# Refinement of concept of support of a module

My rings are commutative and noetherian.

The support of a module is usually defined to be the set of prime ideals of the ring such that localization at that prime does not make the module zero. This is closed under specialization. For finitely generated modules the support is in fact closed and this specialization property is sensible. But for general modules it does not seem to me a reasonable property. I believe that there are several definitions of support for general modules to avoid this property. For this question, I'll use a definiton I probably got from Hopkins and Neeman (if not Thomason), that a prime is in the support of the module if the derived tensor product (or Ext?) of the residue field with the module is not zero. Under this definition, the support of $\mathbb Q_p$ as a $\mathbb Z_p$ module is the generic point and is not closed under specialization. (Deriving the tensor product is necessary so that $\mathbb Q_p/\mathbb Z_p$ has support the special point and not the empty set.)

So this notion of support is just a subset of the prime spectrum, not an ideal in that poset like the old definition, but I don't want to throw out the partial order entirely. I'd like to refine this notion of support to be a subset of the specialization relations (or the arrows of category that is the poset). For example, the two modules $\mathbb Z_p$ and $\mathbb Q_p\oplus \mathbb Q_p/\mathbb Z_p$ both have support both the generic point and the special point, but the first seems to connect the two points, while the second is just the sum of modules with one point supports. Thus the first should have support the whole poset, while the second should have support just the two points, but not their specialization relation. In the derived category, they are both extensions of the same two objects $\mathbb Q_p$ and $\mathbb Q_p/\mathbb Z_p$, up to a shift, but those different extensions express the different refined supports. In the one-dimensional local case, the cone of localization has support just on the special prime and thus every chain complex of modules is the cone of a map from an object supported on generic point to an object supported on the special point. We can turn this into a definition: if that map is the zero map, so that the object is a sum of its local pieces, then it is not supported on the specialization relation, while if it is a nonzero map, the relation is in the support.

Has anyone considered such a refinement of support before?

How does this extend to higher dimensions? I think this allows us to define whether the specialization of adjacent primes should be in the support, but what about more distant specializations? That is, if a point, curve, and surface are in the support, this gives us a definition of whether the specialization of the surface to the curve and the specialization of the curve to the point are in the support, but not the specialization of the surface to the point.

I think the higher adeles / chromatic cubes are relevant.

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You could also think about higher-dimensional simplices in the nerve of the poset of primes. I could imagine that there might be a module whose support includes point, curve and surface and all three pairwise specializations but not the interior of the triangle specializations, if you see what I mean. –  Tom Goodwillie Jun 8 '12 at 22:19
Yes. In constructing the blow up, one considers $k[x,y] \to k[x,t]$, $y\mapsto xt$. The image contains the origin, but not every line through the origin. The image is basically the support of $k[x,t]$ considered as a module. So the refined support probably should have support the specialization from the plane to the origin. (But maybe not! We need a definition!) Taking the sum with other modules that contain the easier specializations, one gets the example you ask for. But before defining support on a triangle, we should define support on an edge. –  Ben Wieland Jun 8 '12 at 23:08