Grothendieck–Serre formula and bigraded Cohen–Macaulay Rees algebras

The Grothendieck–Serre formula for the difference between the Hilbert function and Hilbert polynomial of a graded algebra is generalized for bigraded standard algebras. This is used to get a similar formula for the difference between the Bhattacharya function and Bhattacharya polynomial of two -primary ideals I and J in a local ring in terms of local cohomology modules of Rees algebras of I and J. The cohomology of a variation of the Kirby–Mehran complex for bigraded Rees algebras is studied which is used to characterize the Cohen–Macaulay property of bigraded Rees algebra of I and J for two dimensional Cohen–Macaulay local rings.