![com-alg[1].png](https://i.imgur.com/rv9JiJ9.png "com-alg[1].png")
Studying commutative algebra, what are some cool things further on that I might be able to appreciate now?
AB=BA
>>7981750
/thread
>>7981750
ABBA
>FATHER
ABBA
>FATHER
ABBA
>FATHER
ABBAAAA
FATHER INTO YOUR HANDS I COMMEND MY SPIRIT
FATHER INTO YOUR HANDS
WHY HAVE YOU FORSAKEN ME?
IN YOUR EYES FORSAKEN ME?
IN YOUR THOUGHTS FORSAKEN ME?
IN YOUR HEART FORSAKEN ME?
OHHH TRUST IN MY SELF RIGHTEOUS SUICIDE
WHY CRY WHEN ANGELS DESERVE TO DIE?
INNN MYYY SELF RIGHTEOUS SUICIDE
WHYYY CRYYY WHEN ANGELS DESERVE TO DIIIEEEE
>>7981750
Based
(Exactness of) Localisation, some basic homological algebra (Tor, Ext), basic algebraic geometry (varieties, nullstellensatz)...
>>7981750
this desu senpai
>27 LINES ON A CUBIC SURFACE
[A,B]=0
Are there any fields of upper level math that reject the commutative property with interesting results? Like how abstract algebra relaxes rules on vector spaces?
>>7982252
Lie theory
>>7982252
Group theory, representation theory, operator algebras
>>7982252
Non-commutative geometry is all the rage these days.
>>7981750
nice
>>7982252
Quantum mechanics. This is how we get all the weird juicy stuff.
>>7981643
I find it incredibly hard to come up with purely commutative-algebraic results that are inherently fun. A theorem of Serre says that a local Noetherian ring is regular if and only if all its modules admit a finite resolution by projective modules (direct summands of free modules). Moreover, when this is the case, every module has such a resolution of length equal to at most the Krull dimension of the ring.
>>7981643
What are surfaces like this in commutative algebra? Please try to intuitively explain such surfaces to someone who knows only rudimentary algebra.
>>7983507
>Please try to intuitively explain such surfaces to someone who knows only rudimentary algebra.
That's all you need; they're just solutions to polynomial equations.
>>7983528
With some special properties?
>>7983507
Look up the classical definition of an Algebraic Variety. It is fairly simple.
An Algebraic Surface is simply an Algebraic Variety of dimension 2.
Pictured: the founders of commutative algebra.
>>7983775
Nope, just any old polynomials.
>>7983775
There's really nothing special about commutative algebra
>>7983507
This picture reminds me of a Kummer Surface. This is well documented in the litterature. It can indeed be seen as the set of zeroes of a single multivariate (homogeneous if projective) polynomial of degree 4 in 4 variables. An equation can be derived from the theta coordinates of a general Jacobian Variety of a genus 2, and therefore hyperelliptic, curve, using the fact that it is isomorphic to its quotient by the canonical involution on such varieties (a subclass of Abelian ones).
>>7985169
Here come the dick-swingers
>>7983775
It's not that it's special - just the reverse - it abstract the special cases like C.
By a theorem of Grothendieck, the category of commutative rings is dually equivalent to the category of affine schemes.
What this really means is that you can turn any commutative ring into an actual honest to god topological space with an honest to god sheaf of polynomials on it and form "honest to god" rational "functions" with them, even if the ring seems to have nothing to do with functions and the results get whacked out. In many "nice" cases you still have analogues of nullstellsatz and stuff.
For example, Z will correspond to the terminal affine scheme, since it is initial in CRing, so for any affine scheme, like the complex plane considered with Zariski topology and a new generic point, can be looked at as it maps into SpecZ.
The space of Z will have as "points" prime numbers and the value of the "function" 15 at the "point" p will just be the value of 15 mod p in the field Fp. It abstracts away from functions in this sense in that the same "function" may take a value in a different field at each point, in a way that is compatible with the way the ring works.
The cool thing is that some of what you know about polynomials carries over into these weird number-theoretic schemes.
