Some remarks on Frobenius and Lefschetz in étale cohomology. http://http://www.math.mcgill.ca/ ∼goren/SeminarOnCohomology.html
Bibliographic record
Abstract
In this lecture I will discuss some more or less related issues revolving around the main idea relating (étale) cohomology and the zeta function of a scheme X over Fp, which is: via the Lefschetz trace formula, studying the zeta function amounts to studying the representation of the Frobenius morphism on cohomology. I will start to try to clarify a bit which Frobenius morphism we’re interested in, and then we’ll look explicitely at some examples of 0-dimensional schemes (for which the Lefschetz trace formula takes a particularly simple form!). 1 The absolute Frobenius morphism Let’s start by studying the Frobenius morphism in some generality. The first thing to do is to restrict ourselves to the subcategory of schemes which do admit a Frobenius morphism. Throughout, p will be understood to stand for a fixed prime number. Definition 1.1. A scheme X is said to be of characteristic p if p OX = 0. Of course, a given scheme cannot have two distinct prime characteristics unless its structure sheaf is 0, i.e. unless it is Spec 0, the initial object in the category of schemes. Remark that saying that X is of characteristic p amounts to saying that the (unique!) morphism X → Spec Z factors through Spec Fp.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".