An explicit cycle map for the motivic cohomology of real varieties
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Bibliographic record
Abstract
We provide a direct construction of a cycle map in the level of representing complexes from the motivic cohomology of real (or complex) varieties to the appropriate ordinary cohomology theory. For complex varieties, this is simply integral Betti cohomology, whereas for real varieties the recipient theory is the bigraded \operatorname{Gal}(\mathbb C/\mathbb R) -equivariant cohomology [19]. Using the finite analytic correspondences from [7] we provide a sheaf-theoretic approach to ordinary equivariant RO(G) -graded cohomology for any finite group G . In particular, this gives a complex of sheaves \mathbb Zp_{\omega} on a suitable equivariant site of real analytic manifolds-with-corner whose construction closely parallels that of the Voevodsky's motivic complexes $$\mathbb Zp_{\mathcal M} . Our cycle map is induced by the change of sites functor that assigns to a real variety X its analytic space X(\mathbb C)$ together with the complex conjugation involution.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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