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Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``class …

Fix a filtered monoid, $H=H_0\supsetneq H_1\supsetneq H_2\supsetneq\cdots$. Suppose for any $h\in H$ and any $n\in \Bbb{N}$ exists $h_n\in H$ such that $h\cdot h_n\in H_n$ and $h_n\cdot h\in H_n$. If …

Fix a field of zero characteristic, $k$, e.g. $\Bbb{R}$ or $\Bbb{C}$. Suppose $k$ is normed (and complete for its norm). Consider the ring extensions: $k[x_1,..,x_n]\subset \ k<x_1,..,x_n> \ \subset k …

Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.) Let $M$ be a finitely generated $R$-module o …

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" …

Let $(R,\mathfrak{m})$ be a local ring over a field. Suppose the ring has flat elements, i.e. $\mathfrak{m}^\infty\neq\{0\}$. (The prototype is of course $C^\infty(\Bbb{R}^p,0)$, or a quotient of it, …

Let $R$ be a local Noetherian ring over a field, with the maximal ideal $\mathfrak{m}$. (e.g. $R=k[[x_1,\dots,x_{p>1}]]$) Given two $R$-modules, $N\subset M$, of the same (finite, non-zero) rank. Th …

Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can …

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing do …

Let $R$ be a (commutative, associative, unital) ring, consider a homomorphism of some (finitely generated) free $R$-modules $E\stackrel{A}{\rightarrow}F$. Say $rank(F)=m$. The basic invariants of $A$ …

Let $A\in Mat(m,n;R)$ for $m\le n$ and $R$ a local ring. Consider the $mn\times(m^2+n^2)$ matrix $A\otimes 1_{nn}\oplus 1_{mm}\otimes A^T$, here $1_{mm}$, $1_{nn}$ are identity matrices. I'd like to k …

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ et …

In Artin1968 he considers $\underline{analytic}$ equations, but over the ring $R=k\{x_1,..,x_n\}$. In Artin1969 he works with $R=k\{x_1,..,x_n\}/I$, not necessarily regular, but considers $\underline …

Let $R$ be a local ring (commutative, Noetherian, over an algebraically closed field; if needed Henselian). Suppose one wants to prove some statement. Suppose $R$ happens to be the ring of "functions …

Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,\dotsc,n_1$, $j=1,,n_2$, $k=1,\dotsc n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices. W …