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Let $\mathcal{F}$ and $\mathcal{G}$ be two sheaves (of abelian groups) on a topological space $X$ such that $\mathcal{G}(U)$ is a subgroup of $\mathcal{F}(U)$ for every open set $U$ in $X$. The sheaf …

This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand. Let $K$ be a field and $S=K[X_0,\ldots,X_n]$ a graded ring. Set $X=Proj(S)$ and …

My problem is perhaps a general lack of understanding but it occurred in a special case of a theorem in Eisenbud's and Harris' "The geometry of schemes" (Theorem VI-29). Let $K$ be a field and $n\in\m …

(This question is related to this one) Let $k$ be a field and consider the category $Sch/k$ of schemes over $k$, say also separable and of finite type. The Yoneda embedding $$ Y:Sch/k \to Pre(Sch/k) …

Consider a topos, i.e. the category $Shv$ of sheaves on a Grothendieck site $T$ with values in abelian groups. The category $Shv$ is symmetric monoidal with $\otimes$, the tensor product in every degr …

The definition of a (geometric) vector bundle over a scheme $X$ can be rewritten as follows in terms of 'not-so-geometrical algebra' if $X=Spec R$ is affine and if I am not missing something. A vecto …

Let $R$ be the ring $R[X,Y]/(X^2+Y^2−1)$. The space of $\mathbb{R}$-rational points of the affine scheme associated to $R$ is the topological circle $S^1$. An algebraic vector bundle over $R$ is an $ …

The Stanley-Reisner ring of an abstract simplicial complex $\Delta$ on the vertex set $\{1,...,n\}$ is the $k$-algebra $$ k[X_1,...,X_n]/I_\Delta $$ where $I_\Delta$ is the ideal generated by the $X_{ …

Let $R$ be a, say, noetherian ring and $M$ an $R$-module. The Wikipedia article on 'locally free sheaf' tells me that the following two statements are equivalent: The module $M$ is locally free (Edi …

The first time I got in touch with the abstract notion of a sheaf on a topological space $X$, I thought of it as something which assigns to an open set $U$ of $X$ something like the ring of continuous …