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We present an exact expression for the frequency-dependent Kohn-Shamexact-exchange (EXX) kernel for periodic insulators, which can be employed forthe calculation of electronic response properties within time-dependent (TD)density-functional theory. It is shown that the EXX kernel has along-wavelength divergence behavior of the exact full exchange-correlationkernel and thus rectifies one serious shortcoming of the adiabaticlocal-density approximation and generalized-gradient approximations kernels. Acomparison between the TDEXX and the GW-approximation-Bethe-Salpeter-equationapproach is also made.Comment: two column format 6 pages + 1 figure, to be publisehd in Physical  Review

Exact Kohn-Sham exchange kernel for insulators and its long-wavelength behavior