Matthieu Romagny
Reputation
1,827
Next privilege 2,000 Rep.
 Apr 26 comment On the coherence of formal power series ring ... natural integers. In the former ring, the element $X_1+X_2+\dots$ is not allowed; in the latter it is. Apr 26 comment On the coherence of formal power series ring What I meant is that there is no canonical definition of a ring of formal power series in countably many variables. There are several candidates, each with its own interest depending on the situation. One natural candidate is the completion of the ring of polynomials $F_p[X_1,X_2,\dots]$ with respect to the maximal ideal $(X_1,X_2,\dots)$. Another useful candidate is that introduced in Bourbaki, Algèbre, chapitre IV, whose elements are arbitrary formal sums of monomials $a_iX^i$ with $i$ a finite tuple of... Apr 25 comment On the coherence of formal power series ring The notation is misleading: is $X_\infty$ a variable? Also, you should clarify what you mean by "power series ring with infinitely many variables" since there are several possible definitions (e.g. sometimes the series $X_1+X_2+X_3+\dots$ is allowed and sometimes it is not). Apr 18 comment Finite group action on quasi-projective varieties Dear Ron, the best-known counterexample is due to Hironaka. It appears here and there in the literature; a good source is wikipedia's page for "Hironaka's example". Apr 17 comment Finite group action on quasi-projective varieties If a quotient exists as a variety or a scheme, then the fibres of $\alpha$ will be the same as $G$-orbits, and all contained in an open affine of $X$. But there exist group actions like in your setting where this condition fails, and hence no scheme quotient exists. Thus you must clarify what you mean by "the quotient". Another comment is that since you assume the action is free, if a quotient exists then $\alpha$ will in fact be \'etale everywhere. Apr 4 awarded Favorite Question Feb 1 comment Definition of ind-schemes OK, I wrote down the details for myself... you're right. Sorry. Feb 1 comment Definition of ind-schemes @Martin: I don't think so. In fact, if you restrict to quasi-compact schemes and coverings formed by finite (flat, jointly surjective etc.) families then the inductive limit presheaf is a sheaf. Therefore the map from the presheaf to the sheaf is an isomorphism when evaluated on qc schemes. It is not really a question of glueing things and covering the intersections with affines etc. Feb 1 comment Definition of ind-schemes The equality $\text{Hom}(Y,\varinjlim X_n)=\varinjlim \text{Hom}(Y,X_n)$ extends quite easily from affine schemes $Y$ to quasi-compact ones. Away from qc schemes, I do not think it works. Jan 31 answered A technical question about affine grassmanian Jan 31 comment A technical question about affine grassmanian You're welcome! Jan 30 comment A technical question about affine grassmanian I think you can find more details in Martin Kreidl's thesis, available at uni-due.de/~hx0051/Dissertation.pdf. Best, Jan 26 awarded Necromancer Jan 24 awarded Revival Jan 24 comment Flat cohomology for finite infinitesimal group scheme over a perfect field "some more argument seems to be needed to explain why $X^{perf}$ is a $G^{perf}$-torsor" : perfectization commutes with products, so if the map $G\times X \to X\times X$ is an isomorphism then $G^{perf}\times X^{perf} \to X^{perf}\times X^{perf}$ also. Jan 24 answered Flat cohomology for finite infinitesimal group scheme over a perfect field Jan 3 comment Nisnevich topology on non-(locally) Noetherian schemes "the guy who asked", or "the girl who asked". After all, MO is for female mathematicians also. Jan 2 revised Descent of functions along finite birational morphisms added 306 characters in body Jan 1 revised Pushing-forward morphisms added 5 characters in body Dec 31 revised Pushing-forward morphisms added 16 characters in body