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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300501.png" /> be a [[Field|field]] of characteristic zero and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300502.png" /> be a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300504.png" />-algebra, that is, a homomorphic image of a ring of polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300505.png" />.
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A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300507.png" />-derivation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300508.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z1300509.png" />-linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005010.png" /> that satisfies the Leibniz rule
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005011.png" /></td> </tr></table>
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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 $R = k [ x _ { 1 } , \dots , x _ { n } ] / I$.
  
for all pairs of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005012.png" />.
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A $k$-derivation of $R$ is a $k$-linear mapping $\delta : R \rightarrow R$ that satisfies the Leibniz rule
  
The set of all such mappings is a Lie algebra (often non-commutative; cf. also [[Commutative algebra|Commutative algebra]]) that is a finitely-generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005013.png" />-module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005014.png" />. The algebra and module structures of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005015.png" /> often code aspects of the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005016.png" />.
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\begin{equation*} \delta ( a b ) = a \delta ( b ) + b \delta ( a ) \end{equation*}
  
A more primitive object attached to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005017.png" /> is its module of Kähler differentials, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005019.png" />, of which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005020.png" /> is its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005021.png" />-dual, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005022.png" />.
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for all pairs of elements of $R$.
  
More directly, the structure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005023.png" /> reflects many properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005024.png" />. Thus, the classical Jacobian criterion asserts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005025.png" /> is a smooth algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005026.png" /> exactly when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005027.png" /> is a projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005028.png" />-module (cf. also [[Projective module|Projective module]]).
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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 $R$.
  
For an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005029.png" /> without non-trivial nilpotent elements, local complete intersections are also characterized by saying that the projective dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005030.png" /> (cf. also [[Dimension|Dimension]]) is at most one.
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A more primitive object attached to $R$ is its module of Kähler differentials, $\Omega _ { k } ( R )$, of which $\mathfrak{D}$ is its $R$-dual, $\mathfrak { D } = \operatorname { Hom } _ { R } ( \Omega _ { k } ( R ) , R )$.
  
The technical issues linking these properties are the comparison between the set of polynomials that define <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005031.png" />, represented by the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005032.png" />, and the syzygies of either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005033.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005034.png" /> (cf. also [[Syzygy|Syzygy]]).
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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]]).
  
The Zariski–Lipman conjecture makes predictions about <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005035.png" />, similar to those properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005036.png" />.
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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 most important of these questions is as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005037.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005038.png" />-projective, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005039.png" /> is a [[Regular ring (in commutative algebra)|regular ring (in commutative algebra)]]. More precisely, it predicts that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005040.png" /> is a [[Prime ideal|prime ideal]] for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005041.png" /> is a free <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005042.png" />-module, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005043.png" /> is a regular ring.
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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]]).
  
In [[#References|[a3]]], the question is settled affirmatively for rings of Krull dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005044.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005045.png" /> is defined by a single equation, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005046.png" />.
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The Zariski–Lipman conjecture makes predictions about $\mathfrak{D}$, similar to those properties of $\Omega _ { k } ( R )$.
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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.
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 +
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.
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005047.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005048.png" /> necessarily forces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005049.png" /> to be a local complete intersection. It is not known (as of 2000) whether this is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005050.png" /> 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]]].
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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====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z130/z130050/z13005051.png" />"  ''Commun. Algebra'' , '''6'''  (1978)  pp. 1801–1809</TD></TR></table>
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<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>

Latest revision as of 16:52, 1 July 2020

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
How to Cite This Entry:
Zariski-Lipman conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zariski-Lipman_conjecture&oldid=16673
This article was adapted from an original article by W. Vasconcelos (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article