On the Smoothness of the Quot Functor
For a commutative ring k,we consider free k-modules E, endowing them with k[x_1,...,x_m]-module structuresthrough a ring homomorphism k[x_1,...,x_m] -> End_Z(E). These structures arethen inspected by encoding the actions of the unknowns x_i in matricesX_1,...,X_m. We further introduce the con...
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KTH, Matematik (Avd.)
2013
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ndltd-UPSALLA1-oai-DiVA.org-kth-1236582013-06-20T04:03:51ZOn the Smoothness of the Quot FunctorengWiegandt, SebastianKTH, Matematik (Avd.)2013For a commutative ring k,we consider free k-modules E, endowing them with k[x_1,...,x_m]-module structuresthrough a ring homomorphism k[x_1,...,x_m] -> End_Z(E). These structures arethen inspected by encoding the actions of the unknowns x_i in matricesX_1,...,X_m. We further introduce the concepts of lifts and formal smoothnessfor functors, and define the Quot_{F/A/k}^n functor acting on the category ofk-algebras, taking some k-algebra B to the set of quotients of the form (F ⊗_k B)/N, which are locallyfree as B-modules. Lastly, we find concrete examples of modules showing thatthe functors Hilb_{k[x,y,z]/k}^4 and Quot_{⊕^2 k[x,y]/k[x,y]/k}^2 are not formally smooth Student thesisinfo:eu-repo/semantics/bachelorThesistexthttp://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-123658TRITA-MAT-E ; 2013:29application/pdfinfo:eu-repo/semantics/openAccess |
| collection |
NDLTD |
| language |
English |
| format |
Others
|
| sources |
NDLTD |
| description |
For a commutative ring k,we consider free k-modules E, endowing them with k[x_1,...,x_m]-module structuresthrough a ring homomorphism k[x_1,...,x_m] -> End_Z(E). These structures arethen inspected by encoding the actions of the unknowns x_i in matricesX_1,...,X_m. We further introduce the concepts of lifts and formal smoothnessfor functors, and define the Quot_{F/A/k}^n functor acting on the category ofk-algebras, taking some k-algebra B to the set of quotients of the form (F ⊗_k B)/N, which are locallyfree as B-modules. Lastly, we find concrete examples of modules showing thatthe functors Hilb_{k[x,y,z]/k}^4 and Quot_{⊕^2 k[x,y]/k[x,y]/k}^2 are not formally smooth |
| author |
Wiegandt, Sebastian |
| spellingShingle |
Wiegandt, Sebastian On the Smoothness of the Quot Functor |
| author_facet |
Wiegandt, Sebastian |
| author_sort |
Wiegandt, Sebastian |
| title |
On the Smoothness of the Quot Functor |
| title_short |
On the Smoothness of the Quot Functor |
| title_full |
On the Smoothness of the Quot Functor |
| title_fullStr |
On the Smoothness of the Quot Functor |
| title_full_unstemmed |
On the Smoothness of the Quot Functor |
| title_sort |
on the smoothness of the quot functor |
| publisher |
KTH, Matematik (Avd.) |
| publishDate |
2013 |
| url |
http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-123658 |
| work_keys_str_mv |
AT wiegandtsebastian onthesmoothnessofthequotfunctor |
| _version_ |
1716589397916254208 |