00:43:55 Fangzhou Jin: sorry I was supposed to talk about this in my talk, I prepared this but forgot it the day I gave the talk
00:44:00 Tom Bachmann: FWIW this is Definition 4.4.2 in Adeel's thesis
00:44:04 Tom Bachmann: https://www.preschema.com/papers/thesis.pdf
00:44:10 Tom Bachmann: (but of course also in many other places)
00:45:47 Tom Bachmann: B = B_{cl} x_{A_cl} A ?
00:49:09 Prof Kirsten Wickelgren, Ph.D.: If I shift a locally free sheaf, is it still called locally free?
00:50:36 Tom Bachmann: I think that's a matter of definitions, but saying yes would seem peculiar to me.
00:50:58 Prof Kirsten Wickelgren, Ph.D.: Thanks!
00:51:18 Federico Binda: I agree with Tom
00:51:55 Sean Tilson: That seems really strange to me because it seems like shifting shouldn’t change whether or not it has the universal property. So thanks for clarifying!
00:52:29 Tom Bachmann: the universal property of being locally isomorphic to a certain object not invariant under shifts...?
00:52:37 Sean Tilson: or are our categories of modules always non-negative?
00:53:04 Sean Tilson: I was thinking something closer to the projective basis theorem.
00:53:38 Marc Noel Levine: True, but what does ``free’’ mean (mathematically, not philosophically)? Can’t it mean having a basis, not necessarily in degree 0?
00:54:37 Sean Tilson: right, but that would be invariant under shifts, right? or maybe I am confused.
00:56:11 Tom Bachmann: Isn't N_{Z/X} := pi_0(L_{Z/X}) ?
00:56:38 Tom Bachmann: No it isn't
01:02:09 Tom Bachmann: N_{Z/X} = L_{Z/X}[-1] is a locally free O_Z-module if Z/X is a regular closed immersion. This is what I meant to say. Doesn't 100% explain what surjection means either, but presumably on pi_0 as was said before
01:04:18 Federico Binda: I think it’s really ok if we just put the condition on \pi_0
01:05:59 Federico Binda: Also, you can have Z\to X which is just a quasi smooth closed immersion, like R//0 \to R, but not a regular immersion
01:08:10 Tom Bachmann: Good point. I meant quasi-smooth closed immersion.
01:13:32 Heng Xie: A surjection of commutative rings A -> B means the cokernel is zero. For a surjection of simplicial commutative rings A -> B, we need to make sense what does it mean by the cokernel to be zero. A simplicial commutative ring R is zero if it is contractible (pi_0 R = 0).
01:14:33 Marc Noel Levine: Do cockerels exist is simplicity com rings?
01:14:48 Marc Noel Levine: cokernels (that spell checker!)
01:15:27 Federico Binda: (cockerel was great!)
01:15:43 Sean Tilson: You can just take the upshot, no?
01:15:45 Federico Binda: I mean: I don’t think you can talk about cokernels
01:16:09 Sean Tilson: Ah, sorry, the question is is it a SCR.
01:16:42 Sean Tilson: ignore me.
01:17:39 Heng Xie: Maybe you can talk about cone?
01:18:00 Marc Noel Levine: that sounds better
01:19:09 Sean Tilson: so you can talk about the same upshot in either simplicity abelian groups or simplicity commutative rings. The homotopical notion of these seems like they won’t agree. (the coproduction is different and the notion of cofibrancy is different).
01:58:17 Alessandro D´Angelo: Thanks for the talk!