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Give $R$ the standard grading. Let $I$ be a homogeneous ideal of height $g$. Assume $1 \\leq g \\leq m -1$. Suppose $H^i_I(R) \\neq 0$ for some $i \\geq 0$. We show\n  (1) $H^i_I(R)_n \\neq 0$ for all $n \\leq -m$.\n  (2) if Supp $H^i_I(R) \\neq \\{ (X_1, \\ldots, X_m)\\}$ then $H^i_I(R)_n \\neq 0$ for all $n \\in \\mathbb{Z}$. Furthermore if char $K = 0$ then $\\dim_K H^i_I(R)_n$ is infinite for all $n \\in \\mathbb{Z}$.\n  (3) $\\dim_K H^g_I(R)_n$ is infinite for all $n \\in \\mathbb{Z}$.\n  In fact we prove our results for $\\mathcal{T}(R)$ wher","authors_text":"Tony J. 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