01. Algebraic geometry - Sheaves (Nickolas Rollick)
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- Опубликовано: 3 окт 2024
- Algebraic geometry seminar
Department of Pure Mathematics
University of Waterloo
September 15th, 2016
Following the notes of Ravi Vakil, available at math.stanford.e...
A great lecture! That said, if in the future there is a way to have the camera pointed directly at the board rather than at an angle it would be extremely helpful.
Less interrupting to display vast knowledge and superiority, more interrupting to ask genuine questions and propose clarifications.
Brilliant people have questions, too
I love this guy's enthusiasm.
Nothing like a full board of definitions to begin a topic.
Note that one can turn contravariant functors in to covariant with a simple change of wording: Any functor on the open sets with morphisms as inclusion/subset can be converted in to that of a functor who's morphisms are the superset relationship. This flips the direction of the arrows but more importantly shows that it's all about language. We tend to think in terms of subsets rather than super sets and so when we map that concept to other categories the directions may be inverted from what we think(sorta mirror image) because the natural direction, in this example, would be to use supersets rather than subsets.
thank you very much for making this available to a larger audience! ❤️❤️
Beautiful example at the end of the lecture!
I remember my calculus teacher at the time, talking about sheaf theory and cohomology and the math you need to get there. 6:04, 23:26
Very pleasant video.
The Lecturer is so patient! omg what's the target audience of this class......
Very nice job. I really enjoyed the whole lecture. It cleared up a few things for me.
It was really help full for me,thank you ....Nickolas Rollick sir
so helpful and i love it!
Great presenter
For category, morphisms exist, and compose, but there is no requirement they exists for every pair of objects. For example not every two rings are homomorphic
Actually... Mor(A, B) must exist for every pair A, B, but it could be empty.
and on top of that the constant 0 maps are always homomorphisms
thank you
so good thank you
such a clear ..great series of lectures ..but why ..the camera points at the blackboard from an angle ??? but anyways ..thanks for this informative series of lectures ..very clear lectures ....
Rajarshi Chatterjee I don’t like it either! The room is like that: very wide but very narrow.Thanks for the comments though, I’ll tell Nick you appreciated his lecture.
@@KurtMahler1903 thanks mate ....and yeah Nick is a great teacher ...
Due to the bad camera angle I can't see what is being written.
I'm trying to make notes as he speaks.
THANKS FOR SHEAVES
At 36;10 i think that f-1(W) = Union (f_i)-1(W intersection f_i(Ui)) - i may be wrong as i have never studied sheaves but itfeels like the cover should be involved initially at least
I believe it should be f-1(W) = Union (f_i)-1(W) intersection Ui, as this guarantees that f-1(W) contains only the points that are mapped into W by some f_i and simultaneously belong to Ui so that f will behave like f_i on that point. If we take f-1(W) = Union (f_i)-1(W intersection f_i(Ui)), some extra points slip through the net, namely, those points x that are sent into W by some f_i but do not themeselves belong to Ui. This definition only guarantees that f_i(x) is in f_i(Ui), but this doesn't mean x is in Ui. x may be some other point, outside of Ui, that is sent to the same portion of W that Ui is sent to by f_i. Because x may not be in Ui, f(x) may not equal f_i(x) (as f(x) is defined to behave like f_i only in Ui), so f(x) may not be in W after all (and so x shouldn't be permitted by out construction of f-1(W)), even though f_i(x) most certainly is. I think the difference between these subtly different definitions for f-1(W) essentially stem from the fact that f_i^-1( f_i( Ui ) ) is a superset of Ui - it may not equal Ui, so there can be extra points in f_i^-1(W intersection f_i(Ui) ) who are indeed mapped into W and into f_i(Ui) by f_i, but do not belong to Ui.
Shouldnt one say that the presheaf is a functor from the opposite category of Open(X) to C? Otherwise one might get the impression that a functor always reverses the order here...
He did specify that it’s a contravariant functor. If you don’t add the qualifier then a functor is assumed to be covariant, i.e. non-order-reversing.
@@Bignic2008 was about to comment this!
nice explanation
Scheaves
How is a ring an open set? It's compliment in R is open so a ring is a closed set.
He is not claiming that a ring is an open set, and by your argument that a ring is a closed set, i dont believe that you know what a ring is in this context. It is an algebraic object in which you can add, subtract and multiply (not necessarily divide, such as with integers). A typical ring is not a subset of R so it doesnt have a compliment in R.
Beyond Chase Bender’s response, a set can be both open and closed. A set’s complement can be open and the set itself can still be open, so the closed-ness of a ring wouldn’t properly follow from your reasoning, even ignoring what a ring actually is.
These highly abstract courses being taught with the format of Definition. Theorem. Proof in cycles means the lecturer doesn't really understand the subject itself but are just regurgitating material. It is rare to find an instructor of these courses that truly teach. There is no such thing as teaching, just learning.
Canada, eh?
Sorry
nice.
In scrubs
comon... a sheaf is a presheaf exactly because you are adding more conditions.
Just to be clear: A Sheaf is a presheaf with extra conditions. 24:13
@@j.rogelioperezbuendia720 It is correct language to say it either way depending on how you finish the sentence:
A) A sheaf is a presheaf satisfying blah blah blah
B) A presheaf is a sheaf precisely when it satisfies blah blah blah
Someone also pointed this out to him, but when you're lecturing you're thinking about a lot of things at once and this kind of mistake can happen.
hahahahha I lach mich weg
What an awful blackboard. As expected at least the lower 1/3 is never used, because no-one wants to crawl on the floor half of the time...
probably a result of ADA requirements
Good stuff! From the point of view of motivation, it would be very nice to understand, loosely speaking, what the deficiency is with the old concept of variety. Was it simply too restrictive to account for all the examples?
If you’re still wondering, a big reason for moving to schemes is because there are fewer degeneracies. The variety of the ideal generated by x^2 is the same as that generated by x, but they define different schema.
@@Bignic2008 To add to it, this is much more important than you might think. Take two examples, consider varieties generated by the intersection of (y-x) and (y) and the variety generated by the intersection of (y) and (y-x^2) in a polynomial ring F[x,y], where F is a field. In the first case geometrically the picture is that of an intersection between two straight lines and in the second its a straight line being tangent to a parabola. Varieties regard the two overlaps as the same, when they are clearly not, you are losing out on information. Of course all the information is present in the case of having F be the complex numbers or the reals, you can use calculus to extract it, but it won't be if you take F to be a finite field for instance. Why would you give a shit about this? Well, the reason why is simple - all the arguments and all the information formalized in schemes was already used by Italian algebraic geometers, they would frequently appeal to things like infinitesimals and generic points and other such things while having no formalism for it and ending up proving theorems that are now known to be false. Schemes allowed you to do everything you want in a rigorous way.
Another good property of schemes is that schemes can be made out of almost anything algebraic, you can just plug in a ring and talk about curves over Z or Z[i] or a number field or the p-adic integers or whatever you want as long as it is commutative.
@@mikhailmikhailov8781 I think, that there is a deep moral motivation behind Grothendieck's work. Due to his experience in life and due to pervert, demagogical use of modern mathematics and physics, he found himself living a mission of demonstrating, that mutual understanding is possible; that there are ways to map between different parts of a system and, by extension, for people to find common understanding not only in mathematics...
This is why I experience some sort of joy, while studying this concepts.
@@Suav58 Absolutely. Grothendieck was a brilliant mathematician.