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 real-exponent polynomial rings. But essentially nothing has been known about real-exponent 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 warm-up exercises. In the two-variable real-exponent 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 real-exponent 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 real-exponent 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.

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