Steven Landsburg
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 5h comment if 0→A→A⊕B→B→0 is an exact sequence of finitely generated modules over a commutative Noetherian ring, then the exact sequence does split See my answer here: mathoverflow.net/questions/167701/… 2d comment Please help me for answers to question. best regards Preparing for an exam a day before might or might not be desipicable, but asking others to solve the exam problems for you most certainly is. Feb 8 comment Grothendieck, A Place to Begin I mostly agree with Joe Silverman. You certainly need some commutative algebra first. But I'd start with Mumford's red book rather than Hartshorne. The first part is a very clear presentation of a (somewhat) more classical approach so you understand what schemes are intended to generalize, and the second part is a very clear presentation of scheme theory with emphasis on why it's the right generalization. There's no cohomology, but first things should come first. Feb 6 comment Win/Lose ratios and selection Voted to close as incomprehensible. Jan 29 comment Complete Local Ring and Fermat's Last Theorem I would suggest editing out the final sentence, which has nothing to do with the question. Jan 28 comment If $A$ is the ring of continuous functions on a genus $g$ surface, can the genus of $X$ be seen by simple algebra in $A$? Given the ring $A$, one recovers the space $X$ as the MaxSpec of $A$, and then you can compute the genus (or any other topological invariant) of $X$ any way you like --- for example, via Cech cohomology. The two steps --- going from $A$ to $X$ and going from $X$ to its genus --- are easily described, so it's easy to describe the composition. "Easily described" is not the same thing as "transparent", but I don't see a reason to expect transparency. Jan 24 comment Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid) (CONTINUED) She even said something about trying to figure out what it means for two computations to be essentially different. When I praised her exactly as you propose to praise your students ("Wow! That's exactly the kind of question a real mathematician might ask!"), she said: "Really? A real mathematician might ask a question like this". I said yes. And she said --- this is verbatim, because I wrote it down instantly: "Wow! What a soul-deadening job THAT must be." Jan 24 comment Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid) Not really relevant but I feel compelled to share this: When my daughter was the same age as your students, I found her staring at a piece of paper on which she'd drawn many copies of the same right triangle in a variety of configurations. I asked what she was doing, and she said she was trying to figure out how many different ways there are to compute the area of the same triangle --- she had fit two together to make a rectangle, four together to make a different rectangle, etc. (CONTINUED) Jan 21 comment How do i show that:$\prod\frac{p^2+1}{p^2-1}=\frac{5}{2}$ without using properties of Riemann zeta function? This is probably nonsense but: If you formally expand the factors of this product as power series in $p$, use the fundamental theorem of arithmetic, and make the heroic assumption that the product of infinitely many $-1$'s is $+1$, then the expression becomes $\sum_{n=1}^\infty 2^{\Omega(n)}n^2 = 5/2$, where $\Omega(n)$ is the number of distinct prime factors in $n$. I wonder if this could be true in some sense. For example, set $Z(s)=\sum_{n=1}^\infty 2^{\Omega(n)}n^{-s}$ where this converges, and analytically continue. Then we could hope that $Z(-2)=5/2$. Could this be? Jan 19 comment How is it that you can guess if one of a pair of random numbers is larger with probability > 1/2? Let the two numbers be $a$ and $b$ and denote your strategy by $S$. Your probability of winning is some function $P$ of $a$, $b$ and $S$. The claim is that there exists an $S$ such that for all $a$ and $b$, $P(a,b,S)>1/2$. This is a perfectly meaningful claim and does not rely on any probability distribution for $a$ and $b$. Jan 16 awarded Enlightened Jan 16 awarded Nice Answer Jan 14 comment Description of the equalizer of $\prod _j R/I_j \rightrightarrows \prod _{i,j}R/(I_i+I_j)$ See the first answer here:mathoverflow.net/questions/21782/…. Jan 12 comment What does the axiom of replacement mean and why should I believe it? Should that iterated power set be just a power set? Jan 7 comment What can be said about $A$ and $B$ given the exact sequence $0 \to R^p \to A \to R^r \to R^q \to B \to 0$? If the projective dimension of $B$ is at most $2$, then $A$ is projective. If the projective dimension of $B$ is $3$, then the projective dimension of $A$ is at most $1$. If the projective dimension of $B$ is greater than $3$, then the projective dimension of $A$ is exactly two less than the projective dimension of $B$. Dec 27 comment Grothendieck spectral sequence when one of the functors is contravariant @LisaS. Regarding convergence, etc, Sander Kovacs's answer has convinced me that this is a subtler business than I'd first thought. I see from the comments on that answer that you're hard at work digesting it. Dec 27 comment Grothendieck spectral sequence when one of the functors is contravariant The $E_2$ terms are exactly what you said they are in the second sentence of your post. Dec 27 revised Grothendieck spectral sequence when one of the functors is contravariant added 2 characters in body Dec 27 answered Grothendieck spectral sequence when one of the functors is contravariant Dec 19 comment Bayes statistics precisely formulated Is my answer at mathoverflow.net/questions/187379/… of any help?