^{n}(−1)^{i+1}x_{i}F̄ e_{0} ∧ … ∧ ê_{i} ∧ … ∧ e_{n}, F̄ ∈ R/I.

But, this last expression is 0 if and only if x_{i}F̄ = 0 for all i. This is the same as saying that F̄ is in the socle of R/I. So, calculated this way we obtain that:

T or_{n+1}^{R}(R/I, k) ≃ soc(R/I)(−(n + 1)) (≃ F_{n+1} ⊗_{R} k = ⊕k(−j)^{β_{n+1,j}})

It follows from this discussion that if I is an artinian homogeneous ideal in R = k[x_{0}, …, x_{n}] and A = R/I then:

1) the rank of the last free module in a minimal free resolution of A = R/I, as R-module, is the dimension of the socle of A. So, for artinian algebras, the CM type of the algebra is the same as the dimension of the socle; and

2) If the last summand of the minimal free resolution of A = R/I has a summand R(−j)^{β_{n+1,j}} then the socle of A has β_{n+1,j} minimal generators of degree j − (n + 1).

Let’s apply these observations to some examples that we have already seen:

Examples:

1) If A = k[x, y]/(x^{2}, y^{2}) then the minimal free resolution of A as R = k[x, y] module is:

0 → R(−4) → R(−2)^{2} → R → A → 0

This implies that dim soc(A) = 1 i.e. A is a Gorenstein ring, and the generator of the socle is in degree 4 − 2 = 2.

2) Let A = k[x, y]/(x^{2}, xy, y^{3}), then the minimal free resolution of A as R = k[x, y]-module is:

0 → R(−4) ⊕ R(−3) → R(−3) ⊕ R(−2)^{2} → R → A → 0

From this resolution we see that dim_{k} soc(A) = 2 with generators in degrees 4 − 2 = 2 and 3 − 2 = 1.

Since we can actually “see” the socle in the minimal free resolution, we can begin with an arbitrary CM ring A = R/I and use the fact that the graded Betti numbers in a minimal free resolution of A as R-module don’t change when we move to the artinian reduction, to define the concept of level and Gorenstein for CM rings of any dimension and not only of dimension 0.

So, in this way a CM ring A = R/I is level if the last free module in a minimal free resolution of A as R-module has only one shift. We can say that A is Gorenstein if that last module has rank exactly 1.

Examples:

1) If F_{1}, …, F_{r} form a regular sequence and I = (F_{1}, …, F_{r}), then A = R/I is a Gorenstein ring. See the Koszul complex for the minimal free resolution of A as R-module.

Ideals as above are important enough to merit a special name.

Definition: Let R = k[x_{0}, …, x_{n}] and let I be an ideal of ht. I generated by a regular sequence of (necessarily) ht. I elements. Then I is called a complete intersection ideal.

It’s an easy exercise to show that a height 1 ideal of R is a CM ideal if and only if it is principal. It follows that all height one CM ideals are Gorenstein and, in fact, are complete intersection ideals.

Suppose that I is a Gorenstein ideal of height 2 in R. Then the homological dimension of R/I = 2 and a minimal free resolution of R/I is:

0 → F_{2} → F_{1} → R → R/I → 0

where rk. F_{2} = 1. It follows that rk. F_{1} = 2 and so I = (F, G). Since ht. I = 2 it must be that F and G have no common factor and so form a regular sequence in R. So, all ht. 2 Gorenstein ideals are also complete intersections.

2) Let s = \binom{d+2}{2} and let 𝕏 = \{P_{1}, …, P_{s}\} be points of ℙ^{2} which do not lie on a curve of degree d. Then A = R/I_{𝕏} is a level algebra. In fact, the resolution of A as R-module is:

0 → R(−(d + 2))^{d+1} → R(−(d + 1))^{d+2} → I_{𝕏} → 0

So, A is a level algebra with CM type d + 1 and with socle in degree d + 2 − 2 = d.

