We define the concepts of an injective colouring and the injective chromatic number of a graph and give some upper and lower bounds in general, plus some exact values. We explore in particular the injective chromatic number of the hypercube and put it in the context of previous work on similar concepts, especially the theory of errorcorrecting codes. Finally, we give necessary and sufficient co...

Let G be a finite group and k a field of characteristic p dividing |G|. Then Greenlees has developed a spectral sequence whose E2 term is the local cohomology of H∗(G, k) with respect to the maximal ideal, and which converges to H∗(G, k). Greenlees and Lyubeznik have used Grothendieck’s dual localization to provide a localized form of this spectral sequence with respect to a homogeneous prime i...

We prove that, except in some low-complexity cases, every locally injective simplicial map between pants graphs is induced by a π1-injective embedding between the corresponding surfaces.

Let [Formula: see text] be a ring, class of text]-modules and an integer. We introduce the concepts Gorenstein text]-[Formula: text]-injective text]-flat modules via special finitely presented modules. Besides, we obtain some equivalent properties these on text]-coherent rings. Then investigate relations among text]-injective, text]-flat, injective flat text]-rings (i.e., self rings). Several k...

Let R be a ring and n a fixed positive integer, we investigate the properties of n-strongly Gorenstein projective, injective and flat modules. Using the homological theory , we prove that the tensor product of an n-strongly Gorenstein projective (flat) right R -module and projective (flat) left R-module is also n-strongly Gorenstein projective (flat). Let R be a coherent ring ,we prove that the...