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Let more generally $f : T \to S$ be a fpqc covering and $X,X' \in \mathcal{M}(S)$ with $f^* X \cong f^* X'$ in $\mathcal{M}(T)$. By definition(*) of a prestack in the fpqc-topology, a necessary and su …

If $G$ is a commutative monoid (for your question we will want $G = \mathbb{Z}$), then $A[G]$-comodules identify with $G$-graded $A$-modules. A reference is Demazure, Gabriel, Introduction to Algebrai …

Feed google with "variety whose ring of global sections is not finitely generated". One gets An example of a nice variety whose ring of global sections is not finitely generated by Ravi Vakil.

Here is a sketch: Let $\alpha : R \to \Gamma(X,\mathcal{O}_X)$ be a ring homomorphism. We want to define a morphism $f:X \to \mathrm{Spec}(R)$ which is $\alpha$ on global sections. Let $x \in X$, and …

Let $(X_n)_{n \geq 0}$ be a family of schemes. Let $$X_0 \to X_1 \to X_2 \to \dotsc$$ be a sequence of closed immersions (which therefore gives rise to an ind-scheme). Under which (necessarly and/or s …

The two morphisms are no $G$-morphisms. An $R$-algebra homomorphism $f : R[x]/(x^2-1) \to R[x]/(x^2-1)$ commutes with the $G$-action if and only if the diagram $$\begin{array}{c} R[x]/(x^2-1) & \xrigh …

As already mentioned in the comments, local rings are supposed to abstract the idea of germs of functions. One can make this precise as follows. A ring of germs is defined as a homomorphism $$\mathrm …

The question seems to be about algebraic geometry of commutative semirings (these are rings without subtraction). The theory by Toen-Vaquié (and others) in "Au-dessous de $Spec \mathbb{Z}$" develops …

You can prove essential smallness of the category of finite type $S$-schemes (without further assumptions), where $S$ is any scheme, as follows: If $S$ is affine and we only consider affine finite t …

If $X$ is a scheme, is it always possible to find a basis $\mathcal{B}$ for the topology of $X$ (for example, the affine open subsets) with the following property? For every quasi-coherent sheaf $ …

What are some (intrinsically formulated) properties of the locally ringed topos $(\mathbf{Sh}(X),\mathcal{O}_X)$ for some scheme $X$, which do not hold for arbitrary locally ringed toposes? Is there, …

A cogroupoid object in $\mathsf{CAlg}_R$ is called a Hopf algebroid over $R$. How are cocategory objects in $\mathsf{CAlg}_R$ called? (Unfortunately, bialgebroid is already taken, which seems to mean …

Actually, every scheme is the canonical colimit of all affine schemes mapping into it: $$X = \underset{\substack{U \to X\\U \text{ affine}}}{\mathrm{colim}} U$$ In order to avoid size issues, we can i …

The answer to (2) is no. For example, for every prime $p$, $$\mathbb{Z}[\sqrt{p}] \otimes_{\mathbb{Z}} \mathbb{F}_p = \mathbb{Z}[x]/(x^2-p) \otimes_{\mathbb{Z}} \mathbb{F}_p = \mathbb{F}_p[x]/(x^2)$$ …

One can use idals. The unit $1_{\mathcal{C}} \in \mathcal{C}$ is called reduced if for all idals $e : I \to 1_{\mathcal{C}}$ (i.e. morphisms $e$ satisfying $e \otimes I = I \otimes e$) with $e \otimes …