More generally, if 𝕏 is a family of s = \binom{d+n}{n} points of ℙ^{n} which do not lie on a hypersurface of degree d of ℙ^{n}, then the coordinate ring of the points is a level algebra.

3) Let C be the rational normal curve in ℙ^{n} and ℘ = I_{C} its defining (prime) ideal and A = R/℘ its homogeneous coordinate ring. Then A is also a level algebra.

To see why this is so recall that A is a CM ring whose Hilbert function can be calculated with the Riemann-Roch theorem as: 1 n + 1 ⟶^{+n}. Thus the points of the general hyperplane section of C consist of n points of ℙ^{n−1} which do not lie on a hyperplane. We can apply the previous result to say that the points have coordinate ring which is level and hence the coordinate ring of the rational normal curve in ℙ^{n} is also a level ring.

4) Be careful! The rational quartic curve in ℙ^{3} which we saw earlier does not have a CM coordinate ring, i.e. is not arithmetically CM, in spite of the fact that its resolution ends “nicely”. It’s resolution is:

0 → R(−5) → R(−4)^{4} → R(−2) ⊕ R(−3)^{3} → I → 0

but it is not Gorenstein.

Let’s return to considering artinian rings.

Being a Gorenstein ring puts severe restrictions on the nature of the ring. For example, if A is a Gorenstein ring with socle in degree s then we can identify A_{s} with the field k. Then, the multiplication in the ring induces a bilinear form

A_{i} × A_{s−i} ⟶ A_{s} ≃ k.

Since the socle is concentrated only in degree s of A one can prove that this bilinear form gives a perfect pairing. It follows that

dim_{k} A_{i} = dim_{k} A_{s_{i}}.

I.e. we have proved:

Corollary: Let A be a Gorenstein artinian ring with socle degree s. If the h-vector of A is (1, h_{1}, …, h_{s−i}, 1) then h_{i} = h_{s−i}.

We abbreviate this condition by saying that the h-vector of a Gorenstein ring is symmetric.

Gorenstein schemes enjoy other very interesting properties. E.g. if X is a set of points which is Gorenstein then if Y_{1} and Y_{2} are two subsets of X with the same Hilbert function then Z_{i} = X − Y_{i} are also two sets having the same Hilbert function (follows from Liaison techniques).

This had been observed classically for the case of Complete Intersection Schemes where the Y_{i} were taken to be singleton sets. In that form the theorem was known as the Cayley-Bacharach theorem and a special case (every cubic through 8 of the 9 points of intersection of two plane cubics also passes through the 9th) is very well known.

Examples: If one considers the four points in ℙ^{2}

\begin{aligned}& • \\ • & • & • \end{aligned}

one finds that the Hilbert function is 1 3 4 →, so the artinian reduction has symmetric Hilbert function, but the coordinate ring of these points is not Gorenstein since there are obviously two different subsets of three points with different Hilbert functions and so this set of points doesn’t satisfy the Cayley-Bacharach property.

So, for sets of points in ℙ^{n} we have two obvious necessary conditions in order that the set of points be Gorenstein. They turn out also to be sufficient.

Theorem:(Davis, Geramita, Orecchia) Let I be the ideal of a set, 𝕏, of s distinct points in ℙ^{n} and suppose that the Hilbert function of 𝕏 has first difference which is symmetric and that every subset of 𝕏 having cardinality s − 1 has the same Hilbert function.

Then the homogeneous coordinate ring of 𝕏 is a Gorenstein ring.

We can use this theorem to give some non-trivial examples of Gorenstein ideals which are not complete intersections. As we’ve seen, such examples must have height at least 3.

1) Let 𝕏 be a set of 5 general points in ℙ^{3}. (the argument is the same for n + 2 general points of ℙ^{n}).

Since 𝕏 is general, every subset of 4 points has the same Hilbert function. So, it suffices to note that the Hilbert function of 𝕏 is 1 4 5 → (which has first difference 1 3 1 0 → and so is symmetric). By [DGO] this set of points has Gorenstein coordinate ring.

