By Eben Matlis
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Extra info for 1-Dimensional Cohen-Macaulay Rings
Type. presentable of F
E~le2e~-le2~+l (where er~+l : = e~). Hence we get e~-le2i+l = gl,2 " g2,2 • . . " g'2-1,~ " g~,l and el-le2i = gl,2 • g2,2 " g~-l,~ • gi,1 ' g-1 (where gt+~,~ :-- 1). Now we identify the factors C(g~,l,g~,2) of Co(f, 4) with full m a t r i x rings over C. -. ~ e2 ® ~1 e 1 @ . . @ 1) ¢0(e2i) =/*(2i)¢(E2 ® . . ® ~2 ® ea ® 1 ® . . ® 1) where el or ea, respectively, occurs as the i-th factor. (If n is even, then ¢o(en) = / z ( n ) ¢ ( ¢ 2 ® . . ) This yields the m a t r i x ad,,-,. Using the induction formula (6), it is easy to construct the matrices aa,n.
Mal'tsev (1985) initiated the study of the variety VarB generated by B. Definition. We call a variety V finitely presentable from V is presentable if any finite ring (that is, embeddable into a matrix ring over some commutative ring). A variety W is called almost finitely presentable if any proper subvariety of W is finitely presentable, while W itself is not finitely presentable. Theorem I. (Yu,N. Mal'tsev). T h e variety VarB generated by Bergman's ring B is almost finitely presentable. Y. Finkelstein).
1-Dimensional Cohen-Macaulay Rings by Eben Matlis