Exalcomm (original) (raw)
In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(R,M) are isomorphism classes of commutative k-algebras E with a homomorphism onto the k-algebra R whose kernel is the R-module M (with all pairs of elements in M having product 0). Note that some authors use Exal as the same functor. There are similar functors Exal and Exan for non-commutative rings and algebras, and functors Exaltop, Exantop. and Exalcotop that take a topology into account.
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| dbo:abstract | In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(R,M) are isomorphism classes of commutative k-algebras E with a homomorphism onto the k-algebra R whose kernel is the R-module M (with all pairs of elements in M having product 0). Note that some authors use Exal as the same functor. There are similar functors Exal and Exan for non-commutative rings and algebras, and functors Exaltop, Exantop. and Exalcotop that take a topology into account. "Exalcomm" is an abbreviation for "COMMutative ALgebra EXtension" (or rather for the corresponding French phrase). It was introduced by , 18.4.2). Exalcomm is one of the André–Quillen cohomology groups and one of the Lichtenbaum–Schlessinger functors. Given homomorphisms of commutative rings A → B → C and a C-module L there is an exact sequence of A-modules where DerA(B,L) is the module of derivations of the A-algebra B with values in L. This sequence can be extended further to the right using André–Quillen cohomology. (en) |
| dbo:wikiPageExternalLink | https://books.google.com/books%3Fid=flm-dBXfZ_gC https://web.archive.org/web/20200429024215/https:/math.berkeley.edu/~molsson/MSRISummer07.pdf |
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| dbo:wikiPageWikiLink | dbr:Cambridge_University_Press dbr:Module_(mathematics) dbr:Deformation_theory dbr:Dual_number dbr:André–Quillen_cohomology dbr:Gerbe dbr:Cotangent_complex dbc:Homological_algebra dbr:Lichtenbaum–Schlessinger_functor dbr:Picard_stack |
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| dcterms:subject | dbc:Homological_algebra |
| rdfs:comment | In algebra, Exalcomm is a functor classifying the extensions of a commutative algebra by a module. More precisely, the elements of Exalcommk(R,M) are isomorphism classes of commutative k-algebras E with a homomorphism onto the k-algebra R whose kernel is the R-module M (with all pairs of elements in M having product 0). Note that some authors use Exal as the same functor. There are similar functors Exal and Exan for non-commutative rings and algebras, and functors Exaltop, Exantop. and Exalcotop that take a topology into account. (en) |
| rdfs:label | Exalcomm (en) |
| owl:sameAs | freebase:Exalcomm wikidata:Exalcomm https://global.dbpedia.org/id/4jhbc |
| prov:wasDerivedFrom | wikipedia-en:Exalcomm?oldid=1120865540&ns=0 |
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| is dbo:wikiPageRedirects of | dbr:Exal dbr:Exalcomm_functor dbr:Exalcotop dbr:Exaltop dbr:Exan dbr:Exantop |
| is dbo:wikiPageWikiLink of | dbr:Deformation_(mathematics) dbr:André–Quillen_cohomology dbr:Cotangent_complex dbr:Abelian_2-group dbr:Fibred_category dbr:Exal dbr:Exalcomm_functor dbr:Exalcotop dbr:Exaltop dbr:Exan dbr:Exantop |
| is foaf:primaryTopic of | wikipedia-en:Exalcomm |