
Fibred product of commutative algebras: generators and relations
N. V. Timofeeva^{} ^{} P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl 150003, Russia
Abstract:
The method of direct computation of the 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 finitedimensional vector spaces, the dimension of their
product obeys a Grassmannlike formula. A finitedimensional case
means geometrically the strict version of adding two collections of
points containing a common part.
The method involves description of algebras by generators and
relations on input and returns similar description of the product
algebra. It is "readytoeat" even for computer realization. The
product algebra is welldefined: taking other 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 a fibred product. The input data are 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 an arbitrary way. 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 the surjective case. Also the
way of choosing generators and relations for input algebras is
considered.
Paper is published in the author's wording.
DOI:
https://doi.org/10.18255/1818101520165620634
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UDC:
512.71 Received: 15.04.2016
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Citation:
N. V. Timofeeva, “Fibred product of commutative algebras: generators and relations”, Model. Anal. Inform. Sist., 23:5 (2016), 620–634
Citation in format AMSBIB
\Bibitem{Tim16}
\by N.~V.~Timofeeva
\paper Fibred product of commutative algebras: generators and relations
\jour Model. Anal. Inform. Sist.
\yr 2016
\vol 23
\issue 5
\pages 620634
\mathnet{http://mi.mathnet.ru/mais528}
\crossref{https://doi.org/10.18255/1818101520165620634}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=3569858}
\elib{http://elibrary.ru/item.asp?id=27202311}
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