交换代数的纤维积：生成元与关系
Fibred product of commutative algebras: generators and relations
摘要：
给出并证明了域上具有单位元的有限型交换结合代数范畴中泛（合）积的直接计算方法。系数域不应该是代数封闭的，可以是任何特征。交换结合代数乘积的构造是利用代数几何中的某种等价关系来构造胶合代数方案的代数对应。如果初始代数是有限维向量空间，则它们的乘积的维数服从Grassmann公式。几何上，一维情况意味着将包含某个公共部分的两个点集合相加的严格版本。该方法涉及到代数的描述的生成元和关系的输入和返回类似的描述的产品代数。它是"即食"，甚至为计算机实现。乘积代数是定义良好的：对同一个代数取另一个描述会导致同构的乘积代数。证明了乘积代数具有泛性质，即它确实是可积代数.输入数据是一个代数三元组和一对同态$A_1\stackrel{f_1}{\to}A_0 \stackrel{f_2}{\leftarrow}A_2$。代数和同态可以用任何方式描述。我们证明了计算满积时仅限于$f_i,i=1,2,$是满射的情况是足够的，并描述了如何减少到满射的情况。此外，如何选择生成元和输入代数的关系被认为是。
英文摘要：
The method of direct computation of universal (fibred) product in the category of commutative associative algebras of finite type with unity over a field is given and proven. The field of coefficients is not supposed to be algebraically closed and can be of any characteristic. Formation of fibred product of commutative associative algebras is an algebraic counterpart of gluing algebraic schemes by means of some equivalence relation in algebraic geometry. If initial algebras are finite-dimensional vector spaces the dimension of their product obeys Grassmann-like formula. Finite-dimensional case means geometrically the strict version of adding two collections of points containing some common part. The method involves description of algebras by generators and relations on input and returns similar description of the product algebra. It is "ready-to-eat" even for computer realization. The product algebra is well-defined: taking another descriptions of the same algebras leads to isomorphic product algebra. Also it is proven that the product algebra enjoys universal property, i.e. it is indeed fibred product. The input data is a triple of algebras and a pair of homomorphisms $A_1\stackrel{f_1}{\to}A_0 \stackrel{f_2}{\leftarrow}A_2$. Algebras and homomorphisms can be described in any fashion. We prove that for computing the fibred product it is enough to restrict to the case when $f_i,i=1,2,$ are surjective and describe how to reduce to surjective case. Also the way to choose generators and relations for input algebras is considered.
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