Fundamental Groups etc.

Suppose $R$ is Artinian over a field $k$, so that $R$ has finitely many prime (maximal) ideals $\mathfrak m_1,\ldots,\mathfrak m_n$ and $\mathfrak m_1\cdots\mathfrak m_n$ is nilpotent, say of order $r$. Then by Chineese reminder theorem, $R=R/(\mathfrak m_1\cdots\mathfrak m_n)^r\cong\prod_i R/\mathfrak m_i^r,$ which is a product of Artin local rings.

Finite Maps: An affine map $f:X\to Y$ of schemes is called affine if it is locally given by finitely generated modules, i.e. for affine ${\rm Spec} A\hookrightarrow Y, {\rm Spec} B=f^{-1}({\rm Spec} A)\to {\rm Spec} A$ is given by a finitely generated module $B$.

As such, an algebra $A\to B$ which is a finitely generated module, is also integral. For example, if $x_1,\ldots,x_n$ are spanning elements, then for any $x\in B$, we consider the matrix equation $\sum_j(x\delta_{ij}-\alpha_{ij})x_j=0, i=1\ldots,n$, which shows that ${\rm det}(xI-(\alpha_{ij}))=0$, hence $x$ is integral.

By going up theorem, thus, finite morphisms are closed. Namely, while it is sufficient to check on open affines, if $\phi:A\to B$ happens to be the ring level map and $I$ an ideal in $B$, then $A/\phi^{-1}(I)\hookrightarrow B/I$ is integral, so that the induced scheme level map is surjective by going up theorem, and thus, $\phi^*({\rm Spec} (B/I))={\rm Spec} (A/\phi^{-1}(I))$.

Étale Morphisms: An unramified, flat morphism $f:X\to Y$ is called Étale. Note that being local morphisms, the maps $\mathscr O_{f(x),Y}\to \mathscr O_{x,X}$ are faithfully flat. Flat morphisms are open. Hence, if $Y$ is connected, a finite, Étale map to it is automatically surjective. An Étale covering is a finite, Étale map. If it is surjective as in the above case, then it is faithfully flat, so it is an effective epimorphism in the category of schemes.

Rigid Tensor Categories: Given any two objects $X,Y$ in a tensor category $(\mathcal C,\otimes)$, if the functor $T\mapsto {\rm Hom}(T\otimes X,Y)$ is representable, then we denote by $\underline{\rm Hom}(X,Y)$ the representing object, and denote the poincaré element by $ev_{X,Y}:\underline{\rm Hom}(X,Y)\otimes X\to Y$. Thus, to each $g:T\otimes X\to Y$, there corresponds unique $f: T\to \underline{\rm Hom}(X,Y)$ such that $ev_{X,Y}\circ (f\otimes id_X)=g$. For example, in the category of finite rank vector spaces over a field $k$, $\underline{\rm Hom}(X,Y)$ is given by ${\rm Hom}_k(X,Y)$ itself.

Suppose in $(\mathcal C,\otimes)$, every pair of object $X,Y$ has a $\underline{\rm Hom}(X,Y)$. Then for any pair of families $\{X_i,Y_i\}_i$ we have the morphism $\displaymath{(\otimes_i \underline{\rm Hom}(X_i,Y_i))\otimes (\otimes_i X_i)\cong \otimes_i(\underline{\rm Hom}(X_i,Y_i)\otimes X_i)\stackrel{\otimes ev}\to \otimes_i Y_i}$, which induces a unique morphism $\otimes_i\underline{\rm Hom}(X_i,Y_i)\to \underline{\rm Hom}(\otimes_i X_i,\otimes_i Y_i)$.

If in an a tensor category for every pair of objects $X,Y$ there is a $\underline{\rm Hom}(X,Y)$, and for any pair of finite families $\{X_i,Y_i\}_i$ the above morphism is an isomorphism, then the category will be called a Rigid Tensor Category.

Now we have the theorem [Theorem 2.11, 1]:

Theorem. Let $(\mathcal C,\otimes)$ be a rigid, tensor abelian category with ${\rm End}(\underline 1)=k$, and $\omega: \mathcal C\to {\rm Vec}_k$ be an exact, $k$-linear faithful tensor functor. Then

  1. Then functor $\underline{\rm Aut}^\otimes \omega$ of k-algebras is representable by an affine group scheme $G$ and
  2. $\omega$ defines an equivalence of tensor categories $\mathcal C\to {\rm Rep}_k(G)$.


1. Pierre Deligne and J. S. Milne, Tannakian categories, in Hodge Cycles, Motives, and Shimura Varieties by Pierre Deligne, James S. Milne, Arthur Ogus, Kuang-yen Shih, Lecture Notes in Math. 900, Springer-Verlag, 1982, 414pp.
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