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Assume $A \subseteq B \subseteq C$ are noetherian integral domains, where $A$ and $C$ have the same finite global dimension $n$. Also assume that $C$ is a finitely generated $B$-algebra and $B$ is a finitely generated $A$-algebra. (All rings are not local).

Recall: (1) The global dimension of a ring $R$ is the supremum of the projective dimensions of all $R$-modules (it is a non-negative integer or infinity). (2) The projective dimension of an $R$-module $M$ is the minimal length among all finite projective resolutions of $M$ (if $M$ has a finite projective resolution), or infinity (if $M$ does not have a finite projective resolution).

What can be said about the global dimension of $B$; must it be $n$? (I suspect no) or can it be infinite? (if so, can one please give an example of such $A,B,C$).

Mariano Suárez-Álvarez
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user237522
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1 Answers1

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Consider the inclusions $k[x^2,y^2]\subseteq k[x^2,xy,y^2]\subseteq k[x,y]$.

The middle algebra is isomorphic to $k[u,v,w]/(uv-w^2)$, which has infinite global dimension because its localization "at the origin" is not a local regular ring.

Mariano Suárez-Álvarez
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    The global dimension is very fragile :-) – Mariano Suárez-Álvarez Jun 28 '15 at 01:49
  • :) Thank you very much! Can one find any "reasonable" condition which guarantees a finite global dimension for $B$? – user237522 Jun 28 '15 at 01:54
  • If $R\subseteq S$ are commutative rings and $R$ is a direct summanf of $S$ as an $R$-module, then the global dimension of $R$ is at most the global dimension of $S$ plus the projective dimension of $S$ over $R$. There are several such results (but I do not recall any in which you start with a ring sandwiched between two other rings) You can also use the weaker condition that $S$ be faithfully flat over $R$ plus something else and such things – Mariano Suárez-Álvarez Jun 28 '15 at 01:59
  • Thanks! (I think I have seen the result you mentioned). My problem is that I have $A \subseteq B$ with $B$ a finite free $A$-module, and the above result does not help; it just says that the global dimension of $A$ is at most infinity. Can you please tell me where can I find the other similar results? Can one fine them in the global-dimension questions? – user237522 Jun 28 '15 at 02:33
  • You are interested in commutative rings, and my rings are usually non-commutative. A good reference is the book by McConnell and Robson, which has a whole chapter on gldim. – Mariano Suárez-Álvarez Jun 28 '15 at 02:35
  • I will check the book you recommended. Thanks. – user237522 Jun 28 '15 at 02:37