University of Kashan Mathematics Interdisciplinary Research 2538-3639 6 2 2021 06 01 Big Finitistic Dimensions for Categories of Quiver Representations 139 149 111485 10.22052/mir.2021.240439.1273 EN Roghayeh Bagherian Department of Mathematics, Isfahan University of Technology, Isfahan, I. R. Iran Esmaeil Hosseini Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, I. R. Iran Journal Article 2021 01 31 <span class="fontstyle0">Assume that </span><span class="fontstyle2">A </span><span class="fontstyle0">is a Grothendieck category and </span><span class="fontstyle2">R </span><span class="fontstyle0">is the category of all </span><span class="fontstyle2">A</span><span class="fontstyle0">-representations of a given quiver </span><span class="fontstyle2">Q</span><span class="fontstyle0">. If </span><span class="fontstyle2">Q </span><span class="fontstyle0">is left rooted and </span><span class="fontstyle2">A </span><span class="fontstyle0">has a projective generator, we prove that the big finitistic flat (resp. projective) dimension </span><span class="fontstyle3">FFD(</span><span class="fontstyle2">A</span><span class="fontstyle3">) </span><span class="fontstyle0">(resp. </span><span class="fontstyle3">FPD(</span><span class="fontstyle2">A</span><span class="fontstyle3">)</span><span class="fontstyle0">) of </span><span class="fontstyle2">A </span><span class="fontstyle0">is finite if and only if the big finitistic flat (resp. projective) dimension of </span><span class="fontstyle2">R </span><span class="fontstyle0">is finite. When </span><span class="fontstyle2">A </span><span class="fontstyle0">is the Grothendieck category of left modules over a unitary ring </span><span class="fontstyle4">R</span><span class="fontstyle0">, we prove that if </span><span class="fontstyle3">FPD(</span><span class="fontstyle2">R</span><span class="fontstyle3">) </span><span class="fontstyle4">< </span><span class="fontstyle3">+∞</span><span class="fontstyle2"> </span><span class="fontstyle0">then any representation of </span><span class="fontstyle2">Q </span><span class="fontstyle0">of finite flat dimension has finite projective dimension. Moreover, if </span><span class="fontstyle4">R </span><span class="fontstyle0">is </span><span class="fontstyle4">n</span><span class="fontstyle0">-perfect then we show that </span><span class="fontstyle3">FFD(</span><span class="fontstyle2">R</span><span class="fontstyle3">) </span><span class="fontstyle4">< </span><span class="fontstyle3">+</span>∞ <span class="fontstyle2"> </span><span class="fontstyle0">if and only if </span><span class="fontstyle3">FPD(</span><span class="fontstyle2">R</span><span class="fontstyle3">) </span><span class="fontstyle4">< </span><span class="fontstyle3">+</span>∞<span class="fontstyle0">.</span><br />  https://mir.kashanu.ac.ir/article_111485_e92e83f51ff8b7f5bd417fe9bbecf04e.pdf