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A ring of germs of analytic functions at a point in an [[Analytic space|analytic space]]. The following is a more accurate definition. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124001.png" /> be a field with a non-trivial norm (cf. [[Norm on a field|Norm on a field]]) (which is usually assumed to be complete), and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124002.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124003.png" />-algebra of power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124004.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124005.png" /> which converge on some polycylinder with centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124006.png" />, each series converging on its own polycylinder. A quotient ring of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124007.png" /> is called an analytic ring over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a0124008.png" />, or an analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240010.png" />-algebra; usually, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240011.png" /> is the field of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240012.png" /> or the field of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240013.png" />. Any analytic ring is a local, Noetherian, Hensel ring; its field of residues is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240014.png" />. An analytic ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240015.png" /> is a regular (and a factorial) ring, and its completion in the topology defined by the maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240016.png" /> coincides with the ring of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240017.png" />. The normalization lemma is true: An integral analytic ring is a finite extension of an analytic ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240018.png" />. Algebras that are finite over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240019.png" /> are generally called quasi-analytic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240021.png" />-algebras. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012400/a01240022.png" /> is a perfect field, an analytic ring is an [[Excellent ring|excellent ring]].
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A ring of germs of analytic functions at a point in an [[Analytic space|analytic space]]. The following is a more accurate definition. Let $k$ be a field with a non-trivial norm (cf. [[Norm on a field|Norm on a field]]) (which is usually assumed to be complete), and let $k\{\{X_1,\dots,X_n\}\}$ be the $k$-algebra of power series in $X_1,\dots,X_n$ with coefficients in $k$ which converge on some polycylinder with centre $(0,\dots,0)$, each series converging on its own polycylinder. A quotient ring of the ring $k\{\{X_1,\dots,X_n\}\}$ is called an analytic ring over $k$, or an analytic $k$-algebra; usually, $k$ is the field of real numbers $\mathbf R$ or the field of complex numbers $\mathbf C$. Any analytic ring is a local, Noetherian, Hensel ring; its field of residues is isomorphic to $k$. An analytic ring $k\{\{X_1,\dots,X_n\}\}$ is a regular (and a factorial) ring, and its completion in the topology defined by the maximal ideal $(X_1,\dots,X_n)$ coincides with the ring of formal power series $k[[X_1,\dots,X_n]]$. The normalization lemma is true: An integral analytic ring is a finite extension of an analytic ring $k\{\{X_1,\dots,X_n\}\}$. Algebras that are finite over $k\{\{X_1,\dots,X_n\}\}$ are generally called quasi-analytic $k$-algebras. If $k$ is a perfect field, an analytic ring is an [[Excellent ring|excellent ring]].
  
 
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Latest revision as of 12:10, 25 August 2014

A ring of germs of analytic functions at a point in an analytic space. The following is a more accurate definition. Let $k$ be a field with a non-trivial norm (cf. Norm on a field) (which is usually assumed to be complete), and let $k\{\{X_1,\dots,X_n\}\}$ be the $k$-algebra of power series in $X_1,\dots,X_n$ with coefficients in $k$ which converge on some polycylinder with centre $(0,\dots,0)$, each series converging on its own polycylinder. A quotient ring of the ring $k\{\{X_1,\dots,X_n\}\}$ is called an analytic ring over $k$, or an analytic $k$-algebra; usually, $k$ is the field of real numbers $\mathbf R$ or the field of complex numbers $\mathbf C$. Any analytic ring is a local, Noetherian, Hensel ring; its field of residues is isomorphic to $k$. An analytic ring $k\{\{X_1,\dots,X_n\}\}$ is a regular (and a factorial) ring, and its completion in the topology defined by the maximal ideal $(X_1,\dots,X_n)$ coincides with the ring of formal power series $k[[X_1,\dots,X_n]]$. The normalization lemma is true: An integral analytic ring is a finite extension of an analytic ring $k\{\{X_1,\dots,X_n\}\}$. Algebras that are finite over $k\{\{X_1,\dots,X_n\}\}$ are generally called quasi-analytic $k$-algebras. If $k$ is a perfect field, an analytic ring is an excellent ring.

References

[1] J. Dieudonné, A. Grothendieck, "Critères differentiels de régularité pour les localisés des algèbres analytiques" J. of Algebra , 5 (1967) pp. 305–324
[2] B. Malgrange, "Ideals of differentiable functions" , Tata Inst. (1966)
[3] S.S. Abhyankar, "Local analytic geometry" , Acad. Press (1964)


Comments

References

[a1] H. Grauert, R. Remmert, "Coherent analytic sheaves" , Springer (1984) (Translated from German)
How to Cite This Entry:
Analytic ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_ring&oldid=15652
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article