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fibre product
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(Definition)
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Let $S$ be a scheme, and let $i: X \lra S$ and $j: Y \lra S$ be schemes over $S$ . A fibre product of $X$ and $Y$ over $S$ is a scheme $X \times_S Y$ together with morphisms \begin{eqnarray*} & p: X \times_S Y \lra X & \\ & q: X \times_S Y \lra Y & \end{eqnarray*}such that given any scheme $Z$ with morphisms \begin{eqnarray*} & x: Z \lra X & \\ & y: Z \lra Y & \end{eqnarray*}where $i \circ x = j \circ y$ , there exists a unique
morphism $$ (x,y): Z \lra X \times_S Y $$ making the diagram
commute. In other words, a fiber product is an object $X \times_S Y$ , together with morphisms $p,q$ making the diagram commute, with the universal property that any other collection $(Z,x,y)$ forming such a commutative diagram maps into $(X\times_S Y,p,q)$ .
Fibre products of schemes always exist and are unique up to canonical isomorphism.
Fibre products are also called pullbacks and can be defined in any category using the same definition (but need not exist in general). For example, they always exist in the category of modules over a fixed ring, as well as in the category of groups.
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"fibre product" is owned by djao.
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Cross-references: groups, ring, fixed, modules, category, isomorphism, canonical, maps, commutative diagram, collection, universal property, object, diagram, morphisms, scheme
There are 10 references to this entry.
This is version 7 of fibre product, born on 2002-06-27, modified 2006-06-09.
Object id is 3142, canonical name is FibreProduct.
Accessed 11935 times total.
Classification:
| AMS MSC: | 14A15 (Algebraic geometry :: Foundations :: Schemes and morphisms) |
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Pending Errata and Addenda
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![$\displaystyle \xymatrix{ Z \ar@/_/[ddr]_y \ar@/^/[drr]^x \ar@{.>}[dr]\vert-{(x,y)} \ & X \times_S Y \ar[d]^-q \ar[r]_-p & X \ar[d]^-i \ & Y \ar[r]_-j & S } $](http://images.planetmath.org:8080/cache/objects/3142/js/img1.png "$\displaystyle \xymatrix{ Z \ar@/_/[ddr]_y \ar@/^/[drr]^x \ar@{.>}[dr]\vert-{(x,y)} \ & X \times_S Y \ar[d]^-q \ar[r]_-p & X \ar[d]^-i \ & Y \ar[r]_-j & S } $")