https://en.formulasearchengine.com/w/index.php?action=history&feed=atom&title=Nucleoside_oxidase_%28H2O2-forming%29 2026-08-07T00:34:03Z Revision history for this page on the wiki MediaWiki 1.47.0-wmf.7 https://en.formulasearchengine.com/w/index.php?title=Nucleoside_oxidase_(H2O2-forming)&diff=28431&oldid=prev 2013-11-01T21:14:42Z

<p>Journal cites, added 1 PMC, added 1 issue number using <a href="/w/index.php?title=Testwiki:AWB&amp;action=edit&amp;redlink=1" class="new" title="Testwiki:AWB (page does not exist)">AWB</a> (9488)</p> <p><b>New page</b></p><div>In algebra, &#039;&#039;&#039;Exalcomm&#039;&#039;&#039; is a functor classifying the extensions of a commutative algebra by a [[module (mathematics)|module]]. More precisely, the elements of Exalcomm&lt;sub&gt;&#039;&#039;k&#039;&#039;&lt;/sub&gt;(&#039;&#039;R&#039;&#039;,&#039;&#039;M&#039;&#039;) are isomorphism classes of commutative &#039;&#039;k&#039;&#039;-algebras &#039;&#039;E&#039;&#039; with a homomorphism onto the &#039;&#039;k&#039;&#039;-algebra &#039;&#039;R&#039;&#039; whose kernel is the &#039;&#039;R&#039;&#039;-module &#039;&#039;M&#039;&#039; (with all elements having square 0). There are similar functors &#039;&#039;&#039;Exal&#039;&#039;&#039; and &#039;&#039;&#039;Exan&#039;&#039;&#039; for non-commutative rings and algebras, and functors &#039;&#039;&#039;Exaltop&#039;&#039;&#039;, &#039;&#039;&#039;Exantop&#039;&#039;&#039;. and &#039;&#039;&#039;Exalcotop&#039;&#039;&#039; that take a topology into account.<br /> <br /> &quot;Exalcomm&quot; is an abbreviation for &quot;COMMutative ALgebra EXtension&quot; (or rather for the corresponding French phrase). It was introduced by {{harvtxt|Grothendieck|1964|loc=18.4.2}}.<br /> <br /> Exalcomm is one of the [[André–Quillen cohomology]] groups and one of the [[Lichtenbaum–Schlessinger functor]]s.<br /> <br /> Given homomorphisms of commutative rings &#039;&#039;A&#039;&#039;&amp;nbsp;→&amp;nbsp;&#039;&#039;B&#039;&#039;&amp;nbsp;→&amp;nbsp;&#039;&#039;C&#039;&#039; and a &#039;&#039;C&#039;&#039;-module &#039;&#039;L&#039;&#039; there is an exact sequence of &#039;&#039;A&#039;&#039;-modules {{harv|Grothendieck|1964|loc=20.2.3.1}}<br /> :&lt;math&gt;<br /> 0\rightarrow \operatorname{Der}_B(C,L)\rightarrow \operatorname{Der}_A(C,L)\rightarrow \operatorname{Der}_A(B,L)<br /> \rightarrow \operatorname{Exalcomm}_B(C,L)\rightarrow \operatorname{Exalcomm}_A(C,L)\rightarrow \operatorname{Exalcomm}_A(B,L),<br /> &lt;/math&gt;<br /> where Der&lt;sub&gt;&#039;&#039;A&#039;&#039;&lt;/sub&gt;(&#039;&#039;B&#039;&#039;,&#039;&#039;L&#039;&#039;) is the module of derivations of the &#039;&#039;A&#039;&#039;-algebra &#039;&#039;B&#039;&#039; with values in &#039;&#039;L&#039;&#039;. <br /> This sequence can be extended further to the right using [[André–Quillen cohomology]].<br /> <br /> ==References==<br /> <br /> *{{EGA|book=4-1| pages = 65}}<br /> *{{Citation | last1=Weibel | first1=Charles A. | title=An introduction to homological algebra | url=http://books.google.com/books?id=flm-dBXfZ_gC | publisher=[[Cambridge University Press]] | series=Cambridge Studies in Advanced Mathematics | isbn=978-0-521-43500-0; 978-0-521-55987-4 | id={{MR|1269324}} | year=1994 | volume=38}}<br /> <br /> [[Category:Homological algebra]]</div>

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