Notice that there are 5 quadrics in the ideal of the 5 points, which must be members of a minimal generating set for the ideal of 𝕏. So, since the height of I_{𝕏} is three, this is not a complete intersection ideal. (It is not difficult to show that, in general, the ideal of five general points is generated by the quadrics through the points.)

2) We will now show that the coordinate ring of any 8 points on the rational normal curve C in ℙ^{3} has Gorenstein coordinate ring.

First recall that the coordinate ring of the rational normal curve in ℙ^{3} is generated by the three quadrics which pass through the curve.

Claim: Any 7 points on the rational normal curve has Hilbert function 1 4 7 →.

Pf. If the 7 points lay on a hyperplane that hyperplane would have to contain the curve, by Bezout’s theorem, and so the RNC would lie in a plane – this is not true.

If the 7 points lay on any quadric hypersurface which was NOT in the ideal of the RNC then the quadric hypersurface would meet the RNC in at least 7 points. But, the rational normal curve has degree 3 and the hypersurface has degree 2 so by Bezout’s theorem they should meet in at most 6 distinct points. Thus, any quadric containing the 7 points is in the ideal of the RNC and that ideal has three independent quadrics in it.

It follows from the Claim that any set of 8 points on the RNC has Hilbert function 1 4 7 8 →.

The artinian reduction of the coordinate ring of the 8 points has Hilbert function 1 3 3 1, which is symmetric. The Claim demonstrates that the 8 points satisfy the Cayley-Bacharach property, so by the [DGO] theorem, the 8 points have Gorenstein coordinate ring.

Moreover, the three quadrics in the ideal of the points cannot generate the ideal of the points since they generate the ideal of the RNC. So, the ideal of these 8 points has to have more generators and hence cannot be a complete intersection.

The question as to what can be the h-vector of a Gorenstein ring has attracted a great deal of attention. From the discussion above, the first case that requires attention is the case of Gorenstein ideals in R = k[x_{0}, …, x_{n}] of height 3.

This case was resolved by a notable theorem of D. Buchsbaum and D. Eisenbud who showed that such ideals are always the ideals of Pfaffians of a skew-symmetric matrix.

Using that result, R. Stanley was able to characterize the h-vectors of all the artinian Gorenstein quotients of k[x_{0}, x_{1}, x_{2}]. Stanley’s theorem is the following:

Theorem: Let A = k[x_{0}, x_{1}, x_{2}]/I be a Gorenstein artinian (graded) k-algebra. Let the h-vector of A be

(1, h_{1}, …, h_{s−1}, 1).

Then:

1) h_{1} ≤ 3 and h_{s−i} = h_{i} for every i;

2) the first difference of the sequence 1, h_{1}, …, h_{⌈s/2⌉}, → is an Osequence.

Moreover, any h-vector satisfying 1) and 2) is the h-vector of some Gorenstein quotient of k[x_{0}, x_{1}, x_{2}].

The result of Stanley was extended by several different authors: S. Diesel investigated all the resolutions of Gorenstein ideals that have as h-vector one of those given by Stanley’s theorem.

The symmetry of the h-vector is reflected further in some symmetries among the graded Betti numbers of the Gorenstein ideals. In the case of height 3 Gorenstein ideals this means that given the h-vector, the only important Betti numbers are those which describe the degrees of the minimal generators of the ideal.

That having been said, Diesel was able to prove the following lovely theorem:

Theorem: Fix an h-vector, H, satisfying Stanley’s theorem and for each Gorenstein ideal in k[x_{0}, x_{1}, x_{2}] having h-vector H, write down the degrees of the minimal generating set of the ideal as (d_{1}, d_{2}, …, d_{r}) where d_{1} ≤ d_{2} ≤ .. ≤ d_{r}.

If we partially order these tuples by

(d_{1}, …, d_{r}) ≤ (δ_{1}, …, δ_{s}) ⇔ the multiset {d_{1}, …, d_{r}} ⊆ {δ_{1}, …, δ_{s}},

then there is a unique maximum and unique minimum set of generating degrees.

