When doing module theory over an associative ring, one is often forced to change the ring. In general, there is no hope of providing uniformly applicable techniques for this procedure that would readily yield meaningful information for all kinds of rings. Instead, one usually restricts to various classes of rings. The above also applies to commutative rings, but in that case there is a special change of ring situation when quite a lot of information can be carried over in either of the two directions. This refers to the passage from a ring to its quotient by an ideal generated by a regular sequence. For example, hypersurface rings and, more generally, complete intersections arise this way.
Let \(Q\) be a commutative ring, \(\mathbf {f}:=f_{1},\dots,f_{c}\), a \(Q\)-regular system, \(M\) a \(Q\)-module, \(R:=Q/(\mathbf {f})\) the corresponding quotient ring. In the case when \(\mathbf {f}\) is also \(M\)-regular, the properties of the \(Q\)-module \(M\) and the \(R\)-module \(M/(\mathbf {f})M\) are very similar. This gives a useful tool to do induction arguments. As very instructive examples, one can look at the proofs of the Auslander-Buchsbaum-type formulas.
At the other extreme stands the case when the ideal \((\mathbf {f)}\) lies in the annihilator \(\text{Ann}_{Q}M\) of \(M\). This makes \(M\) an \(R\)-module and one can try to relate the properties of \(M\) over \(R\) and over \(Q\). This is exactly the goal of the paper under review. It has been known for a while that, in the above situation, the formalism of differential graded (DG, for short) algebras and modules becomes relevant. Notice, for example, that the Koszul complex \(\mathbf {K}\) resolving \(R\) as a \(Q\)-module carries a structure of DG algebra. In the paper, the authors develop DG techniques further and systematically use them to provide quantifiable comparisons of homological properties of \(M\) over \(R\) and over \(Q\). The basic technical tool, theorem 1.1, extending the homological product of Cartan and Eilenberg, is stated as follows.
Theorem: If \(R\leftarrow Q\rightarrow k\) are homomorphisms of commutative rings, then \(\text{T}=\operatorname{Tor}_{\ast}^{Q}(R,k)\) is a graded associative and commutative algebra through the pairing \[ \operatorname{Tor}_{i}^{Q}(R,k)\otimes_{Q}\operatorname{Tor}_{j} ^{Q}(R,k)\rightarrow\operatorname{Tor}_{i+j}^{Q}(R,k) \] that extend the canonical product on \(\operatorname{Tor}_{0}^{Q} (R,k)=R\otimes_{Q}k\).
For any \(R\)-module \(M\) and any \(k\)-module \(N\) there are pairings \[ \begin{aligned} \operatorname{Tor}_{i}^{Q}(R,k)\otimes_{Q}\operatorname{Tor}_{j}^{Q}(M,N) & \rightarrow\operatorname{Tor}_{i+j}^{Q}(M,N)\\ \operatorname{Tor}_{i}^{Q}(R,k)\otimes_{Q}\operatorname{Ext}_{Q}^{j}(M,N) & \rightarrow\operatorname{Ext}_{Q}^{j-i}(M,N) \end{aligned} \] that endow \(\operatorname{Tor}_{\ast}^{Q}(M,N)\) and \(\operatorname{Ext} _{Q}^{\ast}(M,N)\) with a structure of graded module over \(\text{T}\), extending the standard actions of \(R\otimes_{Q}k\) on \(M\otimes_{Q}N\) and on \(\operatorname{Hom}_{Q}(M,N)\). The actions of \(\text{T}\) are natural in \(M\) and \(N\); exact sequences of \(R\)-modules or exact sequences of \(k\)-modules induce \(\text{T}\)-linear connecting morphisms.
Each \(N'\) defines a natural morphism of graded \(\operatorname{Tor}_{\ast }^Q (R,k)\)-modules \[ \operatorname{Ext}_{Q}^{\ast}(M,\operatorname{Hom}_{k}(N,N'))\rightarrow\operatorname{Hom}_{k}(\operatorname{Tor}_{\ast}^{Q} (M,N),N') \] that is an isomorphism whenever the \(k\)-module \(N'\) is injective.
This applies, in particular, to the case when, as above, \(R=Q/(\mathbf {f})\). In this case, any \(R\)-module \(M\) has a projective resolution over \(Q\), called a Koszul resolution, which is a DG module over the Koszul complex \(\mathbf {K} \). After recalling its construction, the authors proceed to build explicitly what they call a universal \(R\)-free resolution of \(M\). An important feature of that resolution is that it is a DG module over \(\mathcal{S} \otimes_{Q}\Lambda\), where \(\Lambda=\operatorname{Tor}_{\ast}^{Q} (R,R)=R\langle \xi_{1},\dots ,\xi_{c}\rangle\) is the exterior algebra and \(\mathcal{S}\) is the polynomial algebra \(R[\chi_{1},\dots ,\chi_{c}]\). Here \(c\) is the cardinality of the regular sequence \(\mathbf {f}\). The action of the polynomial algebra on the universal resolution allows to view the letters \(\chi_{i}\) as degree \(2\) operators on cohomology, whereas the action of the exterior algebra on the universal resolution allows to view the letters \(\xi_{i}\) as degree \(1\) operators on homology. This additional structure allows to compute \(\operatorname{Ext}\) in certain cases. At this point a connection is made with the BGG correspondence.
When \(c=2\) the constructions simplify. In this case, the algebra \(\text{T}=\operatorname{Tor}_{\ast}^{Q}(R,k)\) is the exterior algebra \(k\langle \xi_{1},\xi_{2}\rangle\) on two letters (\(k\) is the residue field) and its representation theory is well understood. A graded \(\text{T}\)-module without projective summands is annihilated by \(\xi_{1}\xi_{2}\) thus giving rise to a module over \(\text{T}/(\xi_{1}\xi_{2})\). This amounts to choosing a graded \(k\)-vector space \(N\) and two degree \(1\) endomorphisms with zero composition. A complete list of indecomposable representations can then be produced, based on the work of Kronecker. As expected, this immediately leads to a multitude of explicit results on the structure of cohomology. Still assuming that \(c=2\), it is also shown that a high enough truncation of a minimal resolution of \(M\) is a DG module over the ring \(R[\chi_{1},\chi_{2}]\) of cohomology operators.
Returning to the general \(c\), the authors set up a spectral sequence \[ ^{2}E^{p,q}=\mathcal{S}^{2p+q}\otimes_{R}\operatorname{Ext}_{Q}^{-p} (M,N)\Longrightarrow\operatorname{Ext}_{R}^{p+q}(M,N) \] As an application, they give a short proof of Gulliksen’s result that the noetherian property of \(\operatorname{Ext}_{Q}^{\ast}(M,N)\) as a module over \(\overline{R}:=Q/ (\operatorname{Ann}M+\operatorname{Ann}N)\) implies the noetherian property of \(\operatorname{Ext}_{R}^{\ast}(M,N)\) as a module over \(\overline{R}[\chi_{1},\dots ,\chi_{c}]\).
In the last section of the paper the authors investigate the behavior of the Betti numbers of a module of finite complete intersection dimension over a local ring.