dorsal/arxiv
View SchemaOn modules associated to coalgebra Galois extensions
| Authors | Tomasz Brzezinski |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9712023 |
| URL | https://arxiv.org/abs/q-alg/9712023 |
| Journal | J. Algebra, 215:290-317, 1999 |
Abstract
For a given entwining structure $(A,C)_\psi$ involving an algebra $A$, a coalgebra $C$, and an entwining map $\psi: C\otimes A\to A\otimes C$, a category $\M_A^C(\psi)$ of right $(A,C)_\psi$-modules is defined and its structure analysed. In particular, the notion of a measuring of $(A,C)_\psi$ to $(\tA,\tC)_\tpsi$ is introduced, and certain functors between $\M_A^C(\psi)$ and $\M_\tA^\tC(\tpsi)$ induced by such a measuring are defined. It is shown that these functors are inverse equivalences iff they are exact (or one of them faithfully exact) and the measuring satisfies a certain Galois-type condition. Next, left modules $E$ and right modules $\bar{E}$ associated to a $C$-Galois extension $A$ of $B$ are defined. These can be thought of as objects dual to fibre bundles with coalgebra $C$ in the place of a structure group, and a fibre $V$. Cross-sections of such associated modules are defined as module maps $E\to B$ or $\bar{E}\to B$. It is shown that they can be identified with suitably equivariant maps from the fibre to $A$. Also, it is shown that a $C$-Galois extension is cleft if and only if $A=B\tens C$ as left $B$-modules and right $C$-comodules. The relationship between the modules $E$ and $\bar{E}$ is studied in the case when $V$ is finite-dimensional and in the case when the canonical entwining map is bijective.
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"abstract": "For a given entwining structure $(A,C)_\\psi$ involving an algebra $A$, a\ncoalgebra $C$, and an entwining map $\\psi: C\\otimes A\\to A\\otimes C$, a\ncategory $\\M_A^C(\\psi)$ of right $(A,C)_\\psi$-modules is defined and its\nstructure analysed. In particular, the notion of a measuring of $(A,C)_\\psi$ to\n$(\\tA,\\tC)_\\tpsi$ is introduced, and certain functors between $\\M_A^C(\\psi)$\nand $\\M_\\tA^\\tC(\\tpsi)$ induced by such a measuring are defined. It is shown\nthat these functors are inverse equivalences iff they are exact (or one of them\nfaithfully exact) and the measuring satisfies a certain Galois-type condition.\nNext, left modules $E$ and right modules $\\bar{E}$ associated to a $C$-Galois\nextension $A$ of $B$ are defined. These can be thought of as objects dual to\nfibre bundles with coalgebra $C$ in the place of a structure group, and a fibre\n$V$. Cross-sections of such associated modules are defined as module maps $E\\to\nB$ or $\\bar{E}\\to B$. It is shown that they can be identified with suitably\nequivariant maps from the fibre to $A$. Also, it is shown that a $C$-Galois\nextension is cleft if and only if $A=B\\tens C$ as left $B$-modules and right\n$C$-comodules. The relationship between the modules $E$ and $\\bar{E}$ is\nstudied in the case when $V$ is finite-dimensional and in the case when the\ncanonical entwining map is bijective.",
"arxiv_id": "q-alg/9712023",
"authors": [
"Tomasz Brzezinski"
],
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"q-alg",
"math.QA"
],
"journal_ref": "J. Algebra, 215:290-317, 1999",
"title": "On modules associated to coalgebra Galois extensions",
"url": "https://arxiv.org/abs/q-alg/9712023"
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