Moreover, there is an algorithm for constructing all the “degree sequences” starting with the h-vector H.

Geramita and Migliore showed that the Diesel result held precisely even if one restricted the class of Gorenstein ideals in R = k[x_{0}, …, x_{n}] to the class of radical Gorenstein ideals (i.e. Gorenstein ideals equal to their radicals). Also, DeNegri, Herzog, Trung and Valla showed that one had to add a further condition to the Stanley theorem if one wanted to consider only prime Gorenstein ideals.

Lest you draw the conclusion that all is known in codimension 3, let me quickly dispel that belief. The wide ranging knowledge that we have in codimension 3 has permitted us to ask more penetrating questions and those are just beginning to be understood.

E.G. Iarrobino and Kanev have shown (this is just a specific case) that the Gorenstein artinian quotients of k[x_{0}, x_{1}, x_{2}] = R which have their socle, say, in degree s can be considered as the closed points of the projective space ℙ(R_{s}).

Moreover, if T_{1} = (1, h_{1}, …, h_{s−1}, 1) and T_{2} = (1, g_{1}, …, g_{s−1}, 1) are two h-vectors (guaranteed by Stanley’s theorem) then we can define a partial order on these s + 1-tuples by:

(1, h_{1}, …, h_{s−1}, 1) ≥ (1, g_{1}, …, g_{s−1}, 1) ⇔ h_{i} ≥ g_{i} for all i.

Using that partial order we can consider all the closed points of ℙ(R_{s}) whose hvectors are ≤ T_{1} (say). There is a naturally defined ideal of k[Z_{0}, …, z\binom{s+3}{3}] which defines a subscheme of ℙ(R_{s}) whose closed points are precisely the set we want to describe, but we know very little about the nature of that ideal.

It is a sum of ideals of minors of certain catalecticant matrices which describe this closed subscheme. A former student of mine (T. Derry) has succeeded in showing that all the ideals of minors of the catalecticant matrices are saturated (and this is an important first step), but we would like very much to know if this ideal is radical (examples by M. Boij and A. Iarrobino have shown that it is not, in general, prime).

You can read about this stuff in some lecture notes of mine on Inverse Systems (Queen’s Papers in Pure and Applied Math), some notes based on my lectures that were given by Valla in Barcelona a few years ago and, more completely, in the new Springer Lecture Notes of Iarrobino and Kanev on the subject. I just want to emphasize that there is still lots to know in codimension 3.

Unfortunately, things are much, much worse in codimension ≥ 4. We don’t even know what can be the h-vector of a height 4 Gorenstein ideal. This is, obviously, an area of very active research with work being done by Kustin, Miller, Migliore, Peterson, Nagel, Kleppe, Miro-Roig, Harima, Shin, Ragusa, Zappala, and Geramita (among others). What do we know?

Damm little! is the vulgar but unfortunate truth. However, we know the terrain is more complicated than it might be if we only consider the case of codimension ≤ 3.

The lack of a structure theorem like the one of Buchsbaum and Eisenbud in codimension 3 is, of course, the crucial difference. There may be no such theorem, since the resolution (in codimension 4) depends on two (related) matrices rather than one. Nevertheless, we do have:

lots of h-vectors which we do know are the h-vectors of Gorenstein rings. for those h-vectors which we know, we even have some potentially extremal resolutions.

We also lack a Diesel-like theorem which, once we have a permissible h-vector, tells us what the possible resolutions are that we can associate to it, or even if there is a maximum or a minimum resolution.

Apropos of this last comment, a student of G. Evans, Ben Richert, has recently shown that in codimension 4, and for some h-vectors, we definitely do not have a minimal resolution for all ideals having that h-vector.

For a large family of h-vectors that we know can be the h-vector of a Gorenstein ring, Geramita-Harima-Shin have proposed a maximum resolution (and this proposal has been extended by Migliore-Nagel to a wider class of h-vectors), but as yet there has not been a proof for the existence of maximal resolutions for a family of h-vectors.

