View source for Zariski-Lipman conjecture

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Let $k$ be a [[Field|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|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|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|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|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)|regular ring (in commutative algebra)]]. More precisely, it predicts that if $\text{p}$ is a [[Prime ideal|prime ideal]] for which $\mathfrak { D } _ {\text{p} }$ is a free $R _ { \text{p} }$-module, then $R _ { \text{p} }$ is a regular ring.

In [[#References|[a3]]], the question is settled affirmatively for rings of Krull dimension $1$ (cf. also [[Dimension|Dimension]]), and in all dimensions the rings are shown to be normal (cf. also [[Normal ring|Normal ring]]). Subsequently, G. Scheja and U. Storch [[#References|[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 [[#References|[a2]]] of the graded case.

A related set of questions is collected in [[#References|[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 [[#References|[a1]]].

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