Expository Notes

    These are some short notes that I have written over time. Most of them are things that I have worked out at certain point and I don't want to completely forget. I am posting them here in case they might be useful to someone else. Beware that they are in very rough form.

    Feel free to send me an email if you have comments/questions about these notes.

    (Remark: I have learned that SGA3 has a nice treatment of contents in some of these notes).

    1. A local criterion for smoothness.
    2. We give a local criterion for smoothness of a morphism between smooth schemes in terms of their cotangent spaces. The proof is a small diagram chase argument using the cotangent complex. Corollary 3 in this note can be used to prove the converse in the main proposition in this post in positive characteristic, thus answering a question of Sean Cotner.

    3. My favorite flatness results.
    4. A laundry list of useful criteria for flatness.

    5. A note on scheme-theoretic image for nonquasicompact morphisms.
    6. This contains a rough sketch of a proof that the formation of scheme theoretic image for a (not necessarily quasicompact) morphism commutes with flat base change, under the assumption that the target is quasicompact, quasiseparated, and essentially free over the base. We use the concrete description of coherator for sheaves on a quasicompact quasiseparated scheme.

    7. Resolutions via monoids.
    8. Notes on canonical/bar resolutions, with examples.

    9. Some remarks on equivariant sheaves.
    10. Some expository notes giving intuition on the notion of equivariant sheaf.