One method that has proved useful in studying level algebras in general, and Gorenstein algebras in particular, has been the following “sum” construction which I would like to explain. It first was noticed by Peskine and Szpiro in their seminal paper on Liaison.

Theorem: Let 𝕏 be a set of points in ℙ^{n} having level coordinate ring and let 𝕏 = 𝕐 ⊎ ℤ (disjoint union).

Then I_{𝕐} +I_{𝕏} is a level artinian ideal whose CM type is ≤ the CM type of 𝕏; moreover, if I_{𝕏} is a Gorenstein ideal then so is I_{𝕐} + I_{𝕏}.

Proof: The proof is a simple application of the mapping cone construction and the following short exact sequence. Since I_{𝕏} = I_{𝕐} ∩ I_{ℤ} we have:

0 → I_{𝕏} ⟶^{ϕ} I_{𝕐} ⊕ I_{ℤ} ⟶^{χ} I_{𝕐} + I_{𝕏} → 0

where ϕ(F) = (F, −F) and χ(A, B) = A + B.

Since 𝕐 ∩ ℤ = ∅ we have \sqrt{I_{𝕐} + I_{ℤ}} = m, which has height n + 1.

Let

F = {F_{i}}_{i=1}^{n}, G = {G_{i}}_{i=1}^{n} and G′ = {G′_{i}}_{i=1}^{n}

be minimal free resolutions of I_{𝕏}, I_{𝕐} and I_{ℤ} respectively. Then, using the mapping cone construction, we get a free (not necessarily minimal) resolution H = {H_{i}}_{i=1}^{n+1} of I_{𝕐} + I_{ℤ} where

H_{i} = F_{i−1} ⊕ (G_{i} ⊕ G′_{i}) for i < n and H_{n+1} = F_{n}

Now H_{n+1} ≠ 0 since the height of I_{𝕐} + I_{ℤ} is n + 1, so even though the resolution is not necessarily minimal, we cannot cancel out all of F_{n}. Since F_{n} had all its shifts concentrated in one place we conclude that I_{𝕐} + I_{ℤ} is a level ideal of height n + 1.

In case 𝕏 was Gorenstein, then rk. F_{n} = 1 and so we deduce that I_{𝕐} + I_{ℤ} is also a Gorenstein ideal.

Notice that there is the possibility that a piece of either G_{n} or G′_{n} could cancel off something in F_{n} and so reduce the CM type of the sum. However, the only way that could possibly happen is if the socle degree of the artinian reduction of the coordinate ring of either 𝕐 or ℤ was the same as that of the original set 𝕏. We state that formally as a corollary:

Corollary: With the hypothesis above and assume that the degree of the socle of the artinian reduction of both the coordinate rings of 𝕐 and ℤ are less than that of the artinian reduction of 𝕏, then

CM type of R/I_{𝕏} = CM type of R/(I_{𝕐} + I_{ℤ}) .

Example: Let 𝕏 be a set of 10 general points in ℙ^{2}. Then we have already noted that the coordinate ring of 𝕏 is a level algebra with CM type 4. In fact a resolution for I_{𝕏} is given by:

0 → R^{4}(−5) → R^{5}(−4) → I_{𝕏} → 0

Now write 𝕏 as the disjoint union of 𝕐 (which consists of 6 general points) and ℤ (which consists of 4 general points). We know that the resolutions of these ideals are:

0 → R^{3}(−4) → R^{4}(−3) → I_{𝕐} → 0

and

0 → R(−4) → R^{2}(−2) → I_{ℤ} → 0

The mapping cone construction gives us:

0 → R^{4}(−5) → R^{9}(−4) → R^{2}(−2) ⊕ R^{4}(−3) → I_{𝕐} + I_{ℤ} → 0

Thus, I_{𝕐} + I_{ℤ} is an artinian level algebra of codimension 3 and of CM type 4. Note that the socle degree of this level algebra is 5-3 = 2 and its h-vector is (1, 3, 4).

Note also that we have the minimal free resolution for the ideal we constructed, in this case, since no cancelations are possible.