Let A be an unital, associative, graded algebra over a base ring k. I’m happy to assume that k is a field if need be, and will insist that A free and of finite rank in each degree (locally finite). Further, A is connected: it vanishes in negative degrees and is of rank 1 (generated … Read more
Let R be a commutative ring, let M be a commutative monoid, and let R[M] denote the corresponding monoid algebra. Suppose further that R is universally catenary. One may ask for conditions on M such that the ring R[M] is catenary. I know of the following results: If M is of finite type then R[M] … Read more
I’ll start off a little vauge: Let E be a noncommutative ring which is finitely generated over its noetherian center Z. Denote by modE the category of finitely generated left E-modules and similarly for modZ. We have a functor F:modZ→modE which takes M to E⊗ZM, hence an induced map on (Quillen) K1-groups K1(F):K1(modZ)→K1(modE). I’m interested … Read more
In a paper I recently read something about the “unibranch partial normalization” of a curve. Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it possible to find a minimal ring $R^u$ between $R$ and its normalization $R’$ in $K$ such that all localizations of $R^u$ in the … Read more
Let (R,m) be a local ring over a field. Suppose the ring has flat elements, i.e. m∞≠{0}. (The prototype is of course C∞(Rp,0), or a quotient of it, by some finitely generated ideal.) 1. For which rings and ideals, J⊂R, the following holds. If the completions satisfy ˆJ⊇(ˆm)N then J⊇mN+n, for some finite n. At … Read more
Let R be a noetherian domain with field of fractions F, let V be a finite-dimensional F-vector space, and let M,N⊆V be R-lattices in V (finitely generated R-submodules of V containing a basis for V over F). We define the R-index of N in M, written [M:N]R, to be the R-submodule of F generated by … Read more
Let $k$ be an algebraically closed field of characteristic $0$, let $V=k^n$ be a $k$ vector space of dimension $n$, and let $R=k[V]$ be the ring of polynomial functions on $V$. Suppose that $W\subset\operatorname{GL}(V)$ is a finite reflection group, and let $S\subset W$ be a set of simple reflections so that $(W,S)$ is a finite … Read more
The rational cohomology of a homogeneous space G/K admits a homomorphism from H∗(BK) induced from the classifying map G/K→BK of the principal K-bundle G→G/K. Assume the Lie group is K connected, so that π1(BK)=0; then the space G/K is formal in the sense of rational homotopy theory if and only if H∗(G/K) is a free … Read more
I’m reading Deligne’s paper “Le Groupe Fondamental de la Droite Projective Moins Trois Points”, specifically in the section “Theorie profinie” (sections 15.13 – 15.27) I’m specifically interested in 15.19, where he says (I had to fix some apparent typos): More generally, let $\overline{X}$ be a smooth morphism of relative dimension 1 over a normal scheme … Read more
Some main definitions and basic facts of divided differences: In the symmetric group $S_n$, let $s_i$ denote the adjacent transposition $(i,i+1),i\in \{1,2,\cdots,n-1\}.$ Since $S_n$ is generated by adjacent transpositions, any permutation $w\in S_n$ can be written as $w=s_{i_1}s_{i_2}\cdots s_{i_l}$ and an expression $w=s_{i_1}s_{i_2}\cdots s_{i_l}$ of minimal possible length $l$ is called a reduced decomposition, $l=\ell(w)$ … Read more
