Zariski-Lipman conjecture

Let $k$ be a field of characteristic zero and let $R$ be a finitely-generated $k$-algebra, that is, a homomorphic image of a ring of polynomials $R = k [ x _ { 1 } , \dots , x _ { n } ] / I$.

A $k$-derivation of $R$ is a $k$-linear mapping $\delta : R \rightarrow R$ that satisfies the Leibniz rule

\begin{equation*} \delta ( a b ) = a \delta ( b ) + b \delta ( a ) \end{equation*}

for all pairs of elements of $R$.

The set of all such mappings is a Lie algebra (often non-commutative; cf. also Commutative algebra) that is a finitely-generated $R$-module $\mathfrak { D } = \operatorname { Der } _ { k } ( R )$. The algebra and module structures of $\mathfrak{D}$ often code aspects of the singularities of $R$.

A more primitive object attached to $R$ is its module of Kähler differentials, $\Omega _ { k } ( R )$, of which $\mathfrak{D}$ is its $R$-dual, $\mathfrak { D } = \operatorname { Hom } _ { R } ( \Omega _ { k } ( R ) , R )$.

More directly, the structure of $\Omega _ { k } ( R )$ reflects many properties of $R$. Thus, the classical Jacobian criterion asserts that $R$ is a smooth algebra over $k$ exactly when $\Omega _ { k } ( R )$ is a projective $R$-module (cf. also Projective module).

For an algebra $R$ without non-trivial nilpotent elements, local complete intersections are also characterized by saying that the projective dimension of $\Omega _ { k } ( R )$ (cf. also Dimension) is at most one.

The technical issues linking these properties are the comparison between the set of polynomials that define $R$, represented by the ideal $I$, and the syzygies of either $\Omega _ { k } ( R )$ or $\mathfrak{D}$ (cf. also Syzygy).

The Zariski–Lipman conjecture makes predictions about $\mathfrak{D}$, similar to those properties of $\Omega _ { k } ( R )$.

The most important of these questions is as follows. If $\mathfrak{D}$ is $R$-projective, then $R$ is a regular ring (in commutative algebra). More precisely, it predicts that if $\text{p}$ is a prime ideal for which $\mathfrak { D } _ {\text{p} }$ is a free $R _ { \text{p} }$-module, then $R _ { \text{p} }$ is a regular ring.

In [a3], the question is settled affirmatively for rings of Krull dimension $1$ (cf. also Dimension), and in all dimensions the rings are shown to be normal (cf. also Normal ring). Subsequently, G. Scheja and U. Storch [a4] established the conjecture for hypersurface rings, that is, when $R$ is defined by a single equation, $I = ( f )$.

As of 2000, the last major progress on the question was the proof by M. Hochster [a2] of the graded case.

A related set of questions is collected in [a5]: whether the finite projective dimension of either $\Omega _ { k } ( R )$ or $\mathfrak{D}$ necessarily forces $R$ to be a local complete intersection. It is not known (as of 2000) whether this is true if $\mathfrak{D}$ is projective, a fact which would be a consequence of the Zariski–Lipman conjecture. Several lower dimension cases are known, but the most significant progress was made by L. Avramov and J. Herzog when they solved the graded case [a1].

References

[a1]	L. Avramov, J. Herzog, "Jacobian criteria for complete intersections. The graded case" Invent. Math. (1994) pp. 75–88
[a2]	M. Hochster, "The Zariski–Lipman conjecture in the graded case" J. Algebra , 47 (1977) pp. 411–424
[a3]	J. Lipman, "Free derivation modules" Amer. J. Math. , 87 (1965) pp. 874–898
[a4]	G. Scheja, U. Storch, "Differentielle Eigenschaften der Lokalisierungen analytischer Algebren" Math. Ann. , 197 (1972) pp. 137–170
[a5]	W.V. Vasconcelos, "On the homology of $I / I ^ { 2 }$" Commun. Algebra , 6 (1978) pp. 1801–1809