
Commutative Algebra Webinar

وبینار جبر جابجایی




TITLE

Multigraded Algebra over Polynomial Rings with Real Exponents



LECTURER

Ezra Miller



Duke University







TIME

Thursday, July 29, 2021,


18:00  20:00




VENUE 
Lecture Hall 1, Niavaran Bldg.



SUMMARY 


Commutative algebra over polynomial rings with real exponents
has become important in the past decade because of its entrance
into noncommutative and nonrational toric geometry as well as
applications in topological data analysis. Both of these
settings specifically deal with multigraded modules over
realexponent polynomial rings. But essentially nothing has
been known about realexponent algebra, in part because
noetherian hypotheses fail spectacularly: monomial ideals can
be uncountably generated; the quotient modulo a monomial ideal
can fail to contain even a single copy of the quotient modulo
any prime ideal; all positive powers of the unique maximal
graded ideal are equal to one another. What should Nakayama's
lemma say in a setting where finite generation can never be
expected? How can primary decomposition work when modules need
not contain copies of quotients modulo primes? Even more
fundamentally, what kind of finiteness should prevail given the
abject failure of noetherianity? Thinking geometrically about
the algebra of monomial ideals with real exponents leads the
way to complete, satisfying answers to all of these questions.
The lessons learned reflect back on improved ways to think
about aspects of ordinary noetherian commutative algebra, such
as what it means for a primary decomposition to be minimal and
why that question is directly to dual to Nakayama's lemma.
The mathematics here is a summary of arXiv:math.AT/2008.03819
If you're interested, here are some warmup exercises. In the
twovariable realexponent polynomial ring, find an ideal that
is minimally generated by uncountably many monomials. Instruct
a computer how to store your ideal. Compute a resolution of your
ideal. Does your ideal have a primary decomposition? Show that
the ideal of all realexponent polynomials with constant term 0
is maximal and generated by countably many monomials, although
it does not admit a minimal generating set. Do the same for the
ideal of all realexponent polynomials with no pure powers of
(say) the first variable, but with ``prime" instead of ``maximal".
It is not at all necessary to do these exercises to understand
the talk, but they'll help you appreciate what's going on.
https://us06web.zoom.us/j/9086116889?pwd=TnhBUllnQlkwYXR4OWMxOHpjUSsyUT09
Meeting ID: 908 611 6889
Passcode: 210810




تهران، ضلع جنوبی
ميدان شهيد باهنر (نياوران)، پژوهشگاه
دانشهای بنيادی، پژوهشکده رياضيات
School of
Mathematics, Institute for Research in
Fundamental Sciences (IPM), Niavaran
Bldg., Niavaran Square, Tehran
ipmmath@ipm.ir
♦ +98 21
22290928 ♦
math.ipm.ir 


