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What is...the Why of ringed spaces?

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Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the Why of ringed spaces? Or: Geometry and algebra again.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
In this course I try to cover my favorite topics in algebraic geometry, from classical ideas such as algebraic varieties, to modern ideas such as schemes, to really modern ideas such as tropical varieties. I give a very biased collection of topics, and not nearly all that can be said will be addressed. Sorry for that.
Slides.
www.dtubbenhaue...
Website with exercises.
www.dtubbenhaue...
Thumbnail.
Picture from the first video slides.
Classical algebraic geometry.
en.wikipedia.o...
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Modern algebraic geometry.
en.wikipedia.o...
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Modern algebraic geometry version 2.
en.wikipedia.o...
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Applications of (algebraic) geometry.
math.stackexch...
Pictures used.
www.wikidata.o...
Picture created using reference.wolf...
Another picture created using reference.wolf...
Picture from • What is...Hilbert’s Nu...
Another picture created using reference.wolf...
Some books I am using (I sometimes steal some pictures from there).
agag-gathmann....
www.cambridge....
bertini.nd.edu...
mathoverflow.n...
Computer talk.
magma.maths.us...
reference.wolf...
#algebraicgeometry
#geometry
#mathematics

Опубликовано:

4 сен 2024

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Комментарии : 9

@georgediderrich43 18 дней назад

Your explanations and insights into AG have greatly helped me in my project. Bravo 👏

@VisualMath 18 дней назад

Thanks for the feedback, I am very happy to hear that the videos were helpful. Enjoy your AG journey ☺

@mojedsamad7184 2 месяца назад

Very interesting!

@VisualMath 2 месяца назад

Agreed 😁 I hope you enjoy AG ☺

@Jaylooker 2 месяца назад

That map [t] -> [t,t^-1] looks familiar. Maybe this is the difference between multiplicative group G_m and additive group G_a ([t] -> [t]). Although, over fields of characteristic 0 they are equivalent. See 2.2.2 and 2.2.3 of “Complex cobordism and algebraic topology” (2007) by Morava. It also reminds me of inverting a Lefschetz motive for some reason. I think Emerton’s answer to “Why does one invert G_m in the construction of motivic stable homotopy?” on MathOverflow at least gets some of what I was after with the quote: “… I believe that inverting G_m is same thing as inverting the Lefschetz motive”.

@VisualMath 2 месяца назад

Yes, that should be the difference between the multiplicative group and the additive group. But they are not isomorphic (equivalent) - not sure what you mean with that 🤔

@Jaylooker 2 месяца назад

@@VisualMath Sorry, I think I was wrongly conflating the multiplicative group with the multiplicative group law and the additive group with the additive group law.

@VisualMath 2 месяца назад

@@Jaylooker Ah, no worries. I get confused all the time. The way I remember that they are not the same is via the coordinate rings (polynomials versus Laurent polynomials) 😀

@Jaylooker 2 месяца назад

@@VisualMath Good point. That clarifies things. I think the same maps appear again with Laurent polynomials having rings R[t, t^-1] and polynomials having rings R[t]. These are still related. The localization of a commutative ring S away from an element s ∈ S is a universal way to invert s. One example is localization of polynomial ring Z[t] which is the Laurent polynomial ring Z[t,t^-1] which provides one map. The other map is the identity of a polynomial ring. Localization also apply to categories and this localization of categories is what I had in mind when inverting the Lefschetz motive.