Stack Exchange network consists of 177 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Is there a morphism from a locally free sheaf to the dual twisted by determinant bundle? To be sure this construction makes sense also to define the tensor product of presheaves of $\mathcal{O}$-modules. Hom O Hot Network Questions Preventing certain topics from being raised by a close relative tensor product of sheaves commutes with inverse image.

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However, the tensor product of tilting modules is a tilting module (in the context of finite-dimensional reps of reductive algebraic groups in positive characteristic). The tensor product algebra sheaf Our aim in this section is to define the topological (tensor product),e/-algebra sheaf S®^^. 0.

Pushforward and tensor products. Tensor product of two sheaves on surface. an open source textbook and reference work on algebraic geometry

[4] On topological algebra sheaves 17 2. In Sections 18.9 and 18.11 we defined the change of rings functor by a tensor product construction.

However, we have first to digress by commenting on tensor product topological A-algebras (for details cf. For example, let Fand Gbe two O X-modules. In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product.

Using (4.9) we may de ne various natural operations on sheaves.

Thus, in the context of tensor products, there is always a canonical map $\mathcal F(U)\otimes \mathcal G(U) \to (\mathcal F\otimes\mathcal G)(U)$ (where the target denotes the sheaf tensor product), and the stalk of $\mathcal F\otimes\mathcal G$ at any point is the tensor product of the corresponding stalks of $\mathcal F$ and $\mathcal G$. To be precise, suppose $\mathcal{C}$ is a category, $\mathcal{O}$ is a presheaf of rings, and $\mathcal{F}$, $\mathcal{G}$ are presheaves of … Active 3 years, 2 months ago. Is the presheaf tensor product $\mathcal{F} \otimes_{O_X} \mathcal{G}$ a separated presheaf? Is this true more generally if $(X,O_X)$ is just a ringed space? The tensor product of F and G, denoted F O X G, is the sheaf associated to the presheaf U! Suppose $\\mathcal{A}$ is a quasi coherent sheaf of algebras over a group scheme $\\mathcal{G}$. Then , what can we say about the external tensor product $\\ F (U) O X(U) G(U); and curly hom, denoted Hom O X (F;G), is the sheaf associated to the presheaf U!

Suppose it is generated by global section.

18.26 Tensor product. Sheaves of modules over a ringed space form an abelian category.

$\endgroup$ – Akatsuki Dec 14 '17 at 21:09. add a comment | [14, 16]). This tells you for example, that if $\mathcal{F}, \mathcal{G}$ have some global sections then so does their sheaf tensor product (assuming the presheaf tensor product has global sections).

... $\begingroup$ But tensor product is not a colimit, so I think this is not a stronger claim. Ask Question Asked 8 years, 6 months ago.